Potential game

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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 can be expressed using a single global function called the potential function. The concept was proposed in 1973 by Robert W. Rosenthal.

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 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.

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 payoff for player $i$ given $a_{{-i}}$ .

A simple example

	+1	–1
+1	+b₁+w, +b₂+w	+b₁–w, –b₂–w
–1	–b₁–w, +b₂–w	–b₁+w, –b₂+w
Fig. 1: Potential game 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(s₁, s₂) = b₁ s₁ + b₂ s₂ + w s₁ s₂.

If player 1 moves from −1 to +1, the payoff difference is Δu₁ = u₁(+1, s₂) – u₁(–1, s₂) = 2 b₁ + 2 w s₂.

The change in potential is ΔP = P(+1, s₂) – P(–1, s₂) = (b₁ + b₂ s₂ + w s₂) – (–b₁ + b₂ s₂ – w s₂) = 2 b₁ + 2 w s₂ = Δu₁.

The solution for player 2 is equivalent. Using numerical values b₁ = 2, b₂ = −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	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

Dov Monderer and Lloyd S. Shapley: "Potential Games", Games and Economic Behavior 14, pp. 124–143 (1996).
Emile Aarts and Jan Korst: Simulated Annealing and Boltzmann Machines, John Wiley & Sons (1989) ISBN 0-471-92146-7

External links

Lecture notes of Yishay Mansour about Potential and congestion games

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