hcju/proofwikips huggingface dataset:
Definition:Coin/Coin-Tossing
'''Coin-tossing''' is a technique to select randomly one of two options. The usual scenario is to resolve an issue between two parties. The most usual of these is to determine which of the two parties is to choose whether to take the first move, or otherwise determine the starting arrangement, in a game or sport. A coin is thrown into the air in such a manner that it turns over and over. This can be conveniently achieved by coiling the hand into a loose fist, balancing the coin on the first knuckle of the index finger, and then pinging the thumb up from under the index finger by a physical process derived from the science of catastrophe theory. The nail of the thumb will then impact smartly against the edge of the coin which projects over the edge of the index finger's knuckle, propelling it upwards in such a manner that, by an application of the physics of angular momentum, it will be caused to rotate more-or-less rapidly about a diameter. When it descends, the side which will remain uppermost will be dependent upon both the speed of rotation and the distance travelled, neither of which is easy to gauge precisely enough to make the process predictable. The descending coin will either be caught in the hand by the one flipping it (the "tosser"), to be immediately clasped to the back of the other hand, or allowed to fall directly to the ground. The other party (the "caller") is offered the chance of guessing which of the two sides of the coin is on top. The two sides are usually referred to (in the English-speaking world) as "heads" and "tails". The "heads" side tends to be the one that features the head of state of the nation to which the coin belongs, while the "tails" is the other side. Once the caller has made the call, the uppermost side of the coin is revealed. If the side matches that called by the caller, the caller has won, and is given the option of taking the choice as to the starting configuration of the game. Otherwise the tosser has won, and takes the option.<ref>The play ''{{WP|Rosencrantz_and_Guildenstern_Are_Dead|Rosencrantz and Guildenstern are Dead}}'' by {{WP|Tom_Stoppard|Tom Stoppard}} begins with an unfeasibly long sequence of "heads" calls in a game of '''coin-tossing'''.</ref>
Definition:Coin
A '''coin''' is a device for choosing a random number in the set $\left\{ {0, 1}\right\}$. It has been suggested that this device has other uses, but social anthropologists are divided as to what they might be. Perhaps they can resolve their differences by tossing a coin.
Definition:Coin/Head
The '''head''' of a coin is the side which traditionally bears an image of the head of state of the nation to which the coin is issued.
Definition:Coin/Tail
The '''tail''' of a coin is the opposite side to its head. It traditionally bears an image of cultural or historical significance to the nation to which the coin is issued.
Definition:Game/Rules
A game is specified completely by its '''rules'''. They prescribe for each player a model of rational choice: {{:Definition:Model of Rational Choice}}
Definition:Move
A '''move''' is an action by a player in a game made according to the rules.
Definition:Play
A '''play''' is a particular instance of the operation of a game.
Definition:Blotto
'''Blotto''' is a class of game of which an example has the mechanics as follows: There are two players: '''General $A$''' and '''Colonel $B$(lotto)'''. $A$ commands $4$ companies. $B$ commands $5$ companies. $A$ can reach a town on either of two different roads. It is the aim of $A$ to reach a town, which can be achieved by travelling down either of two different roads. He can send each of his $4$ companies on whatever road he chooses. $B$ can order any of his $5$ companies to defend either of the two roads. $A$ wins if, on either road, he has (strictly) more companies than $B$.
Definition:Card Game with Bluffing
There are two players: $A$ and $B$. First it is $A$'s move. $A$ receives one of two possible cards: either the '''high card''' or the '''low card'''. If he receives the '''high card''', he must bid $2$ credits. If he receives the '''low card''', he has two options, either: :$A (1): \quad$ He may pay $1$ credit, and the play is complete. or :$A (2): \quad$ He may bid $2$ credits. If $A$ has bid $2$ credits, it is $B$'s move. $B$ has two options, either: :$B (1): \quad$ He may pay $1$ credit or :$B (2): \quad$ He may challenge $A$'s bid. If $A$ had the '''high card''', $B$ must pay $2$ credits to $A$. If $A$ had the '''low card''', $A$ must pay $2$ credits to $B$. The play is complete.
