Potential Game - A Simple Example

A Simple Example

	+1	–1
+1
–1
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

$- = -$

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