Recently I refereed a paper on the existence of Berge equilibrium. I must confess that until reading the paper I knew nothing of Berge equilibrium. But in my defence, the equilibrium does not get a mention in any game theory textbook on my shelves and, surely most telling of all, does not get an entry in Wikipedia. So, what is Berge equilibrium and should we hear more about it?

The origins of the equilibrium are a book by French mathematician Claude Berge (who does get a Wikipedia page) on a general theory of n-person games, first published in 1957. But it has seemingly gone pretty much unnoticed from then on, although there is a growing literature on the topic as summarized in a 2017

paper by Larbani and Zhukovskii. The basic idea behind Berge equilibrium seems to be one of altruism or cooperation between players in a group.

To explain, consider a game. Let s

i denote the strategy of player i, s

-i the strategies of everyone other than i and u

i(s

i, s

-i) the payoff of player i given these strategies.

**Nash equilibrium** says that player i

*maximizes his or her payoff* given the strategies of others. So, at a strict Nash equilibrium s

i*, s

-i* we have

u

i(s

i*, s

-i*) > u

i(s

i, s

-i*)

for all i and any other strategy s

i. This says that player i cannot do better by deviating.

**Berge equilibrium** says that each player

*maximizes the payoff of player i* given his or her strategy. So, at a strict Berge equilibrium s

i*, s

-i* we have

u

i(s

i*, s

-i*) > u

i(s

i*, s

-i)

for all i and for any other strategy s

-i. So,

**the other players do their best to maximize the payoff of player i**.

The differences between Nash equilibrium and Berge equilibrium are easy illustrated in the prisoners dilemma. In the game depicted below Fred and William simultaneously have to decide whether to deny or confess. Nash equilibrium says that Fred should Confess because this maximizes his payoff (whatever William does). Berge equilibrium, by contrast, says that Fred should Deny because this maximizes the payoff of William.

Many have argued that Deny is the 'rational' choice in the prisoners dilemma (because both Deny is better than both Confess) and Berge equilibrium appears to capture that idea. Modern game theory, however, provides lots of ways to capture altruism or morality that are arguably more appealing. In particular we can add social preferences into the mix so that if Fred wants to help William then we put that into his payoff function. Then the prisoners dilemma (in material payoffs) is no longer a prisoners dilemma (in social preferences) because Fred maximizes his own payoff by Denying and helping William.

Berge equilibrium only makes sense if

*everyone* is willing to

*fully sacrifice* for others, and that seems a long shot. Unless, that is, players have some connections beyond that usually imagined in non-cooperative game theory. In other words Berge equilibrium may have some bite if we move towards the world of cooperative game theory where Fred and William are part of some coalition. We could, for instance, imagine Fred and William being brothers, part of a criminal gang or players on the same sport team. Here it starts to become more plausible to see full sacrifice. And that it brings us to the concept of

**team reasoning**.

The basic idea behind team reasoning is that players think what is best for

*us*. They act as a cohesive unit, like a family making choices. This looks similar to Berge equilibrium but is actually different. To see the difference consider the coordination game below. For both William and Fred to Cheat is a Berge equillibrium - given that Fred is going to cheat the best thing that William can do for Fred is also to cheat. But mutual Cooperation is clearly better (and also a Berge equilibrium). Team reasoning says unambiguously that both should Cooperate. So, team reasoning is arguably better at picking up sacrifice for the group cause.

Given the tools we have to model social preferences and team reasoning I am skeptical Berge equilibrium will ever get beyond the level of an historical curiosity. But it is still interesting to know that such a concept exists.