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| In [[game theory]], a game is said to be a '''potential game''' if the incentive of all players to change their [[strategy (game theory)|strategy]] can be expressed using a single global function called the '''potential function'''. The concept was proposed in 1973 by [[Robert W. Rosenthal]].
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| The properties of several types of potential games have since been studied. Games can be either ''ordinal'' or ''cardinal'' potential games. In cardinal games, the difference in individual [[payoff]]s for each player from individually changing one's strategy ''[[ceteris paribus]]'' has to have the same value as the difference in values for the potential function. In ordinal games, only the signs of the differences have to be the same.
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| The potential function is a useful tool to analyze equilibrium properties of games, since the incentives of all players are mapped into one function, and the set of pure [[Nash equilibrium|Nash equilibria]] can be found by locating the local optima of the potential function. Convergence and finite-time convergence of an iterated game towards a Nash equilibrium can also be understood by studying the potential function.
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| ==Definition==
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| We will define some notation required for the definition. Let <math>N</math> be the number of players, <math>A</math> the set of action profiles over the action sets <math>A_{i}</math> of each player and <math>u</math> be the payoff function.
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| A game <math>G=(N,A=A_{1}\times\ldots\times A_{N}, u: A \rightarrow \reals^N) </math> is:
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| * an '''exact potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> such that <math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,
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| ::<math> \Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i}) = u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})</math>
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| ::That is: when player <math>i</math> switches from action <math>a'</math> to action <math>a''</math>, the change in the potential equals the change in the utility of that player.
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| * a '''weighted potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> and a vector <math>w \in \reals_{++}^N</math> such that <math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,
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| ::<math> \Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i}) = w_{i}(u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i}))</math>
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| * an '''ordinal potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> such that <math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,
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| ::<math> u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0 \Leftrightarrow
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| \Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i})>0</math>
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| * a '''generalized ordinal potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> such that <math> \forall {a_{-i}\in A_{-i}},\ \forall {a'_{i},\ a''_{i}\in A_{i}}</math>,
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| ::<math> u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0 \Rightarrow
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| \Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i}) >0 </math>
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| *a '''best-response potential game''' if there is a function <math>\Phi: A \rightarrow \reals</math> such that <math>\forall i\in N,\ \forall {a_{-i}\in A_{-i}}</math>,
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| ::<math>b_i(a_{-i})=\arg\max_{a_i\in A_i} \Phi(a_i,a_{-i})</math>
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| where <math>b_i(a_{-i})</math> is the best payoff for player <math>i</math> given <math>a_{-i}</math>.
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| ==A simple example==
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| {{Payoff matrix | Name = Fig. 1: Potential game example
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| | 2L = +1 | 2R = –1 |
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| 1U = +1 | UL = <small>{{nowrap|+b<sub>1</sub>+w, +b<sub>2</sub>+w}}</small> | UR = <small>{{nowrap|+b<sub>1</sub>–w, –b<sub>2</sub>–w}}</small> |
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| 1D = –1 | DL = <small>{{nowrap|–b<sub>1</sub>–w, +b<sub>2</sub>–w}}</small> | DR = <small>{{nowrap|–b<sub>1</sub>+w, –b<sub>2</sub>+w}}</small> }}
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| In a 2-player, 2-strategy game with externalities, individual players' payoffs are given by the function {{nowrap|u<sub>i</sub>(s<sub>i</sub>, s<sub>j</sub>)}} = {{nowrap|b<sub>i</sub> s<sub>i</sub> + w s<sub>i</sub> s<sub>j</sub>}}, where s<sub>i</sub> is players i's strategy, {{nowrap|s<sub>j</sub>}} is the opponent's strategy, and w is a positive [[externality]] from choosing the same strategy. The strategy choices are +1 and −1, as seen in the [[payoff matrix]] in Figure 1.
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| This game has a potential function {{nowrap|P(s<sub>1</sub>, s<sub>2</sub>)}} = {{nowrap|b<sub>1</sub> s<sub>1</sub> + b<sub>2</sub> s<sub>2</sub> + w s<sub>1</sub> s<sub>2</sub>}}.
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| If player 1 moves from −1 to +1, the payoff difference is Δu<sub>1</sub> = {{nowrap|u<sub>1</sub>(+1, s<sub>2</sub>) – u<sub>1</sub>(–1, s<sub>2</sub>)}} = {{nowrap|2 b<sub>1</sub> + 2 w s<sub>2</sub>}}.
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| The change in potential is ΔP = {{nowrap|P(+1, s<sub>2</sub>) – P(–1, s<sub>2</sub>)}} = {{nowrap|(b<sub>1</sub> + b<sub>2</sub> s<sub>2</sub> + w s<sub>2</sub>) – (–b<sub>1</sub> + b<sub>2</sub> s<sub>2</sub> – w s<sub>2</sub>)}} = {{nowrap|2 b<sub>1</sub> + 2 w s<sub>2</sub>}} = Δu<sub>1</sub>.
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| The solution for player 2 is equivalent. Using numerical values b<sub>1</sub> = 2, b<sub>2</sub> = −1, w = 3, this example transforms into a simple [[battle of the sexes (game theory)|battle of the sexes]], as shown in Figure 2. The game has two pure Nash equilibria, (+1, +1) and (−1, −1). These are also the local maxima of the potential function (Figure 3). The only [[stochastically stable equilibrium]] is (+1, +1), the global maximum of the potential function.
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| <center>
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| {|
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| |-
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| | width=50% |
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| {{Payoff matrix | Name = Fig. 2: Battle of the sexes (payoffs)
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| | 2L = +1 | 2R = –1 |
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| 1U = +1 | UL = 5, 2 | UR = –1, –2 |
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| 1D = –1 | DL = –5, –4 | DR = 1, 4 }}
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| | width=50% |
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| {{Payoff matrix | Name = Fig. 3: Battle of the sexes (potentials)
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| | 2L = +1 | 2R = –1 |
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| 1U = +1 | UL = '''4''' | UR = '''0''' |
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| 1D = –1 | DL = '''–6''' | DR = '''2''' }}
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| |}
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| </center>
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| A 2-player, 2-strategy game cannot be a potential game unless
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| :<math>
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| [u_{1}(+1,-1)+u_1(-1,+1)]-[u_1(+1,+1)+u_1(-1,-1)] =
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| [u_{2}(+1,-1)+u_2(-1,+1)]-[u_2(+1,+1)+u_2(-1,-1)]
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| </math>
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| ==References==
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| * Dov Monderer and Lloyd S. Shapley: "Potential Games", ''Games and Economic Behavior'' 14, pp. 124–143 (1996).
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| * Emile Aarts and Jan Korst: ''Simulated Annealing and Boltzmann Machines'', John Wiley & Sons (1989) ISBN 0-471-92146-7
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| ==External links==
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| * Lecture notes of Yishay Mansour about [http://www.math.tau.ac.il/~mansour/course_games/scribe/lecture6.pdf Potential and congestion games]
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| [[Category:Game theory]]
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