Formal definition

Consider a normal-form game ${\displaystyle G=\langle N,S_{i},u_{i}\rangle }$, where ${\displaystyle N=\{1,2,\ldots ,n\}}$ is the set of ${\displaystyle n}$ players, ${\displaystyle S_{i}}$ is the (nonempty) strategy set of player ${\displaystyle i}$ where ${\displaystyle i\in N}$, and ${\displaystyle u_{i}}$ is that player's utility function. Denote a strategy profile as ${\displaystyle s=(s_{1},s_{2},\ldots ,s_{n})\in S}$, and denote an incomplete strategy profile ${\displaystyle s_{-i}=(s_{1},s_{2},\ldots ,s_{i-1},s_{i+1},\ldots ,s_{n})}$. A strategy profile ${\displaystyle s^{\ast }\in S}$ is called a Berge equilibrium if, for any player ${\displaystyle i\in N}$ and any ${\displaystyle s_{-i}\in S_{-i}}$, the strategy profile satisfies ${\displaystyle u_{i}(s_{i}^{\ast },s_{-i})\leq u_{i}(s^{\ast })}$.^[7]

Informal definition

The players in a game are playing a Berge equilibrium if they have chosen a strategy profile such that, if any given player ${\displaystyle i}$ sticks with their chosen strategy while some of the other players change their strategies, then player ${\displaystyle i}$'s payoff will not increase. So, every player in a Berge equilibrium guarantees the best possible payoff for every other player who is playing their Berge equilibrium strategy; this is a contrast with Nash equilibria, in which each player ${\displaystyle i}$ is only concerned about maximizing their own payoffs from their strategy, and no other player cares about the payoff obtained by player ${\displaystyle i}$.^[5]

Berge result

A Berge equilibrium of this game is the situation in which both players pick "cooperate", denote it ${\displaystyle (C,C)}$. This is a Berge equilibrium because each player can only lower the other player's payoff by switching their strategy; if either player switched from "cooperate" to "defect", then they would lower the other player's payoff from 20 down to 5, so they must be in a Berge equilibrium.

Berge versus Nash result

Notice first that the Berge equilibrium ${\displaystyle (C,C)}$ is not a Nash equilibrium, because either the row player or the column player could increase their own payoff from 20 to 25 by switching to "defect" instead of "cooperate".

A Nash equilibrium of this prisoner's dilemma game is the situation in which both players pick "defect", denote it ${\displaystyle (D,D)}$. That strategy pair yields a payoff of 10 to the row player and 10 to the column player, and no player has a unilateral incentive to switch their strategy to maximize their own payoff. However, ${\displaystyle (D,D)}$ is not a Berge equilibrium, because the row player could ensure a higher payoff for the column player by switching strategies and giving the column player a payoff of 25 instead of 10, and the column player could do the same for the row player.

The cooperative nature of the Berge equilibrium therefore avoids the mutual defection problem that has made the prisoner's dilemma a notorious example of the potential for Nash equilibrium reasoning to produce a mutually suboptimal result.^[8]