Chapter 6: Nash Equilibria

Overview

In a Nash equilibrium, all of the players’ expectations are fulfilled and their chosen strategies are optimal.

From a 1994 press release announcing the Nobel Prize winners in economics.^[1]

John Nash has been one of the most important men in developing game theory. Both the book and the Oscar-winning movie A Beautiful Mind are about his life. Nash developed a method of solving games that is appropriately called a “Nash equilibrium,” a no-regrets outcome in which all the players are satisfied with their strategy given what every other player has done. In a Nash equilibrium you are not necessarily happy with the other players’ strategies; rather your strategy is an optimal response to your opponents’ moves. Players in a Nash equilibrium never cooperate and always assume that they can’t alter their opponent’s actions.

Nash equilibrium: No player regrets his strategy, given everyone else’s move.

Consider a simple game with two employees, Tom and Jim, who both want a raise. Assume that if just one employee asks for a raise, he will get it, but if both ask for a salary increase, then their employer will get mad and fire them both. This game has two Nash equilibria: one where just Tom asks for a raise and one where just Jim asks for one. It can’t be a Nash equilibrium for neither employee to ask for a raise because each person would regret not asking for a raise, knowing that the other didn’t request one. It’s also not a Nash equilibrium for both to request a raise because then each would regret getting fired.

The movie, A Beautiful Mind, carelessly provides a perfect example of what is not a Nash equilibrium. In the movie, four attractive women and one truly stunning babe enter a bar. John Nash explains to three of his male schoolmates how they should go about picking up the girls. Nash says that normally all four of the men would simultaneously hit on the babe. Nash claims, however, that following this strategy would be stupid because if all the men went after the same girl, they would get in each others’ way, so none of them would score. Nash predicts that if the four men turned to the merely attractive women after being rejected by the babe, then the merely attractive women would be angry that they were everyone’s second choice, so they, too, would spurn the men. Nash proposes that to avoid involuntary celibacy, the men should cooperate by ignoring the babe and pursuing the merely attractive women.^[2] While the movie never directly states this, it’s implied that Nash’s proposed mating strategy relates to his Nobel-Prize-winning work in economics and thus to the idea of a Nash equilibrium.

Let’s first focus on the pickup strategy that Nash rejects. The four men certainly shouldn’t all pursue the babe. Obviously, if three other men are already hitting on her, and you know that if you too pursue her, you will fail, then it would be in your interest to go for one of the merely attractive women. It’s consequently not a Nash equilibrium for all four men to go for the same woman, regardless of her sex appeal. Each of the four men would regret his choice of pursuing the babe if the three other men also hit on her. He could have done better following the alternative strategy of pursuing one of the merely attractive women. The outcome John Nash rejects is therefore not a Nash equilibrium.

A Beautiful Mind should be stripped of its Oscars because the outcome that John Nash proposes in the movie is also not a Nash equilibrium. Recall that he suggests that the four men should all ignore the babe. Each of the men, however, would regret a strategy of ignoring the babe if everyone else ignored her too. Sure, it might be reasonable not to pursue the best-looking woman in the bar if many other men are hitting on her. If, however, everyone else ignored this stunning babe, then obviously you (assuming you like women) should go for her.

The bar pickup game does have at least one Nash equilibrium, however. In one possible Nash equilibrium, the first, one man pursues the babe while the others go for lesser prizes. The one man going for the gold would clearly be happy with his strategy because he would have the field to himself. The three other men might also be happy with their choice. If this outcome is a Nash equilibrium, then each of the three men going for the merely attractive women would prefer to have a higher chance with one of them than a lower chance of scoring in a two-man competition for the babe. The only Nash equilibrium, however, might be for two or three of the men to go for the babe while the rest pursue the merely attractive women. This outcome would be a Nash equilibrium if the men pursuing the babe would prefer a lower chance of succeeding with her to a higher probability of making it with one of the other girls.

The power of a Nash equilibrium comes from its stability. Everyone is happy with his move, given what everyone else is doing, so no one wants to alter his strategy. Let’s consider the Nash equilibria in Figure 29.

Figure 29

In this game, Player One choosing A and Player Two picking X is a Nash equilibrium. If Player One chooses A, then Player Two’s best choice is X. Thus, given that Player One’s choice is A, Player Two is happy with X. Similarly, if Player Two chooses X, Player One’s optimal choice is A. Obviously, the players would rather be at B,Y than A,X. This doesn’t prevent A,X, however, from being a Nash equilibrium because at A,X each player’s strategy is an optimal response to his opponent’s move. B,Y is also a Nash equilibrium in this game because each player would get his highest possible score at B,Y and thus would obviously be happy with his strategy. B,X is not a Nash equilibrium because both players would regret their choices. If, for example, Player Two chose X, Player One would regret playing B, because had he played A, he would have gotten a higher payoff.

In Figure 30, B,Y is an obvious Nash equilibrium, but there is another: A,X. If Player One chooses A, Player Two would not regret playing X because he will get zero no matter what. For similar reasons Player One would not regret choosing A in response to Player Two picking X. You don’t regret choosing X if, given your opponent’s move, you can’t possibly do better than by playing X.

Figure 30

The outcomes in Figures 29 and 30 show multiple Nash equilibria can coexist, but one could be superior to the rest. Obviously, players in a bad equilibrium should try to move to a better one.

Figure 31 provides another example of a game with multiple Nash equilibria. In this game both players being nice is clearly a Nash equilibrium. If both players are nice, they each get 10. If one person is nice and the other is mean, the mean player gets only 8. It is a stable outcome for both players to be nice, because if one person were nice, the other would want to be nice too. Unfortunately, it’s also a Nash equilibrium for both players to be mean. When both players are mean, they each get a payoff of zero. If, however, one player is nice and the other is mean, then the nice person loses 5. The optimal response to the other person’s meanness is for you also to be mean. Mean, mean is consequently a stable Nash equilibrium.

Figure 31

If you find yourself in a game like Figure 31 where everyone plays mean, the best way to extricate yourself is to convince your opponent that you should both simultaneously start being nice. Keep in mind, however, that if you can’t convince your opponent to change his strategy, you shouldn’t change yours.

If we slightly alter the payoffs in Figure 31, it becomes impossible to achieve the nice, nice outcome. Consider Figure 32. The only difference between this and the previous game is the enhanced benefit of repaying kindness with cruelty. This change, however, results in nice, nice no longer being a Nash equilibrium. If one player is nice, the other is actually better off being mean, so nice, nice is no longer stable.

Figure 32

Consider a less abstract situation where you and a coworker are both being mean and undercutting each other to your boss. The key question for determining whether your game is like Figure 31 or 32 is this: If one of you is nice, should the other be mean or nice in return? It’s possible that if one person were nice, your boss would disapprove if the other was not civil. Repaying kindness with cruelty, however, might be the ideal way of getting ahead in your firm. If both players benefit by being cruel to those who are kind, then you are stuck in a Nash equilibrium where you should be mean. The type of game where everyone is mean is called a prisoners’ dilemma and is the subject of the next chapter.

^[1]The Sveriges Riksbank (Bank of Sweden) Prize in Economic Sciences in Memory of Alfred Nobel, press release, October 11, 1994.

^[2]In the movie, John Nash rushes out of the bar after formulating his strategy so we never see if it could have been successfully implemented.