# Patent

## Title
Bayesian scoring

## Abstract

Players in a gaming environment, particularly, electronic on-line gaming environments, may be scored relative to each other or to a predetermined scoring system. The scoring of each player may be based on the outcomes of games between players who compete against each other in one or more teams of one or more players. Each player's score may be represented as a distribution over potential scores which may indicate a confidence level in the distribution representing the player's score. The score distribution for each player may be modeled with a Gaussian distribution and may be determined through a Bayesian inference algorithm. The scoring may be used to track a player's progress and/or standing within the gaming environment, used in a leaderboard indication of rank, and/or may be used to match players with each other in a future game.

## Background
CROSS-REFERENCE TO RELATED APPLICATION
This application is a continuation of and claims the benefit of prior U.S.patent application Ser. No. 11/041,752 filed Jan. 24, 2005, titled “BAYESIAN SCORING”, which issued as U.S. Pat No. 7,050,868 on May 23, 2006, the contents of which are hereby incorporated by reference.


BRIEF DESCRIPTION OF THE DRAWINGS
The foregoing aspects and many of the attendant advantages of this invention will become more readily appreciated as the same become better understood by reference to the following detailed description, when taken in conjunction with the accompanying drawings, wherein:
FIG. 1 is an example computing system for implementing a scoring system;
FIG. 2 is a dataflow diagram of an example scoring system;
FIG. 3 is an example graph of two latent score distributions;
FIG. 4 is an example graph of the joint distribution of the scores of two players;
FIG. 5 is a flow chart of an example method of updating scores of two players or teams;
FIG. 6 is a flow chart of an example method of matching two players or teams based on their score distributions;
FIG. 7 is a flow chart of an example method of updating scores of multiple teams;
FIG. 8 is a flow chart of an example method of matching scores of multiple teams; and
FIG. 9 is a flow chart of an example method of approximating a truncated Gaussian distribution using expectation maximization.


DETAILED DESCRIPTION
Exemplary Operating Environment
FIG. 1 and the following discussion are intended to provide a brief, general description of a suitable computing environment in which a scoring system may be implemented. The operating environment of FIG. 1 is only one example of a suitable operating environment and is not intended to suggest any limitation as to the scope of use or functionality of the operating environment. Other well known computing systems, environments, and/or configurations that may be suitable for use with a scoring system described herein include, but are not limited to, personal computers, server computers, hand-held or laptop devices, multiprocessor systems, micro-processor based systems, programmable consumer electronics, network personal computers, mini computers, mainframe computers, distributed computing environments that include any of the above systems or devices, and the like.
Although not required, the scoring system will be described in the general context of computer-executable instructions, such as program modules, being executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc., that perform particular tasks or implement particular abstract data types. Typically, the functionality of the program modules may be combined or distributed as desired in various environments.
With reference to FIG. 1, an exemplary system for implementing a scoring system includes a computing device, such as computing device 100. In its most basic configuration, computing device 100 typically includes at least one processing unit 102 and memory 104. Depending on the exact configuration and type of computing device, memory 104 may be volatile (such as RAM), non-volatile (such as ROM, flash memory, etc.) or some combination of the two. This most basic configuration is illustrated in FIG. 1 by dashed line 106. Additionally, device 100 may also have additional features and/or functionality. For example, device 100 may also include additional storage (e.g., removable and/or non-removable) including, but not limited to, magnetic or optical disks or tape. Such additional storage is illustrated in FIG. 1 by removable storage 108 and non-removable storage 110. Computer storage media includes volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information such as computer readable instructions, data structures, program modules, or other data. Memory 104, removable storage 108, and non-removable storage 110 are all examples of computer storage media. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVDs) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to store the desired information and which can be accessed by device 100. Any such computer storage media may be part of device 100.
Device 100 may also contain communication connection(s) 112 that allow the device 100 to communicate with other devices. Communications connection(s) 112 is an example of communication media. Communication media typically embodies computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. The term ‘modulated data signal’ means a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, radio frequency, infrared, and other wireless media. The term computer readable media as used herein includes both storage media and communication media.
Device 100 may also have input device(s) 114 such as keyboard, mouse, pen, voice input device, touch input device, laser range finder, infra-red cameras, video input devices, and/or any other input device. Output device(s) 116 such as display, speakers, printer, and/or any other output device may also be included.
Scoring System
Players in a gaming environment, particularly, electronic on-line gaming environments, may be scored relative to each other or to a predetermined scoring system. As used herein, the score of a player is not a ‘score’ that a player achieves by gaining points or other rewards within a game; but rather, a ranking or other indication of the skill of the player based on the outcome of the game. It should be appreciated that any gaming environment may be suitable for use with the scoring system described further below. For example, players of the game may be in communication with a central server through an on-line gaming environment, directly connected to a game console, play a physical world game (e.g., chess, poker, tennis), and the like.
The scoring may be used to track a player's progress and/or standing within the gaming environment, and/or may be used to match players with each other in a future game. For example, players with substantially equal scores, or scores meeting predetermined and/or user defined thresholds, may be matched to form a substantially equal challenge in the game for each player.
The scoring of each player may be based on the outcomes of games between players who compete against each other in teams of one or more. The outcome of each game may update the score of each player participating in that game. The outcome of a game may be indicated as a particular winner, a ranked list of participating players, and possibly ties or draws. Each player's score on a numerical scale may be represented as a distribution over potential scores which may be parameterized for each player by a mean score μ and a score variance σ2. The variance may indicate a confidence level in the distribution representing the player's score. The score distribution for each player may be modeled with a Gaussian distribution, and may be determined through a Bayesian inference algorithm.
FIG. 2 illustrates an example scoring system for determining scores for multiple players. Although the following example is discussed with respect to one player opposing another single player in a game to create a game outcome, it should be appreciated that following examples will discuss a team comprising one or more players opposing another team, as well as multi-team games. The scoring system 200 of FIG. 2 includes a score update module which accepts the outcome 210 of a game between two or more players. It should be appreciated that the game outcome may be received through any suitable method. For example, the outcome may be communicated from the player environment, such as an on-line system, to a central processor to the scoring system in any suitable manner, such as through a global communication network. In another example, the scores of the opposing player(s) may be communicated to the gaming system of a player hosting the scoring system. In this manner, the individual gaming system may receive the scores of the opposing players in any suitable manner, such as through a global communication network. In yet another example, the scoring system may be a part of the gaming environment, such as a home game system, used by the players to play the game. In yet another example, the game outcome(s) may be manually input into the scoring system if the gaming environment is unable to communicate the game outcome to the scoring system, e.g., the game is a ‘real’ world game such as board chess.
The outcome 210 may be an identification of the winning team, the losing team, and/or a tie. For example, if two players (player A and player B) oppose one another in a game, the game outcome may be one of three possible results, player A wins and player B loses, player A loses and player B wins, and players A and B draw. Each player has a score 212 which may be updated to an updated score 216 in accordance with the possible change over time due to player improvement (or unfortunate atrophy) and the outcome of the game by both the dynamic score module and the score update module. More particularly, where the player score 212 is a distribution, the mean and variance of each player's score may be updated in view of the outcome and the possible change over time due to player improvement (or unfortunate atrophy). The dynamic score module 204 allows the score 212 of one or more players to change over time due to player improvement (or unfortunate atrophy). The score update module 202, through the outcomes of games, learns the score of the player. The player may improve over time, thus, the mean may be increased and/or the variance or confidence in the score may be broadened. In this manner, the score of each player may be modified to a dynamic player score 214 to allow for improvement of the players. The dynamic player scores 214 may then be used as input to the score update module. In this manner, the score of each player may be learned over a sequence of games played between two or more players.
The score of each player may be used by a player match module 206 to create matches between players based upon factors such as player indicated preferences and/or score matching techniques. The matched players, with their dynamic player scores 214 may then oppose one another and generate another game outcome 210.
In some cases, to accurately determine the ranking of a number n of players, at least log(n!), or approximately n log(n) game outcomes may be evaluated. The base of the logarithm depends on the number of unique outcomes between the two players. In this example, the base is three since there are three possible outcomes (player A wins, player A lose, and draw). This lower bound of evaluated outcomes may be attained only if each of the outcomes is fully informative, that is, a priori, the outcomes of the game have a substantially equal probability. Thus, in many games, the players may be matched to have equal strength to increase the knowledge attained from each outcome. Moreover, the players may appreciate a reasonable challenge from a peer player.
It is to be appreciated that although the dynamic score module 204, the score update module 202, the player match module 206 are discussed herein as separate processes within the scoring system 200, any function or component of the scoring system 200 may be provided by any of the other processes or components. Moreover, it is to be appreciated that other scoring system configurations may be appropriate. For example, more than one dynamic scoring module 204, score update module 202, score vector, and/or player match module may be provided, more than one database may be available for storing score, rank, and/or game outcomes, any portion of the modules of the scoring system may be hard coded into software supporting the scoring system, and/or any portion of the scoring system 200 may provided by any computing system which is part of a network or external to a network.
Learning Scores
In a two player game, the outcomes may be player A wins, player A loses, or players A and B draw. The outcome of the game may be indicated in any suitable manner such as through a ranking of the players for that particular game. In accordance with the game outcome, each player of a game may be ranked in accordance with a numerical scale. For example, the rank ri of a player may have a value of 1 for the winner and a value of 2 for a loser. In a tie, the two players will have the same rank.
A player's score si may indicate the player's standing relative to a standard scale and/or other players. The score may be individual to one or more people acting as a player, or to a game type, a game application, and the like. The score si of each player may have a stochastic transitive property. More particularly, if player i is scored above player j, then player i is more likely to win against player j as opposed to player j winning against player i. In mathematical terms:

si≧sj→P(player i wins)≧P(player j wins)  (1)

This stochastic transitive property implies that the probability of player i winning or drawing is greater than or equal to one half because, in any game between two players, there are only three mutually exclusive outcomes (player i wins, loses, or draws).
To estimate the score for each player such as in the score update module 202 of FIG. 2, a Bayesian learning methodology may be used. With a Bayesian approach, the belief in the true score si of a player may be indicated as a probability density of the score (i.e., P(s)). In the following examples, the probability density of the score representing the belief in the true score is selected as a Gaussian with a mean μ and a diagonal covariance matrix (diag(σ2)). The Gaussian density may be shown as:

P(s)=N(s;μ, diag(σ2))  (2)