Definition:Normalization of Game
The operation which reduces the structure of a game to its strategies is known as '''normalizing''' the game.
Definition:Extensive Form of Game
A game is described in its '''extensive form''' as follows: :$(1): \quad$ Each successive move of each player is analysed :$(2): \quad$ The pattern of information is also taken into account.
Definition:Non-Zero-Sum Game/Examples
Examples of non-zero-sum games include: === Trade === Two parties trading: both may simultaneously gain by a transaction. It can be assumed that the buyer attaches a value to goods purchased which is higher than the price paid. At the same time, the price paid to the vendor can also be assumed to be higher than the value, to the vendor, of those goods. Thus, at the end of this transaction, both parties have a positive payoff. === Naval Warfare === The loss of a particular ship may cause a setback of considerably greater value to the enemy than the cost of the materiel used by the attacker to destroy it.
Definition:Zero-Sum Game/Non-Zero
A '''non-zero-sum game''' is a game in which the total of the payoffs to all players is dependent upon the moves made to reach those payoffs.
Definition:Multi-Person Game
A '''multi-person game''' is a game which has (strictly) more than two players.
Definition:Cooperative Game
A '''cooperative game''' is a multi-person game in which the players may form coalitions outside the game.
Definition:Coalition
A '''coalition''' is an agreement among players in a cooperative multi-person game such that they coordinate their strategies so as to confer an advantage to all players in that '''coalition'''.
Definition:Non-Cooperative Game
A '''non-cooperative game''' is a game in which the players cannot form coalitions outside the game. Hence each player is concerned only with his / her own fortunes.
Definition:Payoff Table
Let $G$ be a two-person game. A '''payoff table''' for $G$ is an array which specifies the payoff to each player for each strategy of both players. $G$ is completely defined by its '''payoff table'''. {{PayoffTable|table = $\begin{array} {r {{|}} c {{|}} } & \text{L} & \text{R} \\ \hline \text{T} & w_1, w_2 & x_1, x_2 \\ \hline \text{B} & y_1, y_2 & z_1, z_2 \\ \hline \end{array}$}} The two numbers in the entry formed by row $r$ and column $c$ are the payoffs when the row player's moves is $r$ and the column player's moves is $c$. The first component given is the payoff to the row player. If the names of the players are $1$ and $2$, the convention is that the row player is player $1$ and the column player is player $2$. If the names of the players are $\text{A}$ and $\text{B}$, the convention is that the row player is player $\text{A}$ and the column player is player $\text{B}$.
Definition:Maximising Player
Let $G$ be a two-person game whose players are identified with the symbols $A$ and $B$. It is the convention that the first of the players listed (in this case $A$) is distinguished from the other and referred to as the '''maximising player'''.
Definition:Minimising Player
Let $G$ be a two-person game whose players are identified with the symbols $A$ and $B$. It is the convention that the second of the players listed (in this case $B$) is distinguished from the other and referred to as the '''minimising player'''.
Definition:Distance to Nearest Integer Function/Definition 1
:$\norm \alpha:= \min \set {\size {n - \alpha}: n \in \Z}$
Definition:Distance to Nearest Integer Function/Definition 2
:$\norm \alpha:= \min \set {\set \alpha, 1 - \set \alpha}$ where $\set \alpha$ is the fractional part of $\alpha$.
Definition:Refinement of Cover/Finer Cover
Let $\mathcal V$ be a refinement of $\mathcal U$. Then $\mathcal V$ is '''finer''' than $\mathcal U$.
Definition:Refinement of Cover/Coarser Cover
Let $\mathcal V$ be a refinement of $\mathcal U$. Then $\mathcal U$ is '''coarser''' than $\mathcal V$.
Definition:Maximin Strategy
Let $G$ be a two-person game between players $A$ and $B$. A '''minimax strategy''' for $A$ is one in which: :$(1): \quad A$ assumes that $B$ will make that move which makes $A$'s payoff (or average payoff, in the case of a random element) as small as possible. :$(2): \quad A$ then makes the move which makes this smallest gain the largest possible. Hence it is the strategy that will '''maxi'''mise $A$'s '''min'''imum payoff. The reasoning is: :''I fear that my opponent will make the move which makes my payoff the smallest possible.  Hence I make my move so as to make this smallest payoff as large as possible.''
