Match Prediction Using Learned History Embeddings

Contemporary ranking systems that are based on win/loss history , such as Elo ( Elo , 1978 ) or TrueSkill ( Herbrich et al. , 2007 ) represent each player using a scalar estimate of ability ( plus variance , in the latter case ) . While easily interpretable , this approach has a number of shortcomings : ( i ) latent attributes of a player can not be represented , and ( ii ) it can not seamlessly incorporate contextual information ( e.g . home-field advantage ) . In this work , we propose a simple Transformer-based approach for pairwise competitions that recursively operates on game histories , in addition to modeling players directly . By characterizing each player by its history , rather than an underlying scalar skill estimate , it is able to make accurate predictions even for new players with limited history . When restricted to the same information as several state of the art methods , including Elo and Glicko ( Glickman , 1995 ) , our approach significantly outperforms them on predicting the outcome of real-world Chess , Baseball and Ice Hockey games . 1 INTRODUCTION . Systems for ranking teams or individuals in pairwise competitions have important uses in a wide variety of applications that include : ( online e ) -sports matchups , dating apps , rating sports teams , rating chess players and training populations of agents ( Balduzzi et al. , 2018 ) . The most widely used techniques use as input the outcomes of previous match-ups , this being universally available and requiring no domain insights or meta-data . For easy interpretability , most of them use a single scalar , corresponding to a latent skill attribute , to characterize a player or team . This single value is then used to predict the outcome of match-ups or games . However , this implies that the multitude of attributes that makeup a player can be represented by a single number , which is a questionable assumption . This paper introduces a simple Transformer-based approach for predicting the outcome of competitions from historical data alone . Each player/team is represented by an embedding vector , computed entirely from the history of previous competitions . While a scalar skill rating for each player can not directly be computed , the model can still be used for the same ranking or match-up tasks as existing methods . While the model is naturally able to incorporate contextual information alongside input history , we focus on using historical data alone , restricting our contextual information to home-field advantage . This is the most general scenario and enables direct comparison to many widely used techniques , avoiding the confounding effect of feature engineering . A key feature of our model is that it is not necessary to make use of player identity in order to achieve strong predictions : each player is represented entirely by their previous games and outcomes . In a recursive fashion , the opponents of these games are also represented by their history . The purely relational summary of player history allows it to quickly produce accurate predictions for new players for whom little history exists , or for teams at the start of a new season . Player identity can be included in the model , which enables the modeling of non-transitive relations , as we show in a simple example of rock-paper-scissors . 2 RELATED WORK . Skill-based methods : Skill ranking has a long history in professional sports and competitive games ( Glickman , 1995 ; Dehpanah et al. , 2020 ) . In chess , the most widely-known skill rating system is Elo ( Elo , 1978 ) . This assumes that a player A has a skill rating θA and that its performance pA is distributed according to pA ∼ N ( pA ; θA , σ2 ) where σ is some fixed variance . Then the probability that A beats b is given by Pr ( pA > pB |θA , θB ) = Φ ( θA − θB√ 2σ ) where Φ is the cdf of a unit Gaussian distribution with mean 0 . Replacing Φ with a logistic distribution results in a Bradley-Terry model ( Bradley and Terry , 1952 ) . Both Elo and Bradley-Terry can be made Bayesian by assuming some distribution over the skills of each player , i.e . each player has a variance σA in addition to θA so that each player ’ s performance is modeled as a Gaussian distribution with mean θA and variance σA , rather than a fixed σ . These modifications are made in the Glicko system ( Glickman , 1995 ) and ( Weng and Lin , 2011 ) to Elo and Bradley-Terry , respectively . Both models can be extended to allow for teams of players and more than two person competitions , which is