Selecting the Gaussian allows the distribution to be unimodal with mode μ. In this manner, a player should not be expected to alternate between widely varying levels of play. Additionally, a Gaussian representation of the score may be stored efficiently in memory. In particular, assuming a diagonal covariance matrix effectively leads to allowing each individual score for a player i to be represented with two values: the mean μi and the variance σi2.
The initial and updated scores (e.g., mean μ and variance σ2) of each player may be stored in any suitable manner. For example, the mean and variance of each player may be stored in separate vectors, e.g., a mean vector μ and variance vector σ2, a data store, and the like. If all the means and variances for all possible players are stored in vectors, e.g., μ and σ2, then the update equations may update only those means and variances associated with the players that participated in the game outcome. Alternatively or additionally, the score for each player may be stored in a player profile data store, a score matrix, and the like.
It is to be appreciated that any suitable data store in any suitable format may be used to store and/or communicate the scores and game outcome to the scoring system 200, including a relational database, object-oriented database, unstructured database, an in-memory database, or other data store. A storage array may be constructed using a flat file system such as ACSII text, a binary file, data transmitted across a communication network, or any other file system. Notwithstanding these possible implementations of the foregoing data stores, the term data store and storage array as used herein refer to any data that is collected and stored in any manner accessible by a computer.
The Gaussian model of the distribution may allow efficient update equations for the mean μi and the variance σi2 as the scoring system is learning the score for each player. After observing the outcome of a game, e.g., indicated by the rank r of the players for that game, the belief distribution or density P(s) in the scores s (e.g., score si for player i and score sj for player j) may be updated using Bayes rule given by:




  
    
      
        
          
            
              
                
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where the variable ik is an identifier or indicator for each player of the team k participating in the game. In the two player example, the vector i1 for the first team is an indicator for player A and the vector i2 for the second team is an indicator for player B. In the multiple player example discussed further below, the vector i may be more than one for each team. In the multiple team example discussed further below, the number of teams k may be greater than two. In a multiple team example of equation (3), the probability of the ranking given the scores of the players P(r|si1, . . . , Sik) may be modified given the scores of the team S(sik) which is a function of the scores of the individual players of the team.
The new updated belief, P(s|r,{i1, . . . ik}) is also called the posterior belief (e.g., the updated scores 214, 216) and may be used in place of the prior belief P(s), e.g., the player scores 212 in the evaluation of the next game for those opponents. Such a methodology is known as on-line learning, e.g., over time only one belief distribution P(s) is maintained and each observed game outcome r for the players participating {i1, . . . , ik} is incorporated into the belief distribution.
After incorporation into the determination of the players' scores, the outcome of the game may be disregarded. However, the game outcome r may not be fully encapsulated into the determination of each player's score. More particularly, the posterior belief P((s|r,{i1, . . . ik}) may not be represented in a compact and efficient manner, and may not be computed exactly. In this case, a best approximation of the true posterior may be determined using any suitable approximation technique including expectation propagation, variational inference, assumed density filtering, Laplace approximation, maximum likelihood, and the like. Assumed density filtering (ADF) computes the best approximation to the true posterior in some family that enjoys a compact representation—such as a Gaussian distribution with a diagonal covariance. This best approximation may be used as the new prior distribution. The examples below are discussed with reference to assumed density filtering solved either through numerical integration and/or expectation propagation.
Gaussian Distribution
The belief in the score of each player may be based on a Gaussian distribution. A Gaussian density having n dimensions is defined by:




  
    
      
        
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The Gaussian of N(x) may be defined as a shorthand notation for a Gaussian defined by N(x;0,I). The cumulative Gaussian distribution function may be indicated by Φ(t;μ,σ2) which is defined by:




  
    
      
        
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Again, the shorthand of Φ(t) indicates a cumulative distribution of Φ(t;0,1). The notation of <f(x)>x ̃P denotes the expectation of f over the random draw of x, that is <f(x)>x ̃P=∫f(x)dP(x). The posterior probability of the outcome given the scores or the probability of the scores given the outcome may not be a Gaussian. Thus, the posterior may be estimated by finding the best Gaussian such that the Kullback-Leibler divergence between the true posterior and the Gaussian approximation is minimized. For example, the posterior P(θ|x) may be approximated by N(θ,μx*,Σx*) where the superscript * indicates that the approximation is optimal for the given x. In this manner, the mean and variance of the approximated Gaussian posterior may be given by:

μx*=μ+Σgx  (6)

Σx*=Σ−Σ(gxgxT−2Gx)Σ  (7)

Where the vector gx and the matrix Gx are given by:




  
    
      
        
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and the function Zx is defined by:

Zx(μ,Σ)=∫tx(θ)N(θ;μ;Σ)dθ=P(x)  (10)

Rectified Truncated Gaussians
A variable x may be distributed according to a rectified double truncated Gaussian (referred to as rectified Gaussian from here on) and annotated by x ̃R(x;μ,σ2,α,β) if the density of x is given by:




  
    
      
        
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When taking the limit of the variable β as it approaches infinity, the rectified Gaussian may be denoted as R(x;μ,σ2,α).
The class of the rectified Gaussian contains the Gaussian family as a limiting case. More particularly, if the limit of the rectified Gaussian is taken as the variable α approaches infinity, then the rectified Gaussian is the Normal Gaussian indicated by N(x; μ,σ2) used as the prior distribution of the scores.
The mean of the rectified Gaussian is given by:




  
    
      
        
          
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where the function v(·,α,β) is given by:




  
    
      
        
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The variance of the rectified Gaussian is given by:




  
    
      
        
          
            
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where the function w(·,α,β) is given by:




  
    
      
        
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          ⁡
          
            (
            
              t
              ,
              α
              ,
              β
            
            )
          
        
        =
        
          
            
              v
              2
            
            ⁡
            
              (
              
                t
                ,
                α
                ,
                β
              
              )
            
          
          +
          
            
              
                
                  (
                  
                    β
                    -
                    t
                  
                  )
                
                ⁢
                
                  N
                  ⁡
                  
                    (
                    
                      β
                      -
                      t
                    
                    )
                  
                
              
              -
              
                
                  (
                  
                    α
                    -
                    t
                  
                  )
                
                ⁢
                
                  N
                  ⁡
                  
                    (
                    
                      α
                      -
                      t
                    
                    )
                  
                
              
            
            
              
                Φ
                ⁡
                
                  (
                  
                    β
                    -
                    t
                  
                  )
                
              
              -
              
                Φ
                ⁡
                
                  (
                  
                    α
                    -
                    t
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        16
        )
      
    
  




As β approaches infinity, the functions v(·,α,β) and w(·,α,β) may be indicated as v(·,α) and w(·,α) and determined using:




  
    
      
        
          v
          ⁡
          
            (
            
              t
              ,
              α
            
            )
          
        
        =
        
          
            
              lim
              
                β
                ->
                ∞
              
            
            ⁢
            
              v
              ⁡
              
                (
                
                  t
                  ,
                  α
                  ,
                  β
                
                )
              
            
          
          =
          
            
              N
              ⁡
              
                (
                
                  t
                  -
                  α
                
                )
              
            
            
              Φ
              ⁡
              
                (
                
                  t
                  -
                  α
                
                )
              
            
          
        
      
    
    
      
        (
        17
        )
      
    
  
  
    
      
        
          w
          ⁡
          
            (
            
              t
              ,
              α
            
            )
          
        
        =
        
          
            
              lim
              
                β
                ->
                ∞
              
            
            ⁢
            
              w
              ⁡
              
                (
                
                  t
                  ,
                  α
                  ,
                  β
                
                )
              
            
          
          =
          
            
              v
              ⁡
              
                (
                
                  t
                  ,
                  α
                
                )
              
            
            ·
            
              (
              
                
                  v
                  ⁡
                  
                    (
                    
                      t
                      ,
                      α
                    
                    )
                  
                
                -
                
                  (
                  
                    t
                    -
                    α
                  
                  )
                
              
              )
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




These functions may be determined using numerical integration techniques, or any other suitable technique. The function w(·,α) may be a smooth approximation to the indicator function It≦α and may be always bounded by [0,1]. In contrast, the function v(·,α) may grow roughly like α-t for t<α and may quickly approach zero for t>α.
The auxiliary functions {tilde over (v)}(t,ε) and {tilde over (w)}(t,ε) may be determined using:

{tilde over (v)}(t,ε)=v(t,−ε,ε)  (19)

{tilde over (w)}(t,ε)=w(t,−ε,ε)  (20)

Learning Scores Over Time
A Bayesian learning process for a scoring system learns the scores for each player based upon the outcome of each match played by those players. Bayesian learning may assume that each player's unknown, true score is static over time, e.g., that the true player scores do not change. Thus, as more games are played by a player, the updated player's score 214 of FIG. 2 may reflect a growing certainty in this true score. In this manner, each new game played may have less impact or effect on the certainty in the updated player score 214.
However, a player may improve (or unfortunately worsen) over time relative to other players and/or a standard scale. In this manner, each player's true score is not truly static over time. Thus, the learning process of the scoring system may learn not only the true score for each player, but may allow for each player's true score to change over time due to changed abilities of the player. To account for changed player abilities over time, the posterior belief of the scores P(s|r,{i1, . . . ik}) may be modified over time. For example, not playing the game for a period of time (e.g., Δt) may allow a player's skills to atrophy or worsen. Thus, the posterior belief of the score of a player may be modified based upon the playing history of that player. More particularly, the posterior belief used as the new prior distribution may be represented as the posterior belief P(si|Δt) of the score of the player with index i, given that he had not played for a time of Δt. Thus, the modified posterior distribution may be represented as:




  
    
      
        
          
            
              
                P
                ⁡
                
                  (
                  
                    
                      s
                      i
                    
                    ❘
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                  
                  )
                
              
              =
              
                ∫
                
                  
                    P
                    ⁡
                    
                      (
                      
                        
                          s
                          i
                        
                        ❘
                        μ
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        μ
                        ❘
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                      
                      )
                    
                  
                  ⁢
                  
                    d
                    μ
                  
                
              
            
          
        
        
          
            
              =
              
                ∫
                
                  
                    N
                    ⁡
                    
                      (
                      
                        
                          
                            s
                            i
                          
                          ;
                          
                            μ
                            i
                          
                        
                        ,
                        
                          σ
                          i
                          2
                        
                      
                      )
                    
                  
                  ⁢
                  
                    N
                    ⁡
                    
                      (
                      
                        
                          μ
                          ;
                          
                            μ
                            i
                          
                        
                        ,
                        
                          
                            τ
                            2
                          
                          ⁡
                          
                            (
                            
                              Δ
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            )
                          
                        
                      
                      )
                    
                  
                  ⁢
                  
                    d
                    μ
                  
                
              
            
          
        
        
          
            
              =
              
                N
                ⁡
                
                  (
                  
                    
                      