Definition:Payoff Table/Entry
Each of the values in a '''payoff table''' corresponding to the payoff for a combination of a move by each player is called an '''entry'''.
Definition:Totally Pathwise Disconnected Space/Definition 1
A topological space $T = \struct {S, \tau}$ is '''totally pathwise disconnected'''  all path components of $T$ are singletons.
Definition:Saddle Point (Game Theory)
Let $G$ be a two-player zero-sum game. Let $G$ be defined by a payoff table $T$. Let $E$ be an entry in $T$ such that: :$E$ is the smallest entry in its row :$E$ is the largest entry in its column. Then $E$ is a '''saddle point''' of $G$.
Definition:Specially Strictly Determined Game
Let $G$ be a two-person zero-sum game. Let $G$ have a saddle point. Then $G$ can be referred to as a '''specially strictly determined game'''.
Definition:Eluding Game
'''The eluding game''' is a two-person game whose mechanics are as follows. The two players are $A$ and $B$. $A$ and $B$ each have three possible moves, the values: :$1, 2, 3$ If both choose the same number, $A$ pays to $B$ the amount of the chosen number. Otherwise, $A$ receives the amount of his own number from $B$. === Payoff Table === The payoff table of the '''eluding game''' is as follows: {{:Definition:Eluding Game/Payoff Table}}
Definition:Mixed Strategy
Let $G$ be a game. A '''mixed strategy''' for $G$ is a combination of different strategies for subsequent plays of $G$ which are played with particular frequencies.
Definition:Pure Strategy
Let $G$ be a game. A '''pure strategy''' for $G$ is single strategy for consecutive plays of $G$ which is used exclusively. It can be seen that a '''pure strategy''' is a special case of a mixed strategy for which the frequencies of all strategies are zero except for one, which is used every time.
Definition:Totally Pathwise Disconnected Space/Definition 2
A topological space $T = \struct {S, \tau}$ is '''totally pathwise disconnected'''  the only continuous mappings from the closed unit interval $\closedint 0 1$ to $T$ are constant mappings.
Definition:Solution of Game
Let $G$ be a game. A '''solution''' of $G$ is a systematic description of the outcomes that may emerge in a family of games.
Definition:Optimal Strategy
Let $G$ be a game. Let $P$ be a player of $G$. An '''optimal strategy''' for $P$ is a strategy that contributes towards a solution of $G$.
Definition:Value of Game
Let $G$ be a game. The '''value''' of $G$ is the payoff resulting from a solution of $G$.
Definition:Completely Mixed Game
Let $G$ be a game. Then $G$ is '''completely mixed'''  all players use all their strategies in its optimal solution.
Definition:Dominating Strategy
Let $G$ be a game. Let player $P$ have pure strategies $A_1$ and $A_2$ in $G$. Then $A_1$ '''dominates''' $A_2$ : :for any strategy of an opposing player, $A_1$ is at least as good as $A_2$ :for at least one strategy of an opposing player, $A_1$ is strictly better than $A_2$.
Definition:Strategic Game
A '''strategic game''' is a game in which: :$(1): \quad$ Each player chooses a strategy once per play :$(2): \quad$ The move(s) of each player operates simultaneously.  That is, when choosing a move, each player has any information about the move that another player will make. It consists of: :A finite set $N$ of players :For each player $i \in N$, a non-empty set $A_i$ of moves available to player $i$ :For each player $i \in N$, a preference relation $\succsim_i$ on $A = \displaystyle \prod_{j \mathop \in N} A_j$ of player $i$.
Definition:Extensive Game
An '''extensive game''' is a game in which: :$(1): \quad$ moves are made according to an order of events :$(2): \quad$ A player can consider and reconsider a course of action at any time a decision needs to be made, throughout the play of a game.