done in TrueSkill ( Herbrich et al. , 2007 ) ( Herbrich et al. , 2008 ) and Bayesian-Bradley Terry ( Weng and Lin , 2011 ) . Our approach contrasts with the methods above in that each player is represented by an embedding vector rather than a single number ( plus variance ) . In terms of how player history is utilized , Trueskill ( Herbrich et al. , 2007 ) , is closest to ours but still differs significant ways : ( a ) it uses player-specific representations , whereas ours in agnostic to the player ID ; ( b ) it combines information using a graphical model , rather than a deep learning model and ( c ) it is a flat model , whereas our builds the representation by recursively expanding opponent histories . Embedding approaches : ( Delalleau et al. , 2012 ) used a neural network architecture to learn latent embeddings for players , aiming to predict player enjoyment in online matchups . ( Zhang et al. , 2010 ) learn a factor embedding for each player and take the inner product with a context vector to get a skill rating . ( Balakrishnan and Chopra , 2012 ) use similar methods for user preferences ( `` do you prefer A or B ? '' ) . Our approach differs in that it focuses on the more general no-context scenario . In this setting the data only consists of outcomes ( win/loss or points differential ) and time features , better revealing differences in modeling performance . Non-transitivity : Recent work by ( Balduzzi et al. , 2018 ) proposed a determinant-based framework for modeling non-transitive interactions between RL agents on a leaderboard . Their approach draws on ideas from combinatorial Hodge theory , as laid out in ( Jiang et al. , 2011 ) . In the same spirit as ( Chen and Joachims , 2016a ; b ) who use a 1-layer MLP , we instead rely on the inherent non-linearity of deep neural networks to capture transitive and non-transitive relations ( if player ID is provided ) . ( Chen and Joachims , 2016c ) learn two embeddings for each player , a `` blade '' vector and a `` chest '' vector and use the difference between the two to model the probability of winning that captures intransitivity . Multi-player : Another group of approaches explicitly address multiplayer competitions ( Herbrich et al. , 2007 ; Minka et al. , 2018 ) , e.g . multiplayer online battle area games ( MOBA ) . Many approaches are customized to particular game environments and incorporate domain specific information , e.g . Heros of Newerth ( Suznjevic et al. , 2015 ) and Ghost Recon Online ( Delalleau et al. , 2012 ) . Our method can not naively handle multi-player settings , but there are plausible approaches to incorporating this , as we discuss later . Context : Many recent approaches have shown ways to incorporate contextual information alongside historical outcomes . ( Chen and Joachims , 2016a ) uses context , such as betting odds , to improve predictions . TrueSkill2 ( Minka et al. , 2018 ) refines TrueSkill ( Herbrich et al. , 2007 ) to incorporate context . While not the focus of our paper , since our approach is neural network-based it is able to easily learn useful features from context , if needed . 3 LEARNED HISTORY EMBEDDINGS . 3.1 NETWORK GRAPH . The most intuitive explanation of our approach is from a network graph perspective . Consider a graph where each node n = ( A , B , g ) represents a game between players A and B along with a node attribute , g. We consider the simple case where g = { r , t } where r is a scalar indicating the point differential between A and B and t is the time of the game . There is a directed edge between two nodes n1 and n2 defined by n1 = ( A1 , B1 , g1 ) and n2 = ( A2 , B2 , g2 ) if { A1 , B1 } ∩ { A2 , B2 } 6= ∅1 and t1 > t2 . Thus edges in this graph connect nodes that share one player , and neighbors of a node n are defined to go backward in time . The architecture for our approach relies on Transformer ( Vaswani et al. , 2017 ) layers that use attention to learn embeddings of players , operating on an increasingly large receptive field in this network graph . The `` level 1 '' version of our method will operate on neighbors that are path length one away from the proposed match-up ; the `` level 2 '' version considers all nodes of path length less than two from proposed match-up , and so on . Note that our model acts only on the node features g = ( r , t ) so it has no knowledge of explicit player ID in the match-up , thus it is acting only on the histories of the players . This allows for greater flexibility and forces the model to learn richer representations compared to prior work . 