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. Robert W. Rosenthal created the concept of a congestion game in 1973. Monderer and Shapley (1996) created the concept of a potential game and proved that every congestion game is a potential game.

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

• an exact potential game if there is a function $\Phi: A \rightarrow \reals$ 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 $\Phi: A \rightarrow \reals$ and a vector $w \in \reals_{++}^N$ 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 $\Phi: A \rightarrow \reals$ 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 $\Phi: A \rightarrow \reals$ 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 $\Phi: A \rightarrow \reals$ 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 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

$[u_{1}(+1,-1)+u_1(-1,+1)]-[u_1(+1,+1)+u_1(-1,-1)] = [u_{2}(+1,-1)+u_2(-1,+1)]-[u_2(+1,+1)+u_2(-1,-1)]$

Equilibrium Selection

The existence of pure strategy Nash equilibrium is guaranteed in potential games. There may exist multiple Nash equilibria in potential games. Learning algorithms such as best response, better response can only guarantee that the iterative learning process can converge to one of the Nash equilibra (if multiple). Equilibrium selective learning algorithms aim to design a strategy where convergence to the best Nash equilibrium, with respect to the potential function, is guaranteed. In,[1] the authors propose an equilibrium selective algorithm named MaxLogit, which provably converges to the best Nash equilibrium at the fastest speed in its class, using mixing rate analysis of induced Markovian chains. In a special case where every player shares the same objective function (hence the potential function), and possibly the same action set, the problem is equivalent to distributed combinatorial optimization which arises in many engineering applications. Equilibrium selective learning algorithms such as MaxLogit can be used in such combinatorial optimizations even in a distributed fashion.

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