Definition:Game of Imperfect Information
A '''game of imperfect information''' is a game in which some information about other players is not known by all the players.
Definition:Rational Player
The players of a game are generally assumed to be '''rational''' in the sense that: :$(1): \quad$ Each player is fully aware of all possible choices that can be made :$(2): \quad$ Each player may form expectations about any unknowns :$(3): \quad$ Each player deliberately chooses an action based on a process of optimization.
Definition:Model of Rational Choice
:$(1): \quad$ A set $A$ of moves from which the player may choose :$(2): \quad$ A set $C$ of consequences of each of those moves :$(3): \quad$ A consequence function $g: A \to C$ which maps a consequence to each action :$(4): \quad$ A preference relation $\succsim$, which is a total ordering on $C$.
Definition:Consequence
A '''consequence''' is a state in a game which results from a move made by a player in that game made according to the rules.
Definition:Consequence Function
Let $G$ be a game. Let $P$ be a player of $G$. Let $A$ be the set of moves available to $P$. Let $C$ be the set of consequences of those moves. A '''consequence function''' for $P$ is a mapping from the set $A$ to the set $C$: :$g: A \to C$
Definition:Preference Relation
Let $G$ be a game. Let $P$ be a player of $G$. Let $C$ be the set $C$ of consequences of the moves available to $P$. A '''preference relation''' on $P$ is a total preordering $\succsim$ on $C$ which ranks $C$ according to $P$'s eventual payoff.
Definition:Utility Function
Let $G$ be a game. Let $P$ be a player of $G$. Let $C$ be the set $C$ of consequences of the moves available to $P$. A '''utility function''' on $C$ is a mapping from $C$ to the real numbers $\R$ so as to define a preference relation on $C$: :$U: C \to \R$ by the condition: :$\forall x, y \in C: x \succsim y \iff U \left({x}\right) \ge U \left({y}\right)$
Definition:Rational Decision-Maker
Let $G$ be a game. Let $P$ be a player of $G$. Let $A$ be the set of moves available to $P$. Let $C$ be the set $C$ of consequences of the moves available to $P$. Let $g: A \to C$ be the consequence function on $A$. Let $B \subseteq A$ be a the set of moves which are feasible. $P$ is a '''rational decision-maker'''  $P$ chooses an element $a^* \in B$ such that: :$\forall a \in a^*: g \left({a^*}\right) \succsim g \left({a}\right)$ where $\succsim$ is the preference relation on $C$. That is, that $P$ solves the problem: :$\displaystyle \max_{a \mathop \in B} U \left({g \left({a}\right)}\right)$ where $U$ is the utility function on $C$. It is assumed that $P$ uses the same preference relation for all $B \subseteq A$.
Definition:Steady State Interpretation
Let $G$ be a game. The '''steady state interpretation''' of the solution of $G$ treats $G$ as a model whose purpose is to explain some sort of regularity in a collection of similar situations.
Definition:Deductive Interpretation
Let $G$ be a game. The '''deductive interpretation''' of the solution of $G$ treats $G$ in isolation, as a singular event, and, under the assumption that the players are rational, deduces the outcome of $G$ based on the rational model.
Definition:Positive Real Vector Space
Let $\R_{\ge 0}$ be the set of positive real numbers. Then the $\R_{\ge 0}$-module $\R_{\ge 0}^n$ is called the '''positive real ($n$-dimensional) vector space'''.