                        s
                        i
                      
                      ;
                      
                        μ
                        i
                      
                    
                    ,
                    
                      
                        σ
                        i
                        2
                      
                      +
                      
                        
                          τ
                          2
                        
                        ⁡
                        
                          (
                          
                            Δ
                            ⁢
                            
                              
                            
                            ⁢
                            t
                          
                          )
                        
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        21
        )
      
    
  




where the first term P(si|μ) is the belief distribution of the score of the player with the index i, and the second term P(μ|Δt) quantifies the belief in the change of the unknown true score at a time of length Δt since the last update. The function τ(·) is the variance of the true score as a function of time not played (e.g., Δt). The function τ(Δt) may be small for small times of Δt to reflect that a player's performance may not change over a small period of non-playing time. This function may increase as Δt increases (e.g., hand-eye coordination may atrophy, etc). In the example below, the dynamic score function τ may return a constant value τ0, if the time passed since the last update is greater than zero as this indicates that at least one more game was played. If the time passed is zero, then the function τ may return 0. The constant function τ0 for the dynamic score function τ may be represented as:

τ2(Δt)=IΔt>0τ02  (22)

where I is the indicator function.
Inference
The belief in a particular game outcome may be quantified with all knowledge obtained about the scores of each player, P(s). More particularly, the outcome of a potential game given the scores of selected players may be determined. The belief in an outcome of a game for a selected set of players may be represented as:




  
    
      
        
          
            
              
                P
                ⁡
                
                  (
                  
                    r
                    ❘
                    
                      {
                      
                        
                          i
                          1
                        
                        ,
                        
                          ...
                          ⁢
                          
                            
                          
                          ⁢
                          
                            i
                            k
                          
                        
                      
                      }
                    
                  
                  )
                
              
              =
              
                ∫
                
                  
                    P
                    ⁡
                    
                      (
                      
                        
                          r
                          ❘
                          s
                        
                        ,
                        
                          {
                          
                            
                              i
                              1
                            
                            ,
                            
                              ...
                              ⁢
                              
                                
                              
                              ⁢
                              
                                i
                                k
                              
                            
                          
                          }
                        
                      
                      )
                    
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      
                        s
                        ❘
                        
                          {
                          
                            
                              i
                              1
                            
                            ,
                            
                              ...
                              ⁢
                              
                                
                              
                              ⁢
                              
                                i
                                k
                              
                            
                          
                          }
                        
                      
                      )
                    
                  
                  ⁢
                  
                    d
                    s
                  
                
              
            
          
        
        
          
            
              =
              
                ∫
                
                  
                    P
                    (
                    
                      
                        r
                        ❘
                        
                          S
                          ⁡
                          
                            (
                            
                              s
                              
                                i
                                1
                              
                            
                            )
                          
                        
                      
                      ,
                      ...
                      ⁢
                      
                        
                      
                      ,
                      
                        S
                        ⁡
                        
                          (
                          
                            s
                            
                              i
                              k
                            
                          
                          )
                        
                      
                    
                    }
                  
                  ⁢
                  
                    )
                  
                  ⁢
                  
                    P
                    ⁡
                    
                      (
                      s
                      )
                    
                  
                  ⁢
                  
                    d
                    s
                  
                
              
            
          
        
      
    
    
      
        (
        23
        )
      
    
  




where S(si1), . . . , S(sik) is sA and sB for a two payer game. Such a belief in a future outcome may be used in matching players for future games, as discussed further below.
Two Player Example
With two players (player A and player B) opposing one another in a game, the outcome of the game can be summarized in one variable y which is 1 if player A wins, 0 if the players tie, and −1 if player A loses. In this manner, the variable y may be used to uniquely represent the ranks r of the players. In light of equation (3) above, the update algorithm may be derived as a model of the game outcome y given the scores s1 and s2 as:

P(r|sA,sB)=P(y(r)|sA,sB)  (24)

where y(r)=sign(rB−rA), where rAis 1 and rB is 2 if player A wins, and rA is 2 and rB is 1 if player B wins, and rA and rB are both 1 if players A and B tie.
The outcome of the game (e.g., variable y) may be based on the latent scores of all participating players (which in the two player example are players A and B). The latent score xi may follow a Gaussian distribution with a mean equivalent to the score si of the player with index i, and a fixed latent score variance β2. More particularly, the latent score xi may be represented as N(xi;si, β2). Graphical representations of the latent scores are shown in FIG. 3 as Gaussian curves 302 and 306 respectively. The scores sA and sB are illustrated as lines 304 and 308 respectively.
The latent scores of the players may be compared to determine the outcome of the game. However, if the difference between the teams is small to zero, then the outcome of the game may be a tie. In this manner, a latent tie margin variable ε may be introduced as a fixed number to illustrate this small margin of equality between two competing players. Thus, the outcome of the game may be represented as:
Player A is the winner if:

xA>xB+ε  (25)

Player B is the winner if:

xB>xA+ε  (26)

Player A and B tie if:

|xA−xB|≦ε  (27)

A possible latent tie margin is illustrated in FIG. 3 as the range 310 of width 2ε around zero.
Since the two latent score curves are independent (due to the independence of the latent scores for each player), then the probability of an outcome y given the scores of the individual players A and B, may be represented as:




  
    P
    ⁡
    
      (
      
        
          y
          ❘
          
            s
            A
          
        
        ,
        
          s
          B
        
      
      )
    
  
  ⁢
  
    
  
  ⁢
  
    {
    
      
        
          
            
            ⁢
            
              =
              
                
                  
                    P
                    ⁡
                    
                      (
                      
                        Δ
                        <
                        
                          -
                          ɛ
                        
                      
                      )
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  if
                  ⁢
                  
                    
                  
                  ⁢
                  y
                
                =
                
                  -
                  1
                
              
            
          
        
        
          
            
              
            
            ⁢
            
              (
              28
              )
            
          
        
      
      
        
          
            
            ⁢
            
              =
              
                
                  
                    P
                    ⁡
                    
                      (
                      
                        
                          
                          Δ
                          
                        
                        ≤
                        ɛ
                      
                      )
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  if
                  ⁢
                  
                    
                  
                  ⁢
                  y
                
                =
                0
              
            
          
        
        
          
            
              
            
            ⁢
            
              (
              29
              )
            
          
        
      
      
        
          
            
            ⁢
            
              =
              
                
                  
                    P
                    ⁡
                    
                      (
                      
                        Δ
                        >
                        ɛ
                      
                      )
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  if
                  ⁢
                  
                    
                  
                  ⁢
                  y
                
                =
                
                  +
                  1
                
              
            
          
        
        
          
            
              
            
            ⁢
            
              (
              30
              )
            
          
        
      
    
  




where Δ is the difference between the latent scores xA and xB (e.g., Δ=xA−xB).
The joint distribution of the latent scores for player A and player B are shown in FIG. 4 as contour lines forming a ‘bump’ 402 in a graph with the first axis 410 indicating the latent score of player A and the second axis 412 indicating the latent score of player B. The placement of the ‘bump’ 402 or joint distribution may indicate the likelihood of player A or B winning by examining the probability mass of the area of the region under the ‘bump’ 402. For example, the probability mass of area 404 above line 414 may indicate that player B is more likely to win, the probability mass of area 406 below line 416 limited by lines 414 and 416 may indicate that player A is more likely to win, and the probability mass of area 408 may indicate that the players are likely to tie. In this manner, the probability mass of area 404 under the joint distribution bump 402 is the probability that player B wins, the probability mass of area 406 under the joint distribution bump 402 is the probability that player A wins, and the probability mass of area 408 under the joint distribution bump 402 is the probability that the players tie. As shown in the example joint distribution 402 of FIG. 4, it is more likely that player B will win.
As noted above, the score (e.g., mean μi and variance σi2) for each player i (e.g., players A and B), may be updated knowing the outcome of the game between those two players (e.g., players A and B). More particularly, using an ADF approximation, the update of the scores of the participating players may follow the method 500 shown in FIG. 5. The static variable(s) may be initialized. For example, the latent tie zone ε, the dynamic time update constant τ0, and/or the latent score variation β may be initialized 502. Example initial values for these parameters may be include: β is within the range of approximately 100 to approximately 400 and in one example may be approximately equal to 250, τ0 is within the range of approximately 1 to approximately 10 and may be approximately equal to 10 in one example, and ε may depend on many factors such as the draw probability and in one example may be approximately equal to 50. The score si (e.g., represented by the mean μi and variance σi2) may be received 504 for each of the players i, which in the two player example includes mean μA and variance σA2 for player A and mean μB and variance σB2 for player B.
Before a player has played a game, the score represented by the mean and variance may be initialized to any suitable values. In a simple case, the means may be all initialized at the same value, for example μi=1200. Alternatively, the mean may be initialized to a percentage (such as 20-50%, and in some cases approximately 33%) of the average mean of the established players. The variance may be initialized to indicate uncertainty about the initialized mean, for example σ2=4002. Alternatively, the initial mean and/or variance of a player may be based in whole or in part on the score of that player in another game environment.
If the belief is to be updated based on time, as described above, the variance of each participating player's score may be updated based on the function τ and the time since the player last played. The dynamic time update may be done in the dynamic score module 204 of the scoring system of FIG. 2. As noted above, the output of the dynamic score function τ may be a constant τ0 for all times greater than 0. In this manner, τ0 may be zero on the first time that a player plays a game, and may be the constant τ0 thereafter. The variance of each player's score may be updated 505 by:

σi2←σi2+τ02  (31)

To update the scores based on the game outcome, a parameter c may be computed 506 as the sum of the variances, such that parameter c is:




  
    
      
        c
        =
        
          
            
              (
              
                
                  n
                  A
                
                +
                
                  n
                  B
                
              
              )
            
            ⁢
            
              β
              2
            
          
          +
          
            σ
            A
            2
          
          +
          
            σ
            B
            2
          
        
      
    
    
      
        (
        32
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              2
              ⁢
              
                β
                2
              
            
            +
            
              σ
              A
              2
            
            +
            
              σ
              B
              2
            
          
        
      
    
    
      
        (
        33
        )
      
    
  




where nA is the number of players in team A (in this example 1) and nB is the number of players in team B (in this example 1).
The parameter h may be computed 506 based on the mean of each player's score and the computed parameter c as:




  
    
      
        
          h
          A
        
        =
        
          
            
              μ
              A
            
            -
            
              μ
              B
            
          
          
            c
          
        
      
    
    
      
        (
        34
        )
      
    
  
  
    
      
        
          h
          B
        
        =
        
          
            
