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| | The author is known by the title of Figures Lint. I am a meter reader. To gather badges is what her family and her appreciate. For a while I've been in South Dakota and my mothers and fathers reside nearby.<br><br>Here is my web site [http://ece.modares.ac.ir/mnl/?q=node/770683 home std test] |
| [[Game theory]] is the branch of [[mathematics]] in which [[game (mathematics)|games]] are studied: that is, models describing human behaviour. This is a glossary of some terms of the subject.
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| ==Definitions of a game==
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| === Notational conventions ===
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| ; [[Real numbers]] : <math> \mathbb{R} </math>.
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| ; The set of '''players''' : <math> \mathrm{N} </math>.
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| ; Strategy space : <math> \Sigma\ = \prod_{i \in \mathrm{N}} \Sigma\ ^i </math>, where
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| ; Player i's strategy space : <math> \Sigma\ ^i </math> is the space of all possible ways in which player '''i''' can play the game.
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| ; A strategy for player '''i''' :
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| <math> \sigma\ _i </math>
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| is an element of
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| <math> \Sigma\ ^i </math>.
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| ; Complements :
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| <math> \sigma\ _{-i} </math>
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| an element of <math> \Sigma\ ^{-i} = \prod_{ j \in \mathrm{N}, j \ne i} \Sigma\ ^j </math>, is a tuple of strategies for all players other than '''i'''.
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| ; Outcome space : <math> \Gamma\ </math> is in most textbooks identical to -
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| ; Payoffs : <math> \mathbb{R} ^ \mathrm{N} </math>, describing how much gain (money, pleasure, etc.) the players are allocated by the end of the game.
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| ===Normal form game===
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| A game in normal form is a function:
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| :<math> \pi\ : \prod_{i\in \mathrm{N}} \Sigma\ ^ i \to \mathbb{R}^\mathrm{N}</math>
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| Given the ''tuple'' of ''strategies'' chosen by the players, one is given an allocation of ''payments'' (given as real numbers).
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| A further generalization can be achieved by splitting the '''game''' into a composition of two functions:
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| :<math> \pi\ : \prod_{i \in \mathrm{N}} \Sigma\ ^i \to \Gamma\ </math>
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| the '''outcome function''' of the game (some authors call this function "the game form"), and:
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| :<math> \nu\ : \Gamma\ \to \mathbb{R}^\mathrm{N} </math>
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| the allocation of '''payoffs''' (or '''preferences''') to players, for each outcome of the game.
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| ===Extensive form game===
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| This is given by a [[tree (graph theory)|tree]], where at each [[vertex (graph theory)|vertex]] of the ''tree'' a different player has the choice of choosing an [[graph theory|edge]]. The ''outcome'' set of an extensive form game is usually the set of tree leaves.
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| ===Cooperative game===
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| A game in which players are allowed form coalitions (and to enforce coalitionary discipline). A cooperative game is given by stating a ''value'' for every coalition:
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| :<math> \nu\ : 2^{\mathbb{P}(N)} \to \mathbb{R}</math>
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| It is always assumed that the empty coalition gains nil. ''Solution concepts'' for cooperative games
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| usually assume that the players are forming the ''grand coalition'' <math> N </math>, whose value <math> \nu(N) </math> is then divided among the players to give an allocation.
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| ===Simple game===
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|
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| A Simple game is a simplified form of a cooperative game, where the possible gain is assumed to be either '0' or '1'. A simple game is couple ('''N''', '''W'''), where '''W''' is the list of "winning" '''coalitions''', capable of gaining the loot ('1'), and '''N''' is the set of players.
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| ==Glossary==
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| ; Acceptable game : is a '''game form''' such that for every possible '''preference profiles''', the game has '''pure nash equilibria''', all of which are '''pareto efficient'''.
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| ; Allocation of goods : is a function <math> \nu\ : \Gamma\ \to \mathbb{R} ^\mathrm{N} </math>. The allocation is a '''cardinal''' approach for determining the good (e.g. money) the players are granted under the different outcomes of the game.
