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 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.

+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

- = -

Read more about this topic:  Potential Game

Famous quotes containing the word simple:

    “‘Tis the gift to be simple ‘tis the gift to be free
    ‘Tis the gift to come down where you ought to be
    And when we find ourselves in the place just right
    ‘Twill be in the valley of love and delight.”
    —Unknown. ‘Tis the Gift to Be Simple.

    AH. American Hymns Old and New, Vols. I–II. Vol. I, with music; Vol. II, notes on the hymns and biographies of the authors and composers. Albert Christ-Janer, Charles W. Hughes, and Carleton Sprague Smith, eds. (1980)