              μ
              B
            
            -
            
              μ
              A
            
          
          
            c
          
        
      
    
    
      
        (
        35
        )
      
    
  




which, indicates that hA=−hB. The parameter ε′ may be computed 506 based on the number of players, the latent tie zone ε, and the parameter c as:




  
    
      
        
          ɛ
          ′
        
        =
        
          
            ɛ
            ⁡
            
              (
              
                
                  n
                  A
                
                +
                
                  n
                  B
                
              
              )
            
          
          
            2
            ⁢
            
              c
            
          
        
      
    
    
      
        (
        36
        )
      
    
  




And for the two player example, this leads to:




  
    
      
        
          ɛ
          ′
        
        =
        
          ɛ
          
            c
          
        
      
    
    
      
        (
        37
        )
      
    
  




The outcome of the game between players A and B may be received 508. For example, the game outcome may be represented as the variable y which is −1 if player B wins, 0 if the players tie, and +1 if player A wins. To change the belief in the scores of the participating players, such as in the score update module of FIG. 2, the mean and variance of the each score may be updated 510. More particularly, if the player A wins (e.g., y=1), then the mean μA of the winning player A may be updated as:




  
    
      
        
          μ
          A
        
        ←
        
          
            μ
            A
          
          +
          
            
              
                σ
                A
                2
              
              
                c
              
            
            ⁢
            
              v
              ⁡
              
                (
                
                  
                    h
                    A
                  
                  ,
                  
                    ɛ
                    ′
                  
                
                )
              
            
          
        
      
    
    
      
        (
        38
        )
      
    
  




The mean μB of the losing player B may be updated as:




  
    
      
        
          μ
          B
        
        ←
        
          
            μ
            B
          
          -
          
            
              
                σ
                B
                2
              
              
                c
              
            
            ⁢
            
              v
              ⁡
              
                (
                
                  
                    h
                    A
                  
                  ,
                  
                    ɛ
                    ′
                  
                
                )
              
            
          
        
      
    
    
      
        (
        39
        )
      
    
  




The variance σi2 of each player i (A and B) may be updated when player A wins as:




  
    
      
        
          σ
          i
          2
        
        ←
        
          
            σ
            i
            2
          
          (
          
            1
            -
            
              
                
                  σ
                  i
                  2
                
                c
              
              ⁢
              
                w
                ⁡
                
                  (
                  
                    
                      h
                      A
                    
                    ,
                    
                      ɛ
                      ′
                    
                  
                  )
                
              
            
          
          )
        
      
    
    
      
        (
        40
        )
      
    
  




However, if player B wins (e.g., y=−1), then the mean μA of the losing player A may be updated as:




  
    
      
        
          μ
          A
        
        ←
        
          
            μ
            A
          
          -
          
            
              
                σ
                A
                2
              
              
                c
              
            
            ⁢
            
              v
              ⁡
              
                (
                
                  
                    h
                    B
                  
                  ,
                  
                    ɛ
                    ′
                  
                
                )
              
            
          
        
      
    
    
      
        (
        41
        )
      
    
  




The mean μB of the winning player B may be updated as:




  
    
      
        
          μ
          B
        
        ⁢
        
          
        
        ←
        
          
        
        ⁢
        
          
            μ
            B
          
          ⁢
          
            
          
          +
          
            
          
          ⁢
          
            
              
                σ
                B
                2
              
              
                c
              
            
            ⁢
            
              
            
            ⁢
            
              v
              ⁡
              
                (
                
                  
                    h
                    B
                  
                  ,
                  
                    
                  
                  ⁢
                  
                    ɛ
                    ′
                  
                
                )
              
            
          
        
      
    
    
      
        (
        42
        )
      
    
  




The variance σi2 of each player i (A and B) may be updated when player B wins as:




  
    
      
        
          σ
          i
          2
        
        ←
        
          
            σ
            i
            2
          
          (
          
            1
            -
            
              
                
                  σ
                  i
                  2
                
                c
              
              ⁢
              
                w
                ⁡
                
                  (
                  
                    
                      h
                      B
                    
                    ,
                    
                      ɛ
                      ′
                    
                  
                  )
                
              
            
          
          )
        
      
    
    
      
        (
        43
        )
      
    
  




If the players A and B draw, then the mean μA of the player A may be updated as:




  
    
      
        
          μ
          A
        
        ←
        
          
            μ
            A
          
          +
          
            
              
                σ
                A
                2
              
              
                c
              
            
            ⁢
            
              
                v
                ~
              
              ⁡
              
                (
                
                  
                    h
                    A
                  
                  ,
                  
                    ɛ
                    ′
                  
                
                )
              
            
          
        
      
    
    
      
        (
        44
        )
      
    
  




The mean μB of the player B may be updated as:




  
    
      
        
          μ
          B
        
        ←
        
          
            μ
            B
          
          +
          
            
              
                σ
                B
                2
              
              
                c
              
            
            ⁢
            
              
                v
                ~
              
              ⁡
              
                (
                
                  
                    h
                    B
                  
                  ,
                  
                    ɛ
                    ′
                  
                
                )
              
            
          
        
      
    
    
      
        (
        45
        )
      
    
  




The variance σA2 of player A may be updated when the players tie as:




  
    
      
        
          σ
          A
          2
        
        ←
        
          
            σ
            A
            2
          
          (
          
            1
            -
            
              
                
                  σ
                  A
                  2
                
                c
              
              ⁢
              
                
                  w
                  ~
                
                ⁡
                
                  (
                  
                    
                      h
                      A
                    
                    ,
                    
                      ɛ
                      ′
                    
                  
                  )
                
              
            
          
          )
        
      
    
    
      
        (
        46
        )
      
    
  




The variance σB2 of player B may be updated when the players tie as:




  
    
      
        
          σ
          B
          2
        
        ←
        
          
            σ
            B
            2
          
          (
          
            1
            -
            
              
                
                  σ
                  B
                  2
                
                c
              
              ⁢
              
                
                  w
                  ~
                
                ⁡
                
                  (
                  
                    
                      h
                      B
                    
                    ,
                    
                      ɛ
                      ′
                    
                  
                  )
                
              
            
          
          )
        
      
    
    
      
        (
        47
        )
      
    
  




In equations (38-47) above, the functions v( ), w( ), {tilde over (v)}( ), and {tilde over (w)}( ) may be determined from the numerical approximation of a Gaussian. Specifically, functions v( ), w( ), {tilde over (v)}( ), and {tilde over (w)}( ) may be evaluated using equations (17-20) above using numerical methods such as those described in Press et al., Numerical Recipes in C: the Art of Scientific Computing (2d. ed.), Cambridge, Cambridge University Press, ISBN-0-521-43108-5, which is incorporated herein by reference, and by any other suitable numeric or analytic method.
The updated values of the mean and variance of each player's score from the score update module 202 of FIG. 2 may replace the old values of the mean and variance (scores 212). The newly updated mean and variance of each player's score incorporate the additional knowledge gained from the outcome of the game between players A and B.
The updated beliefs in a player's score may be used to predict the outcome of a game between two potential opponents. For example, a player match module 206 shown in FIG. 2 may use the updated and/or maintained scores of the players to predict the outcome of a match between any potential players and match those players meeting match criteria, such as approximately equal player score means, player indicated preferences, approximately equal probabilities of winning and/or drawing, and the like.
To predict the outcome of a game, the probability of a particular outcome y given the mean scores and standard deviations of the scores for each potential player, e.g., P(y|sA,sB) may be computed. Accordingly, the probability of the outcome (P(y)) may be determined from the probability of the outcome given the player scores with the scores marginalized out.
FIG. 6 illustrates an example method 600 of predicting a game outcome between two potential players (player A and player B). The static variable(s) may be initialized 602. For example, the latent tie zone ε, the dynamic time update constant τ0, and/or the latent score variation β may be initialized. The score si (e.g., represented by the mean μi and variance σi2) may be received 604 for each of the players i who are participating in the predicted game. In the two player example, the player scores include mean μA and variance σA2 for player A, and mean μB and variance σB2 for player B.
Parameters may be determined 606. The parameter c may be computed 606 as the sum of the variances using equation (32) or (33) above as appropriate. Equations (32) and (33) for the parameter c may be modified to include the time varying aspects of the player's scores, e.g., some time Δt has passed since the last update of the scores. The modified parameter c may be computed as:

c=(nA+nB)β2+σA2+σB2+(nA+nB)τ0   (48)

where nA is the number of players in team A (in this example 1 player) and nB is the number of players in team B (in this example 1 player). The parameter ε′ may be computed using equation (36) or (37) above as appropriate.
The probability of each possible outcome of the game between the potential players may be determined 608. The probability of player A winning may be computed using:




  
    
      
        
          P
          ⁡
          
            (
            
              y
              =
              1
            
            )
          
        
        =
        
          Φ
          ⁡
          
            (
            
              
                
                  μ
                  A
                
                -
                
                  μ
                  B
                
                -
                
                  ɛ
                  ′
                
              
              
                c
              
            
            )
          
        
      
    
    
      
        (
        49
        )
      
    
  




The probability of player B winning may be computed using:




  
    
      
        
          P
          ⁡
          
            (
            
              y
              =
              
                -
                1
              
            
            )
          
        
        =
        
          Φ
          ⁡
          
            (
            
              
                
                  μ
                  B
                
                -
                
                  μ
                  A
                
                -
                
                  ɛ
                  ′
                
              
              
                c
              
            
            )
          
        
      
    
    
      
        (
        50
        )
      
    
  




As noted above, the function Φ indicates a cumulative Gaussian distribution function having an argument of the value in the parentheses and a mean of zero and a standard deviation of one. The probability of players A and B having a draw may be computed using:

P(y=0)=1−P(y=1)−P(y=−1)  (51)

The determined probabilities of the outcomes may be used to match potential players for a game, such as comparing the probability of either team winning or drawing with a predetermined or user provided threshold or other preference. A predetermined threshold corresponding to the probability of either team winning or drawing may be any suitable value such as approximately 25%. For example, players may be matched to provide a substantially equal distribution over all possible outcomes, their mean scores may be approximately equal (e.g., within the latent tie margin), and the like. Additional matching techniques which are also suitable for the two player example are discussed below with reference to the multi-team example.
Two Teams
The two player technique described above may be expanded such that ‘player A’ includes one or more players in team A and ‘player B’ includes one or more players in team B. For example, the players in team A may have any number of players indicated by nA, and team B may have any number of players indicated by nB. A team may be defined as one or more players whose performance in the game achieve a single outcome for all the players on the team.
Each player of each team may have an individual score si represented by a mean μi and a variance σi2. More particularly, the players of team A may be indicated with the indices iA, and the players of team B may be indicated with the indices iB.
Since there are only two teams, like the two player example above, there may be three possible outcomes to a match, i.e., team A wins, team B wins, and teams A and B tie. Like the latent scores of the two player match above, a team latent score t(i) of a team with players having indices i is a linear function of the latent scores xj of the individual players of the team. For example, the team latent score t(i) may equal b(i)Tx with b(i) being a vector having n elements. Thus, the outcome of the game may be represented as:
Team A is the winner if:

t(iA)>t(iB)+ε  (52)

