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 ui(si, sj) = bi si + w si sj, where si is players i's strategy, sj 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.
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A 2-player, 2-strategy game cannot be a potential game unless
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