Definition:Absolute Real Vector Ordering
Let $x$ and $y$ be elements of the real vector space $\R^n$. The '''absolute real vector ordering''' is the partial ordering $\ge$ defined on the real vector space $\R^n$ as: :$\forall x, y \in \R^n: x \ge y \iff \forall i \in \left\{ {1, 2, \ldots, n}\right\}: x_i \ge y_i$
Definition:Absolute Real Vector Strict Ordering
Let $x$ and $y$ be elements of the real vector space $\R^n$. The '''absolute real vector strict ordering''' is the strict partial ordering $\ge$ defined on the real vector space $\R^n$ as: :$\forall x, y \in \R^n: x \ge y \iff \forall i \in \left\{ {1, 2, \ldots, n}\right\}: x_i \ge y_i$
Definition:Maximizers of Real Function
Let $f: X \to \R$ be a real-valued function. The '''set of maximisers''' of a $f$ are denoted: :$\displaystyle \arg \max_{x \mathop \in X} f \left({x}\right)$ {{definition wanted|The notation is defined in my source work, but not the concept.}}
Definition:Profile
Let $G$ be a game. Let $N$ be the set of players of $G$. Let $V$ be a variable defining some aspect of each of the players. The family of values of $V$ is referred to as a '''profile''', denoted: :$\family {x_i}_{i \mathop \in N}$ If the fact that $i \in N$ is understood, then $\family {x_i}$ can be used.
Definition:Profile/Less Player
Let $x = \left\langle{x_i}\right\rangle_{i \mathop \in N}$ be a profile of $V$. Then $x_{-i}$ is used to denote $x$ for all players except for $i$: :$x_{-i} := \left\langle{x_j}\right\rangle_{j \mathop \in N \setminus \left\{ {i}\right\} }$
Definition:Continuous Total Preordering
Let $S$ be a set. Let $\precsim$ be a total preordering on $S$. Let $\precsim$ be such that: :$a \precsim b$ whenever there exist sequences $\left\langle{a^k}\right\rangle_k$ and $\left\langle{b^k}\right\rangle_k$ that converge to $a$ and $b$ respectively for which $a^k \precsim b^k$ for all $k$. Then $\precsim$ is '''continuous'''. {{explain|The above has been rendered more-or-less verbatim from the source cited, but it is not well explained. Its context appears to be important in the discipline of game theory.}}
Definition:Quasi-Concave Total Preordering
Let $R^n$ be a real vector space. Let $\precsim$ be a total preordering on $\R^n$. Let $\precsim$ be such that: :for all $b \in \R^n$, the set $\left\{ {a \in \R^n: b \precsim a}\right\}$ is convex. Then $\precsim$ is '''quasi-concave'''.
Definition:Strictly Quasi-Concave Total Preordering
Let $R^n$ be a real vector space. Let $\precsim$ be a total preordering on $\R^n$. Let $\precsim$ be such that: :for all $b \in \R^n$, the set $\left\{ {a \in \R^n: b \precsim a}\right\}$ is strictly convex. Then $\precsim$ is '''strictly quasi-concave'''.
Definition:Pareto Efficiency
Let $N$ be a finite set. Let $X \subseteq \R^N$ be a set. {{explain|what does $\R^N$ mean in this context? This definition has been rendered verbatim from the source work and needs amplification.}} Then $x \in X$ is '''Pareto efficient'''  there exists no $y \in X$ for which $x_i < y_i$ for all $i \in N$. {{NamedforDef|Vilfredo Federico Damaso Pareto|cat = Pareto}}
Definition:Strong Pareto Efficiency
Let $N$ be a finite set. Let $X \subseteq \R^N$ be a set. {{explain|what does $\R^N$ mean in this context? This definition has been rendered verbatim from the source work and needs amplification.}} Then $x \in X$ is '''strongly Pareto efficient'''  there exists no $y \in X$ for which $x_i \le y_i$ for all $i \in N$ and for which $x_i < y_i$ for at least one $i \in N$. {{NamedforDef|Vilfredo Federico Damaso Pareto|cat = Pareto}}
Definition:Limit of Vector-Valued Function/Definition 1
Let: :$\mathbf r: t \mapsto \begin{bmatrix} f_1\left({t}\right) \\ f_2\left({t}\right) \\ \vdots \\ f_n\left({t}\right) \end{bmatrix}$ be a vector-valued function. The '''limit''' of $\mathbf r$ as $t$ approaches $c$ is defined as follows: {{begin-eqn}} {{eqn | l = \lim_{t \to c} \ \mathbf r \left({t}\right)       | o = :=       | r = \begin{bmatrix} \lim_{t \to c} \ f_1\left({t}\right) \\ \lim_{t \to c} \ f_2\left({t}\right) \\ \vdots \\ \lim_{t \to c} \ f_n\left({t}\right)\end{bmatrix} }} {{end-eqn}} where each $\lim$ on the RHS is a limit of a real function. The limit is defined to exist precisely when all the respective limits of the component functions exist.