Team B is the winner if:

t(iB)>t(iA)+ε  (53)

Team A and B tie if:

|t(iA)−t(iB)|≦ε  (54)

where ε is the latent tie margin discussed above. The probability of the outcome given the scores of the teams siA and siB is shown in equations (28-30) above. However, in the team example, the term Δ of equations (28-30) above is the difference between the latent scores of the teams t(iA) and t(iB). More particularly, the term Δ may be determined as:

Δ=t(iA)−t(iB)=(b(iA)−b(iB))Tx=aTx  (55)

where x is a vector of the latent scores of all players and the vector a comprises linear weighting coefficients.
The linear weighting coefficients of the vector a may be derived in exact form making some assumptions. For example, one assumption may include if a player in a team has a positive latent score, then the latent team score will increase; and similarly, if a player in a team has a negative latent score, then the latent team score will decrease. This implies that the vector b(i) is positive in all components of i. The negative latent score of an individual allows a team latent score to decrease to cope with players who do have a negative impact on the outcome of a game. For example, a player may be a so-called ‘team killer.’ More particularly, a weak player may add more of a target to increase the latent team score for the other team than he can contribute himself by scoring. The fact that most players contribute positively can be taken into account in the prior probabilities of each individual score. Another example assumption may be that players who do not participate in a team (are not playing the match and/or are not on a participating team) should not influence the team score. Hence, all components of the vector b(i) not in the vector i should be zero (since the vector x as stored or generated may contain the latent scores for all players, whether playing or not). In some cases, only the participating players in a game may be included in the vector x, and in this manner, the vector b(i) may be non-zero and positive for all components (in i). An additional assumption may include that if two players have identical latent scores, then including each of them into a given team may change the team latent score by the same amount. This may imply that the vector b(i) is a positive constant in all components of i. Another assumption may be that if each team doubles in size and the additional players are replications of the original players (e.g., the new players have the same scores si, then the probability of winning or a draw for either team is unaffected. This may imply that the vector b(i) is equal to the inverse average team size in all components of i such that:




  
    
      
        
          b
          ⁡
          
            (
            i
            )
          
        
        =
        
          
            2
            
              
                n
                A
              
              +
              
                n
                B
              
            
          
          ⁢
          
            
              ∑
              
                j
                ∈
                i
              
            
            ⁢
            
              e
              j
            
          
        
      
    
    
      
        (
        56
        )
      
    
  




where the vector e is the unit n-vector with zeros in all components except for component j which is 1, and the terms nA and nB are the numbers in teams A and B respectively. With the four assumptions above, the weighting coefficients a are uniquely determined.
If the teams are equal sized, e.g., nA+nB, then the mean of the latent player scores, and hence, the latent player scores x, may be translated by an arbitrary amount without a change in the distribution Δ. Thus, the latent player scores effectively form an interval scale. However, in some cases, the teams may have uneven numbering, e.g., nA and nB are not equal. In this case, the latent player scores live on a ratio scale in the sense that replacing two players each of latent score x with one player of latent score 2x does not change the latent team score. In this manner, a player with mean score s is twice as good as a player with mean score s/2. Thus, the mean scores indicate an average performance of the player. On the other hand, the latent scores indicate the actual performance in a particular game and exist on an interval scale because in order to determine the probability of winning, drawing, and losing, only the difference of the team latent scores is used, e.g., t(iA)−t(iB).
The individual score si represented by the mean μi and variance σi2 of each player i in a team participating in a game may be updated based upon the outcome of the game between the two teams. The update equations and method of FIG. 5 for the two player example may be modified for a two team example. With reference to the method 500 of FIG. 5, the latent tie zone ε, the dynamic time update constant τ0, and the latent score variation β may be initialized 502 as noted above. Similarly, the score si (e.g., represented by the mean μi and variance σi2) may be received 504 for each of the players i in each of the two teams, which in the two team example includes mean μAi and variance σAi2 for the players i in team A and mean μBi and variance σui2 for the players i in team B.
Since the update to the belief based on time depends only on the variance of that player (and possibly the time since that player last played), the variance of each player may be updated 505 using equation (31) above. As noted above, the update based on time may be accomplished through the dynamic score module 204 of FIG. 2.
With reference to FIG. 5, the parameters may be computed 506 similar to those described above with some modification to incorporate the team aspect of the scores and outcome. The parameter c may be computed 506 as the sum of the variances, as noted above. However, in a two team example where each team may have one or more players, the variances of all players participating in the game must be summed. Thus, for the two team example, equation (32) above may be modified to:




  
    
      
        c
        =
        
          
            
              (
              
                
                  n
                  A
                
                +
                
                  n
                  B
                
              
              )
            
            ⁢
            
              
            
            ⁢
            
              β
              2
            
          
          +
          
            
              ∑
              
                i
                =
                1
              
              
                n
                A
              
            
            ⁢
            
              σ
              
                A
                i
              
              2
            
          
          +
          
            
              ∑
              
                i
                =
                1
              
              
                n
                B
              
            
            ⁢
            
              σ
              
                B
                i
              
              2
            
          
        
      
    
    
      
        (
        57
        )
      
    
  




The parameters hA and hB may be computed 506 as noted above in equations (34-35) based on the mean of each team's score μA and μB. The team mean scores μA and μB for teams A and team B respectively may be computed as the sum of the means of the player(s) for each team as:




  
    
      
        
          μ
          A
        
        =
        
          
            ∑
            
              i
              =
              1
            
            
              n
              A
            
          
          ⁢
          
            μ
            
              A
              i
            
          
        
      
    
    
      
        (
        58
        )
      
    
  
  
    
      
        
          μ
          B
        
        =
        
          
            ∑
            
              i
              =
              1
            
            
              n
              B
            
          
          ⁢
          
            μ
            
              B
              i
            
          
        
      
    
    
      
        (
        59
        )
      
    
  




The parameter ε′ may be computed 506 as




  
    ɛ
    ′
  
  =
  
    
      ɛ
      ⁢
      
        
      
      ⁢
      
        (
        
          
            n
            A
          
          +
          
            n
            B
          
        
        )
      
    
    
      2
      ⁢
      
        c
      
    
  




where nA is the number of players in team A, nB is the number of players in team B.
The outcome of the game between team A and team B may be received 508. For example, the game outcome may be represented as the variable y which is equal to −1 if team B wins, 0 if the teams tie, and +1 if team A wins. To change the belief in the probability of the previous scores of each participating player of each team, the mean and variance of each participating player may be updated 510 by modifying equations (38-46) above. If team A wins the game, then the individual means may be updated as:




  
    
      
        
          μ
          
            A
            i
          
        
        ←
        
          
            μ
            
              A
              i
            
          
          +
          
            
              
                σ
                
                  A
                  i
                
                2
              
              
                c
              
            
            ⁢
            v
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  h
                  A
                
                ,
                
                  ɛ
                  ′
                
              
              )
            
          
        
      
    
    
      
        (
        60
        )
      
    
  
  
    
      
        
          μ
          
            B
            i
          
        
        ←
        
          
            μ
            
              B
              i
            
          
          -
          
            
              
                σ
                
                  B
                  i
                
                2
              
              
                c
              
            
            ⁢
            v
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  h
                  A
                
                ,
                
                  ɛ
                  ′
                
              
              )
            
          
        
      
    
    
      
        (
        61
        )
      
    
  




The variance σi2 of each player i (of either team A or B) may be updated when team A wins as shown in equation (40) above.
However, if team B wins (e.g., y=−1), then the mean μAi of each participating player may be updated as:




  
    
      
        
          μ
          
            A
            i
          
        
        ←
        
          
            μ
            
              A
              i
            
          
          -
          
            
              
                σ
                
                  A
                  i
                
                2
              
              
                c
              
            
            ⁢
            v
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  h
                  B
                
                ,
                
                  ɛ
                  ′
                
              
              )
            
          
        
      
    
    
      
        (
        62
        )
      
    
  
  
    
      
        
          μ
          
            B
            i
          
        
        ←
        
          
            μ
            
              B
              i
            
          
          +
          
            
              
                σ
                
                  B
                  i
                
                2
              
              
                c
              
            
            ⁢
            v
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  h
                  B
                
                ,
                
                  ɛ
                  ′
                
              
              )
            
          
        
      
    
    
      
        (
        63
        )
      
    
  




The variance σi2 of each player i (of either team A or B) may be updated when team B wins as shown in equation (43) above.
If the teams A and B draw, then the mean μhd Ai and μBi of each player of the teams A and B respectively may be updated as:




  
    
      
        
          μ
          
            A
            i
          
        
        ←
        
          
            μ
            
              A
              i
            
          
          +
          
            
              
                σ
                
                  A
                  i
                
                2
              
              
                c
              
            
            ⁢
            
              v
              ~
            
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  h
                  A
                
                ,
                
                  ɛ
                  ′
                
              
              )
            
          
        
      
    
    
      
        (
        64
        )
      
    
  
  
    
      
        
          μ
          
            B
            i
          
        
        ←
        
          
            μ
            
              B
              i
            
          
          +
          
            
              
                σ
                
                  B
                  i
                
                2
              
              
                c
              
            
            ⁢
            
              v
              ~
            
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  h
                  B
                
                ,
                
                  ɛ
                  ′
                
              
              )
            
          
        
      
    
    
      
        (
        65
        )
      
    
  




The variance σAi2 A of each player in team A may be updated when the teams tie as:




  
    
      
        
          σ
          
            A
            i
          
          2
        
        ←
        
          
            σ
            
              A
              i
            
            2
          
          ⁡
          
            (
            
              1
              -
              
                
                  
                    σ
                    
                      A
                      i
                    
                    2
                  
                  c
                
                ⁢
                
                  w
                  ~
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      h
                      A
                    
                    ,
                    
                      ɛ
                      ′
                    
                  
                  )
                
              
            
            )
          
        
      
    
    
      
        (
        66
        )
      
    
  




The variance σBi2 of each player in team B may be updated when the teams tie as:




  
    
      
        