3.2 LEVEL 1 HISTORY . We wish to predict the outcome of a proposed match occurring between players A and B , which will result in either a win or loss ( draws are discussed later ) . Our model makes this prediction using the T previous match-ups of each player ZA : = { gA1 , ... , gAT } and ZB : = { gB1 , ... , gBT } respectively . These histories are considered level 1 since they correspond to the nodes in the network graph that are depth one away from each player . When computing the probability that A beats B we must respect the underlying symmetry that p ( A beats B ) = 1 − p ( B beats A ) . To do this we stack the inputs as Z1 = ( [ ZA ZB ] ) and Z2 = ( [ ZB ZA ] ) and run them through a multi-layer Transformer ζ such that o ( A , B ) : = ζ ( Z1 ) − ζ ( Z2 ) . The resulting logit o ( A , B ) has the desired property o ( A , B ) + o ( B , A ) = 0 and is trained with cross entropy loss . This level 1 model is only able to learn fairly simple weighted averages over the history , or generic comparisons of the form `` team A is doing better than team B in its past x games '' . 3.3 LEVEL 2 HISTORY . The model becomes more powerful once we consider all nodes in the graph that are at most path length two away from the current node . These nodes correspond to `` histories of history '' , where all previous games in the proposed match-up history include the history of each opponent involved . Consider two players A and B with T most recent match histories ZA = { gA1 , ... , gAT } and ZB = { gB1 , ... , gBT } respectively . We now also use the ith opponent ’ s own history , looking at their T previous match-ups . These are represented by features hij , from the match-up with the jth opponent in player i ’ s history , i being an opponent in A ’ s history . We learn an embedding for each opponent i in player A ’ s history by passing the features hij , j ∈ 1 .. T through a multi-layer Transformer and an MLP . This step summarizes the history of nodes at depth 2 into an embedding vector ĥi . The level 1 history of playerA is sumarized using the same Transformer + MLP to produce ĝ . The level 1 and 2 history embeddings are combined to produce a summary embedding ZA for player A . The equivalent summary for player B is computed in the same fashion using the same set of attention + MLP layers . Figure 1 shows this two level hierarchy . Formally , for a given player , we have the level 1 history X ∈ RT×k , each row of which corresponds to gi , k being the number of features per match-up ( k = 2 for the case g = r , t ) . We also have level 2 history Y ∈ RT×T×k , which is a tensor that holds hij . Both X and Y are processed with 1i.e . A1 = A2 or A1 = B2 or B1 = A2 or B1 = B2 . the same multi-layer Transformer φ ( . ) : RT×k → RT×d , d being the output embedding dimension ( a hyper-parameter ) . The result is then passed to an MLP which ψ ( . ) : RT×d → Re summarizes the embeddings . Thus , the level 1 and 2 histories are summarized into e dimensional embeddings ĝ = ψ ( φ ( X ) ) and ĥi = ψ ( φ ( Y ) ) , respectively . These are are combined into a matrix Z ( of size 2e× T ) : Z = [ ĝ . . . ĝ ĥ1 . . . ĥT ] Here the ith column corresponds to an embedding representing the ith match-up in a player ’ s history . The first part of the embedding , ĝ , captures the player ’ s own history ( level 1 ) , which is shared for each game in the history , and the second part of the embedding , ĥi , represents the history of opponent in the ith game . Computing Z for each player A and B yields ZA and ZB , which we combine using a multi-layer Transformer ζ ( . ) : R2e×2T → R to produce a scalar output . Optionally , contextual information c about the encounter between A and B can be added at this stage by appending it to ZA and ZB . Using the same procedure as in Section 3.2 to enforce the underlying symmetry , output logit o is defined to be : o = ζ ZAZB c − ζ ZBZA c ( 1 )

Summary: The paper presents an approach to predicting match outcomes between two individuals (teams or players) by comparing embeddings of previous match outcomes of both individuals. Embeddings are generated using a transformer architecture and can be applied recursively to embed the history of opponents of previous matches. Logits are produced by stacking embeddings and training with cross entropy after pushing outputs through an MLP. Experiments compare the method to established pairwise rating algorithms and the results show strong performance on chess, hockey, and baseball datasets. A toy demonstration shows the model can capture intransitive relationships in rock-paper-scissors.