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| ; Best reply : the best reply to a given complement <math> \sigma\ _{-i} </math> is a strategy <math> \tau\ _i </math> that maximizes player '''i''''s payment. Formally, we want: <br> <math>
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| \forall \sigma\ _i \in\ \Sigma\ ^i \quad \quad
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| \pi\ (\sigma\ _i ,\sigma\ _{-i} ) \le \pi\ (\tau\ _i ,\sigma\ _{-i} )
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| </math>.
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| ; Coalition : is any subset of the set of players: <math> \mathrm{S} \subseteq \mathrm{N} </math>.
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| ; Condorcet winner : Given a '''preference''' ''ν'' on the '''outcome space''', an outcome '''a''' is a condorcet winner if all non-dummy players prefer '''a''' to all other outcomes.
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| ; Dictator: A player is a ''strong dictator'' if he can guarantee any outcome regardless of the other players. <math>m \in \mathbb{N}</math> is a ''weak dictator'' if he can guarantee any outcome, but his strategies for doing so might depend on the complement strategy vector. Naturally, every strong dictator is a weak dictator. Formally: <br> ''m'' is a ''Strong dictator'' if: <br> <math> \forall a \in \mathrm{A}, \; \exist \sigma\ _n \in \Sigma\ ^n \; s.t. \; \forall \sigma\ _{-n} \in \Sigma\ ^{-n}: \; \Gamma\ (\sigma\ _{-n},\sigma\ _n) = a </math> <br> ''m'' is a ''Weak dictator'' if: <br> <math> \forall a \in \mathrm{A}, \; \forall \sigma\ _{-n} \in \Sigma\ ^{-n} \; \exist \sigma\ _n \in \Sigma\ ^n \; s.t. \; \Gamma\ (\sigma\ _{-n},\sigma\ _n) = a </math>
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| Another way to put it is: <br> a ''weak dictator'' is <math>\alpha</math>-effective for every possible outcome. <br> A ''strong dictator'' is <math>\beta</math>-effective for every possible outcome. <br> A game can have no more than one ''strong dictator''. Some games have multiple ''weak dictators'' (in ''rock-paper-scissors'' both players are ''weak dictators'' but none is a ''strong dictator''). <br> See ''Effectiveness''. Antonym: ''dummy''.
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| ; Dominated outcome : Given a '''preference''' ''ν'' on the '''outcome space''', we say that an outcome '''a''' is dominated by outcome '''b''' (hence, '''b''' is the ''dominant'' strategy) if it is preferred by all players. If, in addition, some player strictly prefers '''b''' over '''a''', then we say that '''a''' is '''strictly dominated'''. Formally: <br> <math>
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| \forall j \in \mathrm{N} \; \quad \nu\ _j (a) \le\ \nu\ _j (b)
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| </math> for domination, and <br> <math>
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| \exists i \in \mathrm{N} \; s.t. \; \nu\ _i (a) < \nu\ _i (b)
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| </math> for strict domination. <br> An outcome '''a''' is (strictly) '''dominated''' if it is (strictly) '''dominated''' by some other '''outcome'''. <br> An outcome '''a''' is dominated for a '''coalition''' '''S''' if all players in '''S''' prefer some other outcome to '''a'''. See also '''Condorcet winner'''.
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| ; Dominated strategy : we say that strategy is (strongly) dominated by strategy <math> \tau\ _i </math> if for any complement strategies tuple <math> \sigma\ _{-i} </math>, player ''i'' benefits by playing <math> \tau\ _i </math>. Formally speaking: <br> <math>
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| \forall \sigma\ _{-i} \in\ \Sigma\ ^{-i} \quad \quad
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| \pi\ (\sigma\ _i ,\sigma\ _{-i} ) \le \pi\ (\tau\ _i ,\sigma\ _{-i} )
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| </math> and <br> <math>
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| \exists \sigma\ _{-i} \in\ \Sigma\ ^{-i} \quad s.t. \quad
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| \pi\ (\sigma\ _i ,\sigma\ _{-i} ) < \pi\ (\tau\ _i ,\sigma\ _{-i} )
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| </math>. <br> A strategy '''σ''' is (strictly) '''dominated''' if it is (strictly) '''dominated''' by some other '''strategy'''.