          σ
          
            B
            i
          
          2
        
        ←
        
          
            σ
            
              B
              i
            
            2
          
          ⁡
          
            (
            
              1
              -
              
                
                  
                    σ
                    
                      B
                      i
                    
                    2
                  
                  c
                
                ⁢
                
                  w
                  ~
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      h
                      B
                    
                    ,
                    
                      ɛ
                      ′
                    
                  
                  )
                
              
            
            )
          
        
      
    
    
      
        (
        67
        )
      
    
  




As with equations (38-43), the functions v( ), w( ), {tilde over (v)}( ), and {tilde over (w)}( ) may be evaluated using equations (17-20) above using numerical methods. In this manner, the updated values of the mean and variance of each player's score may replace the old values of the mean and variance to incorporate the additional knowledge gained from the outcome of the game between teams A and B.
Like the scoring update equations above, the matching method of FIG. 6 may be modified to accommodate two teams of one or more players each. Like above, the static variables may be initialized 602. The score si (represented by the mean μAi and μBi and the variance σAi2 and σBi2 for each player i of each respective team A and B) may be received 604 for each of the players. In addition, the matchmaking criteria may take into account the variability of scores within the team. For example, it may be desirable to have teams comprising players having homogeneous scores, because in some cases they may better collaborate.
The parameters may be determined 606 as noted above. For example, the parameter c may be computed using equation (57), the mean of each team μA and μB may be computed using equations (58) and (59), and ε′ may be computed using equation (36).
The probability of each possible outcome of the game between the two potential teams may be determined 608. The probability of team A winning may be computed using equation (49) above. The probability of team B winning may be computed using equation (50) above. The probability of a draw may be computed using equation (51) above. The determined probabilities of the outcomes may be used to match potential teams for a game, such as comparing the probability of either team winning and/or drawing, the team and/or player ranks, and/or the team and/or player scores with a predetermined or user provided threshold.
Multiple Teams
The above techniques may be further expanded to consider a game that includes multiple teams, e.g., two or more opposing teams which may be indicated by the parameter. The index j indicates the team within the multiple opposing teams and ranges from 1 to k teams, where k indicates the total number of opposing teams. Each team may have one or more players i, and the jth team may have a number of players indicated by the parameter nj and players indicated by ij. Knowing the ranking r of all k teams allows the teams to be re-arranged such that the ranks rj of each team may be placed in rank order. For example, the rank of each team may be placed in rank-decreasing order such that r(1)≦r(2)≦ . . . ≦r(k) where the index operator ( ) is a permutation of the indices j from 1 to k. Since in some cases, the rank of 1 is assumed to indicate the winner of the game, the rank-decreasing order may represent a numerically increasing order. In this manner, the outcome r of the game may be represented in terms of the permutation of team indices and a vector y∈{0,+1}k−1. For example, (yj=+1) if team (j) was winning against team (j+1), and (yj=0) if team (j) was drawing against team (j+1). In this manner, the elements of the vector y may be indicated as yj=sign(r(j+1)−r(j)).
Like the example above with the two teams, the outcome of the game may be based upon the latent scores of all participating players. The latent score xi may follow a Gaussian distribution with a mean equivalent to the score si of the player with index i, and a fixed latent score variance β2. In this manner, the latent score xi may be represented by N(xi;si, β2). The latent score t(i) of a team with players having indices in the vector i may be a linear function of the latent scores x of the individual players. In this manner, the latent scores may be determined as t(i)=b(i)Tx with b(i) as described above with respect to the two team example. In this manner, given a sample x of the latent scores, the ranking is such that the team with the highest latent team score t(i) is at the first rank, the team with the second highest team score is at the second rank, and the team with the smallest latent team score is at the lowest rank. Moreover, two teams will draw if their latent team scores do not differ by more than the latent tie margin ε. In this manner, the ranked teams may be re-ordered according to their value of the latent team scores. After re-ordering the teams based on latent team scores, the pairwise difference between teams may be considered to determine if the team with the higher latent team score is winning or if the outcome is a draw (e.g., the scores differ by less than ε).
To determine the re-ordering of the teams based on the latent scores, a k−1 dimensional vector Δ of auxiliary variables may be defined where:

Δj:=t(i(j))−t(i(j+1))=ajTx.  (68)

In this manner, the vector Δ may be defined as:




  
    
      
        Δ
        =
        
          
            
              A
              T
            
            ⁢
            x
          
          =
          
            
              [
              
                
                  
                    
                      a
                      1
                      T
                    
                  
                
                
                  
                    ⋯
                  
                
                
                  
                    
                      a
                      
                        k
                        -
                        1
                      
                      T
                    
                  
                
              
              ]
            
            ⁢
            x
          
        
      
    
    
      
        (
        69
        )
      
    
  




Since x follows a Gaussian distribution (e.g., x ̃N(x;s,β2I), the vector Δ is governed by a Gaussian distribution (e.g., Δ ̃N(Δ;ATs,β2ATA). In this manner, the probability of the ranking r (encoded by the matrix A based on the permutation operator ( ) and the k−1 dimensional vector y) can be expressed by the joint probability over Δ as:




  
    
      
        
          P
          ⁢
          
            
          
          ⁢
          
            (
            
              
                y
                |
                
                  s
                  
                    i
                    1
                  
                
              
              ,
              ...
              ⁢
              
                
              
              ,
              
                s
                
                  i
                  k
                
              
            
            )
          
        
        =
        
          
            ∏
            
              j
              =
              1
            
            k
          
          ⁢
          
            
              
                (
                
                  P
                  ⁡
                  
                    (
                    
                      
                        Δ
                        j
                      
                      >
                      ɛ
                    
                    )
                  
                
                )
              
              
                y
                j
              
            
            ⁢
            
              
                (
                
                  P
                  ⁡
                  
                    (
                    
                      
                        
                        
                          Δ
                          j
                        
                        
                      
                      ≤
                      ɛ
                    
                    )
                  
                
                )
              
              
                1
                -
                
                  y
                  j
                
              
            
          
        
      
    
    
      
        (
        70
        )
      
    
  




The belief in the score of each player (P(si)) which is parameterized by the mean scores μ and variances σ2 may be updated given the outcome of the game in the form of a ranking r. The belief may be determined using assumed density filtering with standard numerical integration methods (for example, Gentz, et al., Numerical Computation of Multivariate Normal Probabilities, Journal of Computational and Graphical Statistics 1, 1992, pp. 141-149.), the expectation propagation technique (see below), and any other suitable technique. In the special case that there are two teams (e.g., k=2), the update equations reduce to the algorithms described above in the two team example. And similarly, if each of the two teams has only one player, the multiple team equations reduce to the algorithms described above in the two player example.
In this example, the update algorithms for the scores of players of a multiple team game may be determined with a numerical integration for Gaussian integrals. Similarly, the dynamic update of the scores based on time since the last play time of a player may be a constant τ0 for non-play times greater than 0, and 0 for a time delay between games of 0 or at the first time that a player plays the game.
FIG. 7 illustrates an example method 700 of updating the scores of players playing a multiple team game. The latent tie zone ε, the dynamic time update constant τ0, and the latent score variation β may be initialized 702 as noted above. In addition, the matrix A having k−1 columns and n rows (i.e., the total number of players in all teams) may be initialized 702 with any suitable set of numbers, such as 0. The score si (e.g., represented by the mean μi and variance σi2 ) may be received 704 for each of the players i in each of the teams, which in the multiple team example includes mean μji and variance σji2 for the players i in each team j.
Since the update to the belief based on time depends only on the variance of that player (and possibly the time since that player last played), the variance of each player may be updated 706 using equation (31) above. In this manner, for each player in each team, the dynamic update to the variance may be determined before the game outcome is evaluated. More particularly, the update to the variance based on time since the player last played the game, and the player's skill may have changed in that period of time before the current game outcome is evaluation. Alternatively, the belief based on time may be done after the scores are updated based on the game outcome.
The scores may be rank ordered by computing 708 the permutation ( ) according to the ranks r of the players participating in the game. For example, the ranks may be placed in decreasing rank order.
The ranking r may be encoded 710 by the matrix A. More particularly, for each combination of the n(j) and n(j+1) players of team (j) and (j+1), the matrix element Arow,j may be determined as:

Arow,j=2/(n(j)+n(j+1))  (71)

where the row variable is defined by the player i(j), the column variable is defined by the index j which varies from 1 to k−1 (where k is the number of teams), and

Arow+1,j=−2/(n(j)+n(j+1))  (72)

where the row variable is defined by the player i(j+1), the column variable is defined by the index j which varies from 1 to k−1 (where k is the number of teams), n(j) is the number of players on the (j)th team, and n(j+1) is the number of players on the (j+1)th team. If the (j)th team is of the same rank as the (j+1) team, then the lower and upper limits a and b of a truncated Gaussian may be set as:

ai=−ε  (73)

bi=ε  (74)

Otherwise, if the (j)th team is not of the same rank as the (j+1) team, then the lower and upper limits a and b of a truncated Gaussian may be set as:

ai=ε  (75)

bi=∞  (76)

The determined matrix A may be used to determine 712 interim parameters. Interim parameters may include a vector u and matrix C using the equations:

u=ATμ  (77)

C=AT(β2I+diag(σ2))A  (78)

where the vector μ is a vector containing the means of the layers, β is the latent score variation, and σ2 is a vector containing the variances of the players. The vector μ and σ2 may contain the means of the participating players or of all the players. If the vector contain the score parameters for all the players, then, the construction of A may provide a coefficient of) for each non-participating player.
The interim parameters u and C may be used to determine 714 the mean z and the covariance Z of a truncated Gaussian representing the posterior with parameters u, C, and integration limits of the vectors a and b. The mean and covariance of a truncated Gaussian may be determined using any suitable method including numerical approximation (see Gentz, et al., Numerical Computation of Multivariate Normal Probabilities, Journal of Computational and Graphical Statistics 1, 1992, pp. 141-149.), expectation propagation (see below), and the like. Expectation Propagation will be discussed further below with respect to FIG. 9.
Using the computed mean z and covariance Z, the score defined by the mean μi and the variance σi2 of each player participating in the multi-team game may be updated 716. In one example, the function vector v and matrix W may be determined using:

v=AC−1(z−u)  (79)

W=AC−1(C−Z)C−1AT  (80)

Using the vector v and the matrix W, the mean μji and variance σji2 of each player i in each team j may be updated using:

μji←μji+σji2vji  (81)

σji2←σji2(1−σji2Wji,ji)  (82)

