As we worked through our first session on Nash equilibrium Friday in the graduate intro game theory course, we inevitably hit the point of the class becoming rather frustrated at the idea of multiple equilibria. “If we can’t get a deterministic prediction, what’s the point?” Or, more commonly, it’s “There’s got to be some reason to expect one equilibrium over another! Surely, that one is better than the two that also exist.” Then, of course, the inevitable response is that the definition of a Nash equilibrium gives us no traction on that idea.
So what are we to do? Well, if we want to make sharper empirical predictions, then we move to refinements of Nash in a richer game form, one that allows us to generate, at a minimum, fewer equilibria and, ideally, a single one. But that’s only if we want to make precise empirical predictions. We can also learn quite a bit, as it happens, from the outcomes that don’t occur, those that, despite the fact that they meet some standard of normative desirability, cannot be an equilibrium.
The example that came up in class was Chicken, where each player has a choice of turning or going straight, and where the mutually worst outcome is a head-on collision. Should one player go straight while the other turns, the one that stays straight gets its best possible outcome, while the turner gets the next-to-worst outcome. Finally, should both turn, they avoid head-on collisions and get their second-best outcome. Here’s a simple illustration:
If a Nash equilibrium is a set of mutual best responses, such that no player has an incentive to unilaterally deviate from its strategy given the other player’s strategy, then there are two equilibria of this game: (1) 1 turns and 2 doesn’t, or (2) 2 turns and 1 doesn’t. Put simply, if I know the other player will turn, then I’ll go straight, and vice versa. The question of which of these two equilibria will occur in a given play of the game is, obviously, up in the air.
However, the story doesn’t end here. By plenty of standards, students typically point to the turn-turn outcome as somehow socially desirable, because it ensures both players do as well as possible together. Another way of stating it would be that it maximizes the welfare of the least well-off individual, and that jives with plenty of students’ sense of what they’d like to see. But this outcome is, of course, impossible in this setup, because a player would have an incentive to deviate, to refuse to turn, if it knew that the other player would turn. So turn-turn can’t be a Nash equilibrium here, and we can learn a lot from that fact.
In essence, we can use this model to learn something about why social optima (however one might define them) are often difficult to achieve, i.e. why some particular proposed social scheme (here, no one trying to prove another a chicken) is utopian. Given the competitive preferences of the players in Chicken, the “cooperative” outcome in which neither takes advantage of the other simply isn’t self-enforcing, because neither player has an incentive to follow through on it. So while we don’t derive a specific prediction over what will happen here, we do gain powerful insight into why outcomes with some particularly desirable feature won’t happen, and that’s enormously useful, no?
So what makes a model useful? It depends on what we ask of it—predictions or insight—and a social scientist can always do with more of each…