Friday, 11 December 2015

What is the difference between a Nash equilibrium and evolutionary stable strategy?

Nash equilibrium is the main game theoretic concept used in economics. Evolutionary stable strategy, or ESS, is the main game theoretic concept using in biology. The issue I want to look at here is the relationship between the two. This is an issue that appears to cause much confusion amongst students (and some academics).
         The main thing that I would suggest needs to be recognised is that Nash equilibrium and ESS are fundamentally different. They were conceived completely independently with very contrasting objectives in mind. The Nash equilibrium was developed by John Nash in the 1940/50s as an equilibrium concept for non-cooperative games. And note that the genius of Nash was not so much to develop the Nash equilibrium but to recognise the importance of non-cooperative games. The ESS, by contrast, was developed by John Maynard Smith and George Price in the 1970s as an equilibrium concept for evolutionary games. Again, the genius was not so much the ESS but the recognition that evolutionary forces can be modelled using game theory.
        Even though Nash equilibrium and ESS were designed with contrasting objectives in mind they end up looking similar in certain respects. It is here that the confusion starts. So, let us define each concept in turn and then make sense of the similarities.
         The basic idea with a non-cooperative game is that we have set of players who each, independently, choose a strategy. The payoff of each player is then determined by the strategies that each player has chosen. To illustrate consider the hawk-dove game depicted below. In this game, Gary and Dawn have to simultaneously choose whether to play hawk or dove. The numbers give the respective payoffs to all combinations of outcome. For instance, if Gary chooses Dove and Dawn chooses Hawk then Gary gets 0 and Dawn gets 4.

          A Nash equilibrium is a list of strategies for each player such that no player has any incentive to change their strategy (given the strategies of others). In the hawk-dove game above there are three Nash equilibria. (1) If Gary plays Dove then Dawn does best to play Hawk. Similarly, if Dawn plays Hawk then Gary does best to play Dove. So, Gary plays Dove and Dawn plays Hawk is a Nash equilibrium. (2) By a symmetric argument, Gary plays Hawk and Dawn plays Dove is a Nash equilibrium. (3) Suppose that Dawn tosses a coin to decide what to do. If the coin comes down heads she plays Hawk, if tails she plays Dove. Then the expected payoff of Gary if he plays Hawk is (0.5)(-2) + (0.5)(4) = 1. His expected payoff if he plays Dove is (0.5)(0) + (0.5)(2) = 1. So, Gary is indifferent what he chooses. By a similar logic, Dawn is indifferent if Gary tosses a coin to decide what to do. This means there is a Nash equilibrium where Gary and Dawn independently toss a coin and randomize between Hawk and Dove.
          It is worth noting that the first two Nash equilibrium we looked at are asymmetric - Gary chooses a different strategy to Dawn. The third equilibria is symmetric - both Gary and Dawn choose the same strategy, namely to toss a coin and choose Hawk if and only if their coin comes down heads.
         Let us look now at an evolutionary game. The most common formulation (there are many alternatives) goes something like this: Over time, individuals from a large population randomly meet each other and play a game. For instance, individuals might meet in pairs and play the hawk-dove game. Gary and Dawn are now just two of the many people we are interested in. Keeping track of everyone's strategy would be burdensome and so we focus on a population strategy. You can think of the population strategy as detailing the probability that a randomly chosen member of the population will choose Hawk, and the probability they play Dove.
           An evolutionary stable strategy is a population strategy that is immune to invasion. This means that if any small subset of the population deviate from the ESS then they will get a lower payoff than those who did not deviate. To illustrate, consider the population strategy in which everyone plays Dove and gets payoff 2. A mutant who plays Hawk would get a payoff of  4, because he only ever plays against individuals choosing Dove. This beats the population strategy and so everyone play Dove is not an ESS. In fact, the only ESS in the hawk-dove game, given above, is the population strategy in which Hawk is played with probability one half and Dove with probability one half.
           You might have noticed that this ESS is 'similar' to the third Nash equilibrium. Let us think what that means. In the evolutionary game we have a population strategy - Hawk is played with probability one half and Dove with probability one half. In the two player game between Gary and Dawn we have a symmetric Nash equilibrium - Gary and Dawn independently choose Hawk with probability one half and Dove with one half. More generally, if a strategy, call it X, is an ESS of an evolutionary game, where players are matched to play some game G, then the strategy profile where both players play X is a symmetric Nash equilibrium of game G. So, there is a sense in which ESS 'must be' a Nash equilibrium.
          Note that the reciprocal is not true. In the hawk-dove game there are three Nash equilibria and only one of those maps into an ESS. The basic issue here is one of symmetry. A population strategy has to be symmetric, by definition, because it treats all individuals in the population equally. For instance, if some play Hawk and some Dove then we cannot rule out the possibility that two individuals will meet who both play Hawk or both play Dove. The Nash equilibrium where Gary plays Hawk and Dawn plays Dove cannot, therefore, be equated with a population strategy.
           Textbooks and journals are full of results connecting ESS and Nash equilibrium. And from a theoretical point of view it is clearly of interest to study the relationship between the two. On a practical level, though, it is not really clear to me that we can learn much from knowing the two are related. This doubt comes from the fundamentally differences between a non-cooperative and evolutionary game. In the evolutionary hawk-dove game, for instance, forces will push towards the ESS. So, the ESS is the natural thing to study. But in the non-cooperative game between Gary and Dawn the equilibrium where both toss a coin to decide what to does not seem very appealing. It gives low payoffs. And it is hard to imagine this is how Gary and Dawn would think about the game. The two asymmetric equilibria seem more likely to be most relevant.
           My claim, therefore, is that we get too caught up in the similarities between ESS and Nash equilibrium. They are different concepts and there is no harm in thinking of them as such.  

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