The above equations and methods for a multiple team game may be reduced to the two team and the two player examples given above.
In this manner, the update to the mean of each player's score may be a linear increase or decrease based on the outcome of the game. For example, if in a two player example, player A has a mean greater than the mean of player B, then player A should be penalized and similarly, player B should be rewarded. The update to the variance of each player's score is multiplicative. For example, if the outcome is unexpected, e.g., player A's mean is greater than player B's mean and player A loses the game, then the variance of each player may be reduced more because the game outcome is very informative with respect to the current belief about the scores. Similarly, if the players' means are approximately equal (e.g., their difference is within the latent tie margin) and the game results in a draw, then the variance may be little changed by the update since the outcome was to be expected.
As discussed above, the scores represented by the mean μ and variance σ2 for each player may be used to predict the probability of a particular game outcome y given the mean scores and standard deviations of the scores for all participating players. The predicted game outcome may be used to match players for future games, such as by comparing the predicted probability of the outcome of the potential game with a predetermined threshold, player indicated preferences, ensuring an approximately equal distribution over possible outcomes (e.g., within 1-25%), and the like. The approximately equal distribution over the possible outcomes may depend on the number of teams playing the game. For example, with two teams, the match may be set if each team has an approximately 50% chance of winning or drawing. If the game has 3 teams, then the match may be made if each opposing team has an approximately 30% chance of winning or drawing. It is to be appreciated that the approximately equal distribution may be determined from the inverse of number of teams playing the game.
In one example, one or more players matched by the player match module may be given an opportunity to accept or reject a match. The player's decision may be based on given information such as the challenger's score and/or the determined probability of the possible outcomes. In another example, a player may be directly challenged by another player. The challenged player may accept or deny the challenge match based on information provided by the player match module.
The probability of a game outcome may be determined by computing the probability of a game outcome y (P(y)) from the probability of the outcome given the scores (P(y|si1, . . . , sik) where the attained knowledge over the scores si1, . . . , sik represented by the mean and variance of each player is marginalized out.
Like the scoring update equations above, the matching method of FIG. 6 may be modified to accommodate multiple teams of one or more players each. An example modified method 800 of determining the probability of an outcome is shown in FIG. 8. Like above, the static variables, such as the latent score variation β, the latent tie zone ε, the constant dynamic τ0, and the matrix A, may be initialized 802. The matrix A may be initialized to a matrix containing all zeros.
The score si (represented by the mean μi and the variance σi2 for each participating player i) may be received 804 for each of the players. The ranking r of the k teams may be received 806. For each player participating, the variance σi2 may be updated 808 for each participating player based upon the time since that player has last played the game, e.g., dynamic update based on time. In this manner, the variance for each potential participating player i, the variance may be updated using equation (31) above.
The scores of the teams may be rank ordered by computing 810 the permutation ( ) according to the ranks r of the players. For example, as noted above, the ranks may be placed in decreasing rank order.
The encoding of the ranking may be determined 812. The encoding of the ranking may be determined using the method described with reference to determining the encoding of a ranking 710 of FIG. 7 and using equations (71-76). Interim parameters u and C may be determined 814 using equations (77-78) above and described with reference to determining interim parameters 712 of FIG. 7.
The probability of the game outcome may be determined 816 by evaluation of the value of the constant function of a truncated Gaussian with mean u and variance C. As noted above, the truncated Gaussian may be evaluated in any suitable manner, including numerical approximation (see Gentz, et al., Numerical Computation of Multivariate Normal Probabilities, Journal of Computational and Graphical Statistics 1, 1992, pp. 141-149.), expectation propagation, and the like.
Numerical Approximation
One suitable technique of numerical approximation is discussed in Gentz, et al., Numerical Computation of Multivariate Normal Probabilities, Journal of Computational and Graphical Statistics 1, 1992, pp. 141-149. In one example, if the dimensionality (e.g., the number of players nj in a team j) of the truncated Gaussian is small, then the approximated posterior may be estimated based on uniform random deviates, based on a transformation of random variables which can be done iteratively using the cumulative Gaussian distribution Φ discussed above.
Since the normalization constant Zr(u,C) equals the probability of the ranking r, then the normalization constant may be determined by integrating the equation:




  
    
      
        
          
            Z
            r
          
          ⁢
          
            
          
          ⁢
          
            (
            
              μ
              ,
              σ
            
            )
          
        
        =
        
          
            ∫
            a
            b
          
          ⁢
          
            N
            ⁢
            
              
            
            ⁢
            
              (
              
                
                  z
                  ;
                  u
                
                ,
                C
              
              )
            
            ⁢
            
              d
              z
            
          
        
      
    
    
      
        (
        83
        )
      
    
  




The mean z may be determined using ADF by:




  
    
      
        ⁣
        
          
            
              〈
              z
              〉
            
            
              z
              ∼
              
                R
                ⁡
                
                  (
                  z
                  )
                
              
            
          
          =
          
            
              u
              ⁢
              
                
              
              ⁢
              
                (
                μ
                )
              
            
            +
            
              
                C
              
              ⁡
              
                [
                
                  
                    
                      v
                      ⁡
                      
                        (
                        
                          
                            
                              u
                              ⁢
                              
                                
                              
                              ⁢
                              
                                (
                                μ
                                )
                              
                            
                            
                              C
                            
                          
                          ⁢
                          
                            ɛ
                            
                              C
                            
                          
                        
                        )
                      
                    
                    ·
                    
                      v
                      ~
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      (
                      
                        
                          
                            u
                            ⁡
                            
                              (
                              μ
                              )
                            
                          
                          
                            C
                          
                        
                        ⁢
                        
                          ɛ
                          
                            C
                          
                        
                      
                      )
                    
                    
                      1
                      -
                      y
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        84
        )
      
    
  




Numerically approximating the above equations will provide the mean and normalization constant which may be used to numerically approximate a truncated Gaussian.
Expectation Propagation
Rather than numerical approximation, expectation propagation may be used to update and/or predict the score of a player. In the case of multiple teams, the update and prediction methods may be based on an iteration scheme of the two team update and prediction methods. To reduce the number of inversion s calculated during the expectation propagation, the Gaussian distribution may be assumed to be rank 1 Gaussian, e.g., that the likelihood ti,r is some function of the one-dimensional projection of the scores s. The efficiency over the general expectation approximation may be increased by assuming that the posterior is a rectified, truncated Gaussian distribution.
For example, FIG. 9 shows an example method 1200 of approximating a truncated Gaussian with expectation propagation.
The mean μ and covariance Σ of a non-truncated Gaussian may be received 1202. The mean may have n elements, and the covariance matrix may be dimensioned as nxn. The upper and lower truncation points of the truncated Gaussian may be received. For example, if the jth team is of the same rank as the j+1 team, then the lower and upper limits a and b of a truncated Gaussian may be set for each j and j+1 player as:

ai=−ε  (85)

bi=ε  (86)

Otherwise, if the jth team is not of the same rank as the j+1 team, then the variables a and b may be set for each j and j+1 player as:

ai=ε  (87)

bi=∞

The parameters of the expectation propagation may be initialized 1206. More particularly, for each i from 1 to n, the mean μi may be initialized to zero or any other suitable value, the parameter πi may be initialized to zero or any other suitable value, the parameter ζs may be initialized to 1 or any other suitable value. The approximated mean μ* may be initialized to the received mean μ, and the approximated covariance Σ* may be initialized to the received covariance Σ.
An index j may be selected 1208 from 1 to n. The approximate mean and covariance (μ* and Σ*) may be updated 1210. More particularly, the approximate mean and covariance may be updated by:




  
    
      
        
          μ
          *
        
        =
        
          
            μ
            *
          
          +
          
            
              
                
                  
                    π
                    j
                  
                  ⁢
                  
                    (
                    
                      
                        μ
                        j
                        *
                      
                      -
                      
                        μ
                        j
                      
                    
                    )
                  
                
                +
                
                  α
                  j
                
              
              
                e
                j
              
            
            ⁢
            
              t
              j
            
          
        
      
    
    
      
        (
        88
        )
      
    
  
  
    
      
        
          Σ
          *
        
        =
        
          
            Σ
            *
          
          +
          
            
              
                
                  
                    π
                    j
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    e
                    j
                  
                
                -
                
                  β
                  j
                
              
              
                e
                j
                2
              
            
            ⁢
            
              t
              j
            
            ⁢
            
              t
              j
              T
            
          
        
      
    
    
      
        (
        89
        )
      
    
  




where tj is determined by:

tj=[Σ1,j*,Σ2,j*, . . . ,Σn,j*]  (90)

and the factors dj and ej are determined by:

dj=πiΣj,j*  (91)

ej=1−dj  (92)

The factors αj and βj may be determined by:

αj=v(φj′,aj′,bj′)/√{square root over (ψj)}  (93)

βj=w(φj′,aj′,bj′)/√{square root over (ψj)}  (94)

where the function v( ) and w( ) may be evaluated using equations (17-18) above and the parameters (φj′,aj′,bj′, and ψj may be evaluated using:

φj=μj*+dj(μj*−μj)/ej  (95)

ψj=Σj,j*/ej  (96)

φj′=φj/√{square root over (ψj)}  (97)

ψj′=ψj/√{square root over (ψj)}  (98)

aj′=aj/√{square root over (ψj)}  (99)

bj′=bj/ψ  (100)

The factors πj, μj, and ζj may be updated 1212. More particularly, the factors may be updated using:




  
    
      
        
          π
          j
        
        =
        
          1
          /
          
            (
            
              
                β
                j
                
                  -
                  1
                
              
              -
              
                ψ
                j
              
            
            )
          
        
      
    
    
      
        (
        101
        )
      
    
  
  
    
      
        
          μ
          j
        
        =
        
          
            φ
            j
          
          +
          
            
              α
              j
            
            /
            
              β
              j
            
          
        
      
    
    
      
        (
        102
        )
      
    
  
  
    
      
        
          ς
          j
        
        =
        
          
            
              (
              
                
                  Φ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        b
                        j
                        ′
                      
                      -
                      
                        φ
                        j
                        ′
                      
                    
                    )
                  
                
                -
                
                  Φ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      
                        a
                        j
                        ′
                      
                      -
                      
                        φ
                        j
                        ′
                      
                    
                    )
                  
                
              
              )
            
            ·
            exp
          
          ⁢
          
            
          
          ⁢
          
            
              α
              j
              2
            
            
              2
              ⁢
              
                
                  β
                  j
                
                (
                
                  
                    1
                    -
                    
                      
                        ψ
                        j
                      
                      ⁢
                      
                        β
                        j
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        103
        )
      
    
  




The termination criteria may then be evaluated 1214. For example, the termination condition Δz may be computed using:

Δz=|Z*−Z*old|  (104)

