Potential game

In game theory, a game is said to be a potential game if the incentive of all players to change their strategy can be expressed using a single global function called the potential function. The concept originated in a 1996 paper by Dov Monderer and Lloyd Shapley.^[1]

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 payoffs for each player from individually changing one's strategy, other things equal, 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.

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 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.

Definition

We will define some notation required for the definition. Let $N$ be the number of players, $A$ the set of action profiles over the action sets $A_{i}$ of each player and $u$ be the payoff function.

A game $G=(N,A=A_{{1}}\times \ldots \times A_{{N}},u:A\rightarrow \mathbb{R} ^{N})$ is:

an exact potential game if there is a function $\Phi :A\rightarrow \mathbb{R}$ such that $\forall {a_{{-i}}\in A_{{-i}}},\ \forall {a'_{{i}},\ a''_{{i}}\in A_{{i}}}$ ,

\Phi (a'_{{i}},a_{{-i}})-\Phi (a''_{{i}},a_{{-i}})=u_{{i}}(a'_{{i}},a_{{-i}})-u_{{i}}(a''_{{i}},a_{{-i}})

That is: when player

i

switches from action

a'

to action

a''

, the change in the potential equals the change in the utility of that player.

a weighted potential game if there is a function $\Phi :A\rightarrow \mathbb{R}$ and a vector $w\in \mathbb{R} _{{++}}^{N}$ such that $\forall {a_{{-i}}\in A_{{-i}}},\ \forall {a'_{{i}},\ a''_{{i}}\in A_{{i}}}$ ,

\Phi (a'_{{i}},a_{{-i}})-\Phi (a''_{{i}},a_{{-i}})=w_{{i}}(u_{{i}}(a'_{{i}},a_{{-i}})-u_{{i}}(a''_{{i}},a_{{-i}}))

an ordinal potential game if there is a function $\Phi :A\rightarrow \mathbb{R}$ such that $\forall {a_{{-i}}\in A_{{-i}}},\ \forall {a'_{{i}},\ a''_{{i}}\in A_{{i}}}$ ,

u_{{i}}(a'_{{i}},a_{{-i}})-u_{{i}}(a''_{{i}},a_{{-i}})>0\Leftrightarrow \Phi (a'_{{i}},a_{{-i}})-\Phi (a''_{{i}},a_{{-i}})>0

a generalized ordinal potential game if there is a function $\Phi :A\rightarrow \mathbb{R}$ such that $\forall {a_{{-i}}\in A_{{-i}}},\ \forall {a'_{{i}},\ a''_{{i}}\in A_{{i}}}$ ,

u_{{i}}(a'_{{i}},a_{{-i}})-u_{{i}}(a''_{{i}},a_{{-i}})>0\Rightarrow \Phi (a'_{{i}},a_{{-i}})-\Phi (a''_{{i}},a_{{-i}})>0

a best-response potential game if there is a function $\Phi :A\rightarrow \mathbb{R}$ such that $\forall i\in N,\ \forall {a_{{-i}}\in A_{{-i}}}$ ,

b_{i}(a_{{-i}})=\arg \max _{{a_{i}\in A_{i}}}\Phi (a_{i},a_{{-i}})

where $b_{i}(a_{{-i}})$ is the best action for player $i$ given $a_{{-i}}$ .

A simple example

In a 2-player, 2-strategy game with externalities, individual players' payoffs are given by the function $u i (s i, s j) = b i s i + w s i s j$ , where $s i$ is players i's strategy, $s j$ 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.

This game has a potential function $P(s 1, s 2) = b 1 s 1 + b 2 s 2 + w s 1 s 2$ .

If player 1 moves from −1 to +1, the payoff difference is $Δ u 1 = u 1 (+1, s 2) - u 1 (-1, s 2) = 2 b 1 + 2 w s 2$ .

The change in potential is $ΔP = P(+1, s 2) - P(-1, s 2) = (b 1 + b 2 s 2 + w s 2) - (- b 1 + b 2 s 2 - w s 2) = 2 b 1 + 2 w s 2 = Δ u 1$ .

The solution for player 2 is equivalent. Using numerical values $b 1 = 2$ , $b 2 = - 1$ , $w = 3$ , this example transforms into a simple 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.

	+1	–1
+1	$+ b 1 + w, + b 2 + w$	$+ b 1 - w, - b 2 - w$
–1	$- b 1 - w, + b 2 - w$	$- b 1 + w, - b 2 + w$
Fig. 1: Potential game example

	+1	–1
+1	5, 2	–1, –2
–1	–5, –4	1, 4
Fig. 2: Battle of the sexes (payoffs)

	+1	–1
+1	4	0
–1	–6	2
Fig. 3: Battle of the sexes (potentials)

A 2-player, 2-strategy game cannot be a potential game unless

[u_{{1}}(+1,-1)+u_{1}(-1,+1)]-[u_{1}(+1,+1)+u_{1}(-1,-1)]=[u_{{2}}(+1,-1)+u_{2}(-1,+1)]-[u_{2}(+1,+1)+u_{2}(-1,-1)]

References

↑ Monderer, Dov; Shapley, Lloyd (1996). "Potential Games". Games and Economic Behavior. 14: 124–143. doi:10.1006/game.1996.0044.

External links

Lecture notes of Yishay Mansour about Potential and congestion games
Section 19 in: Vazirani, Vijay V.; Nisan, Noam; Roughgarden, Tim; Tardos, Éva (2007). Algorithmic Game Theory (PDF). Cambridge, UK: Cambridge University Press. ISBN 0-521-87282-0.
Non technical exposition by Huw Dixon of the inevitability of collusion Chapter 8, Donut world and the duopoly archipelago,Surfing Economics.

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[MS-1] Monderer, Dov; Shapley, Lloyd (1996). "Potential Games". Games and Economic Behavior. 14: 124–143. doi:10.1006/game.1996.0044.

Topics in game theory
Definitions	Cooperative game Determinacy Escalation of commitment Extensive-form game First-player and second-player win Game complexity Graphical game Hierarchy of beliefs Information set Normal-form game Preference Sequential game Simultaneous game Simultaneous action selection Solved game Succinct game
Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium
Strategies	Dominant strategies Pure strategy Mixed strategy Strategy-stealing argument Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy
Classes of games	Symmetric game Perfect information Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global game Strictly determined game Potential game
Games	Chess Infinite chess Checkers Tic-tac-toe Prisoner's dilemma Optional prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock–paper–scissors Pirate game Dictator game Public goods game Blotto game War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Prisoners and hats puzzle Trust game Princess and Monster game Rendezvous problem
Theorems	Arrow's impossibility theorem Aumann's agreement theorem Folk theorem Minimax theorem Nash's theorem Purification theorem Revelation principle Zermelo's theorem
Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Claude Shannon Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens Jennifer Tour Chayes John Harsanyi John Maynard Smith Antoine Augustin Cournot John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Axelrod Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey
See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition First-move advantage in chess Game mechanics Glossary of game theory List of game theorists List of games in game theory No-win situation Solving chess Topological game Tragedy of the commons Tyranny of small decisions

Potential game

Definition

A simple example

See also

References

External links