Definition:Finite Game
A '''finite game''' is a game in which the set $A_i$ of moves available to player $i$ is finite for each player $i$.
Definition:Profile of Preference Relations
Let $G$ be a game. Let $P$ be a player of $G$. Let $C$ be the set $C$ of consequences of the moves available to $P$. The '''profile $\left\langle{\succsim_i^*}\right\rangle$ of preference relations over $C$''' is defined as: :$\forall a, b \in A: \left({a \succsim_i b}\right) \iff \left({g \left({a}\right) \succsim_i^* g \left({b}\right)}\right)$
Definition:Consequence Function with Probability
Let $G$ be a game. Let $P$ be a player of $G$. Let $A$ be the set of moves available to $P$. Let $C$ be the set of consequences of those moves. Let the consequences of those moves be affected by a random variable on a probability space $\Omega$ whose realization is not known to the players before they make their moves. A '''consequence function''' for $P$ is a mapping from $A \times \Omega$ to $C$: :$g: A \times \Omega \to C$ interpreted as that $g \left({a, \omega}\right)$ is the consequence when the move is $a \in A$ and the realization $\omega$ of the random variable is $\omega \in \Omega$.
Definition:Lottery
A '''lottery''' is a game in which the consequence of each move is determined by the realization of a random variable. Let $X$ denote the set of prizes which a player may receive. Let $\Omega$ denote the set of possible states. A '''lottery''' is a mapping $f: X \times \Omega \to \R_{\ge 0}$ such that: :$\forall t \in \Omega: \displaystyle \sum_{x \mathop \in X} f \left({x, t}\right) = 1$ === Probability Model === {{:Definition:Lottery/Probability Model}} === State-Variable Model === {{:Definition:Lottery/State-Variable Model}}
Definition:Lottery Induced by Preference Relation
Let $G$ be a game. Let $N$ be the set of players of $G$. Let $A$ be the set of moves available to player $i \in N$. Let $C$ be the set of consequences of those moves. Let the consequences of those moves be affected by a random variable on a probability space $\Omega$ whose realization is not known to the players before they make their moves. Let $g: A \times \Omega \to C$ be the consequence function for player $i$. Then the '''lottery on $C$ induced by the profile of preference relations over $C$''' is defined by: :$\forall a, b \in A: \left({a \succsim_i b}\right) \iff \left({g \left({a, \omega_a}\right) \succsim_i^* g \left({b, \omega_b}\right)}\right)$ where $\succsim_i$ is the preference relation for player $i$.
Definition:Payoff Function
Let $G$ be a game. Let $N$ be the set of players of $G$. Let $A$ be the set of moves available to player $i \in N$. A '''payoff function''' on $A$ is a mapping $u_i$ from $A$ to the real numbers $\R$: :$u_i: A \to \R$ defined by the condition: :$\forall a, b \in A: a \succsim_i b \iff u_i \left({a}\right) \ge u_i \left({b}\right)$ where $\succsim_i$ denotes the preference relation for player $i$.
Definition:Payoff Table/Zero-Sum
Let $G$ be a two-person zero-sum game. A '''payoff table''' for $G$ is an array which specifies the payoff to (conventionally) the maximising player for each strategy of both players. As $G$ is zero-sum, there is no need to specify the payoff to the minimising player, as it will be the negative of the payoff to the maximising player. {{PayoffTable|table = $\begin{array} {r {{|}} c {{|}} } & \text{L} & \text{R} \\ \hline \text{T} & w & x \\ \hline \text{B} & y & z \\ \hline \end{array}$}}
Definition:Consecutive Strategic Games
The interpretation of a strategic game $G$ can be that a particular play of $G$ is one of a sequence of such plays. In such a case, it is assumed that a player can form an expectation of the behaviour of other players through experience of previous plays.