Or nay other suitable termination condition which may indicate convergence of the approximation. The determined termination condition Δz may be compared to a predetermined termination toleration criterion δ. If the absolute value of the determined termination condition is less than or equal to the termination toleration criterion, then the approximated mean μ*, variance Σ*, and normalization constant Z* may be considered converged. If the termination criteria is not fulfilled, then the method may return to selecting an index 1208. If the termination criteria is fulfilled, then the approximated mean and covariance may be returned. In addition, the normalization constant Z* may be evaluated 1216. More particularly, the normalization constant may be evaluated using:




  
    
      
        
          
            
              
                Z
                *
              
              =
              
              ⁢
              
                
                  (
                  
                    
                      ∏
                      
                        i
                        =
                        1
                      
                      n
                    
                    ⁢
                    
                      ς
                      i
                    
                  
                  )
                
                ·
                
                  
                    
                    
                      
                        ∑
                        *
                      
                      ⁢
                      
                        ∑
                        
                          -
                          1
                        
                      
                    
                    
                  
                
                ·
              
            
          
        
        
          
            
              
              ⁢
              
                exp
                ⁡
                
                  (
                  
                    
                      -
                      
                        1
                        2
                      
                    
                    ⁢
                    
                      (
                      
                        
                          
                            ∑
                            
                              i
                              =
                              1
                            
                            n
                          
                          ⁢
                          
                            
                              π
                              i
                            
                            ⁢
                            
                              μ
                              i
                              2
                            
                          
                        
                        +
                        
                          
                            μ
                            T
                          
                          ⁢
                          
                            
                              ∑
                              
                                -
                                1
                              
                            
                            ⁢
                            μ
                          
                        
                        -
                        
                          
                            μ
                            
                              *
                              T
                            
                          
                          ⁢
                          
                            
                              ∑
                              
                                *
                                
                                  -
                                  1
                                
                              
                            
                            ⁢
                            
                              μ
                              *
                            
                          
                        
                      
                      )
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        105
        )
      
    
  




Matchmaking and Leaderboards
As noted above, the probability of the outcome may be used to match players such that the outcome is likely to be challenging to the teams, in accordance with a predetermined threshold. Determining the predicted outcome of a game may be expensive in some cases in terms of memory to store the entire outcome distribution for more than four teams. More particularly, there are O(2k-1k!) outcomes where k is the number of teams and where O( ) means ‘order of’ e.g., the function represented by O( ) can only be different by a scaling factor and/or a constant. In addition, the predicted outcomes may not distinguish between players with different standard deviations σi if their means μi are identical. In some cases, it may be computationally expensive to compute the distance between two outcome distributions. Thus, in some cases it may be useful to compute the score gap between the scores of two players. For example, the score gap may be defined as the difference between two scores si and sj. The expected score gap E(si−sj) or E[(si−sj)2] may be determined using:




  
    
      
        
          
            E
            ⁡
            
              [
              
                
                
                  
                    s
                    i
                  
                  -
                  
                    s
                    j
                  
                
                
              
              ]
            
          
          =
          
            
              2
              ⁢
              
                σ
                ij
                2
              
              ⁢
              
                
              
              ⁢
              N
              ⁢
              
                
              
              ⁢
              
                (
                
                  
                    
                      μ
                      ij
                    
                    ;
                    0
                  
                  ,
                  
                    σ
                    ij
                    2
                  
                
                )
              
            
            +
            
              
                μ
                ij
              
              ⁡
              
                (
                
                  
                    2
                    ⁢
                    
                      Φ
                      ⁡
                      
                        (
                        
                          
                            μ
                            ij
                          
                          
                            σ
                            ij
                          
                        
                        )
                      
                    
                  
                  -
                  1
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        or
      
    
    
      
        (
        106
        )
      
    
  
  
    
      
        
          E
          ⁡
          
            [
            
              
                (
                
                  
                    s
                    i
                  
                  -
                  
                    s
                    j
                  
                
                )
              
              2
            
            ]
          
        
        =
        
          
            μ
            ij
            2
          
          +
          
            σ
            ij
            2
          
        
      
    
    
      
        (
        107
        )
      
    
  




where μij is the difference in the means of the players (i.e., μij=μi−μj) and where σij2 is the sum of the variances of the players i and j (i.e., σij2=σj2+σj2). The expectation of the gap in scores may be compared to a predetermined threshold to determine if the player i and j should be matched. For example, the predetermined threshold may be in the range of approximate 3 to approximately 6, and may depend on many factors including the number of players available for matching. More particularly, the more available players, the lower the threshold may be set.
Moreover, the score belief of player i can be used to compute a conservative score estimate as μi−l·σi where l is a positive number that quantifies the level of conservatism. Any appropriate number for l may be selected to indicate the level of conservatism, such as the number 3, may be used for leaderboards. The advantage of such a conservative score estimate is that for new players, the estimate it can be zero (due to the large initial variance σi2) which is often more intuitive for new players (“starting at zero”).
Having now described some illustrative embodiments of the invention, it should be apparent to those skilled in the art that the foregoing is merely illustrative and not limiting, having been presented by way of example only. Numerous modifications and other illustrative embodiments are within the scope of one of ordinary skill in the art and are contemplated as falling within the scope of the invention. In particular, although the above example are described with reference to modeling the prior and/or the posterior probability with a Gaussian, it is to be appreciated that the above embodiments may be expanded to allowing arbitrary distributions over players' scores, which may or may not be independent. Moreover, although many of the examples presented herein involve specific combinations of method operations or system elements, it should be understood that those operations and those elements may be combined in other ways to accomplish the same objectives. Operations, elements, and features discussed only in connection with one embodiment are not intended to be excluded from a similar role in other embodiments. Moreover, use of ordinal terms such as “first” and “second” in the claims to modify a claim element does not by itself connote any priority, precedence, or order of one claim element over another or the temporal order in which operations of a method are performed, but are used merely as labels to distinguish one claim element having a certain name from another element having a same name (but for use of the ordinal term) to distinguish the claim elements.

## Claims

1. A method comprising:
a) receiving a draw outcome of a game between at least a first team opposing a second team, the first team including at least one player and the second team including at least one other player;
b) receiving a first score including a first mean and a first variance representing a distribution associated with the at least one player of the first team;
c) receiving a second score including a second mean and a second variance representing a distribution associated with the at least one other player of the second team;
d) updating the first mean and the first variance based on the draw outcome of the game;
e) updating the first mean and the first variance of the first score due to changed abilities of the at least one player of the first team based on a dynamic score function of a period of time since the first team last played the game; and
f) providing an indication of the updated first mean and the updated first variance to the at least one player of the first team.

2. The method of claim 1, wherein the dynamic score function is a constant value for periods of time greater than zero.

3. The method of claim 2, wherein updating the first variance based on a dynamic score function includes summing the first variance with a square of the constant value.

4. The method of claim 1, wherein receiving an outcome includes receiving an outcome of the game between the first team, the second team, and a third team.

5. The method of claim 1, wherein updating the first mean based on the draw outcome of the game depends on a predetermined latent tie margin defining a difference in scores.

6. The method of claim 1, wherein updating based on the draw outcome of the game includes modeling a posterior probability of the scores as a truncated rectified Gaussian distribution.

7. The method of claim 1, wherein updating based on the draw outcome of the game includes estimating a posterior probability of the scores of the first and second teams using expectation propagation.

8. The method of claim 1, further comprising determining a score estimate of the at least one player of the first team based on a difference between the updated first mean based on the draw outcome of the game and a conservative level indicator multiplied by a square root of the updated first variance based on the draw outcome of the game.

9. The method of claim 8, further comprising comparing the score estimate of the at least one player of the first team with a score estimate of another player, and ranking the at least one player of the first team and the another player based on the respective score estimate.

10. The method of claim 1, wherein updating the first mean and the first variance based on the draw outcome of the game includes decreasing the first variance to indicate an increase in a confidence level of the first variance.

11. The method of claim 1, further comprising determining a first conservative score estimate based on a difference between the first mean and a level of conservatism parameter multiplied by the first variance, and ranking the first player in a leaderboard based on the conservative score estimate.

12. One or more computer readable media containing computer executable instructions that, when implemented, perform a method comprising:
a) determining a first score including a first mean and a first variance representing a distribution associated with a first player of a first team;
b) determining a second score including a second mean and a second variance representing a distribution associated with a second player of a second team, wherein the first score and the second score are updated based on a draw outcome of a game between the first team and the second team;
c) determining an expectation value of a gap between the first score and the second score based on the first mean, the first variance, the second mean, and the second variance;
d) matching the first team with the second team based on a comparison of the expectation value with a predetermined threshold, wherein the predetermined threshold is based on the number of players available for matching; and
e) providing an indication of the match to at least one of the first team and the second team.

13. The computer readable media of claim 12, wherein determining the expectation of a gap includes evaluating




  
    
      2
      ⁢
      
        σ
        ij
        2
      
      ⁢
      
        
      
      ⁢
      N
      ⁢
      
        
      
      ⁢
      
        (
        
          
            
              μ
              ij
            
            ;
            0
          
          ,
          
            σ
            ij
            2
          
        
        )
      
    
    +
    
      
        μ
        ij
      
      ⁡
      
        (
        
          
            2
            ⁢
            
              Φ
              ⁡
              
                (
                
                  
                    μ
                    ij
                  
                  
                    σ
                    ij
                  
                
                )
              
            
          
          -
          1
        
        )
      
    
  
  ,




where μij is the difference in the means of the teams (i.e., μij=μi−μj) and where σij2 is the sum of the variances of the teams i and j (i.e., σij2=σj2+σj2).

14. The computer readable media of claim 12, wherein determining the expectation of a gap includes summing the first variance, the second variance, and a square of a difference between the first mean and the second mean.

15. One or more computer readable media containing computer executable components comprising:
a) means for updating a first score of a first player of a first team and a second score of a second player of a second team based on a draw outcome of a game between the first team and the second team, wherein each of the first score and the second score is modeled as a distribution; and
b) means for updating a first mean and a first variance of the first score due to changed abilities of the first player of the first team based on a dynamic score function of a period of time since the first player last played the game.

16. The computer readable media of claim 15, further comprising means for updating a third score of a third player of the first team, the third score being modeled as a distribution.

17. The computer readable media of claim 15, further comprising means for updating a fourth score of a fourth player of a third team a win outcome of the game, the fourth score being modeled as a distribution, the game being between the first team, the second team, and the third team, and the win outcome indicating the third team winning over both the first team and the second team.

18. The computer readable media of claim 15, further comprising means for ranking the first player and the second player based on the first score and the second score.