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| ; Dummy : A player '''i''' is a dummy if he has no effect on the outcome of the game. I.e. if the outcome of the game is insensitive to player '''i''''s strategy.
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| Antonyms: ''say'', ''veto'', ''dictator''.
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| ; Effectiveness : A coalition (or a single player) '''S''' is ''effective for '' '''a''' if it can force '''a''' to be the outcome of the game. '''S''' is α-effective if the members of '''S''' have strategies s.t. no matter what the complement of '''S''' does, the outcome will be '''a'''.
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| '''S''' is β-effective if for any strategies of the complement of '''S''', the members of '''S''' can answer with strategies that ensure outcome '''a'''.
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| ; Finite game : is a game with finitely many players, each of which has a finite set of '''strategies'''.
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| ; Grand coalition : refers to the coalition containing all players. In cooperative games it is often assumed that the grand coalition forms and the purpose of the game is to find stable imputations.
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| ; Mixed strategy : for player '''i''' is a probability distribution '''P''' on <math> \Sigma\ ^i </math>. It is understood that player '''i''' chooses a strategy randomly according to '''P'''.
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| ; Mixed Nash Equilibrium : Same as '''Pure Nash Equilibrium''', defined on the space of '''mixed strategies'''. Every finite game has '''Mixed Nash Equilibria'''.
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| ; Pareto efficiency : An '''outcome''' ''a'' of '''game form''' ''π'' is (strongly) '''pareto efficient''' if it is '''undominated''' under all '''preference profiles'''.
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| ; Preference profile : is a function <math> \nu\ : \Gamma\ \to \mathbb{R} ^\mathrm{N} </math>. This is the '''ordinal''' approach at describing the outcome of the game. The preference describes how 'pleased' the players are with the possible outcomes of the game. See '''allocation of goods'''.
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| ; Pure Nash Equilibrium : An element <math> \sigma\ = (\sigma\ _i) _ {i \in \mathrm{N}} </math> of the strategy space of a game is a ''pure nash equilibrium point'' if no player '''i''' can benefit by deviating from his strategy <math> \sigma\ _i </math>, given that the other players are playing in <math> \sigma\ </math>. Formally: <br> <math>
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| \forall i \in \mathrm{N} \quad \forall \tau\ _i \in\ \Sigma\ ^i \quad
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| \pi\ (\tau\ ,\sigma\ _{-i} ) \le \pi\ (\sigma\ )
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| </math>. <br> No equilibrium point is dominated.
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| ; Say : A player '''i''' has a '''Say''' if he is not a ''Dummy'', i.e. if there is some tuple of complement strategies s.t. π (σ_i) is not a constant function.
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| Antonym: ''Dummy''.
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| ; Value : A '''value''' of a game is a rationally expected '''outcome'''. There are more than a few definitions of '''value''', describing different methods of obtaining a solution to the game.
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| ; Veto : A veto denotes the ability (or right) of some player to prevent a specific alternative from being the outcome of the game. A player who has that ability is called '''a veto player'''.
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| Antonym: ''Dummy''.
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| ; Weakly acceptable game : is a game that has '''pure nash equilibria''' some of which are '''pareto efficient'''.
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| ; [[Zero sum]] game : is a game in which the allocation is constant over different '''outcomes'''. Formally: <br> <math>
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| \forall \gamma\ \in \Gamma\ \sum_{i \in \mathrm{N}} \nu\ _i (\gamma\ ) = const. </math> <br> w.l.g. we can assume that constant to be zero. In a zero sum game, one player's gain is another player's loss. Most classical board games (e.g. [[chess]], [[checkers]]) are '''zero sum'''.
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| {{Game theory}}
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| {{DEFAULTSORT:Glossary Of Game Theory}}
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| [[Category:Game theory| ]]
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| [[Category:Glossaries of mathematics|Game theory]]
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