## “The Explanatory Relevance of Nash Equilibrium: One-Dimensional Chaos in Boundedly Rational Learning,” E. Wagner (2013)

The top analytic philosophy journals publish a surprising amount of interesting game and decision theory; the present article, by Wagner in the journal Philosophy of Science, caught my eye recently.

Nash equilibria are stable in a static sense, we have long known; no player wishes to deviate given what others do. Nash equilibria also require fairly weak epistemic conditions: if all players are rational and believe the other players will play the actual strategies they play with probability 1, then the set of outcomes is the Nash equilibrium set. A huge amount of work in the 80s and 90s considered whether players would “learn” to play Nash outcomes, and the answer is by and large positive, at least if we expand from Nash equilibria to correlated equilibria: fictitious play (I think what you do depends on the proportion of actions you took in the past) works pretty well, rules that are based on the relative payoffs of various strategies in the past work with certainty, and a type of Bayesian learning given initial beliefs about the strategy paths that might be used generates Nash in the limit, though note the important followup on that paper by Nachbar in Econometrica 2005. (Incidentally, a fellow student pointed out that the Nachbar essay is a great example of how poor citation measures are for theory. The paper has 26 citations on Google Scholar mainly because it helped kill a literature; the number of citations drastically underestimates how well-known the paper is among the theory community.)

A caution, though! It is not the case that every reasonable evolutionary or learning rule leads to an equilibrium outcome. Consider the “continuous time imitative-logic dynamic”. A continuum of agents exist. At some exponential time for each agent, a buzzer rings, at which point they randomly play another agent. The agent imitates the other agent in the future with probability exp(beta*pi(j)), where beta is some positive number and pi(j) is the payoff to the opponent; if imitation doesn’t occur, a new strategy is chosen at random from all available strategies. A paper by Hofbauer and Weibull shows that as beta grows large, this dynamic is approximately a best-response dynamic, where strictly dominated strategies are driven out; as beta grows small, it looks a lot like a replicator dynamic, where imitation depends on the myopic relative fitness of a strategy. A discrete version of the continuous dynamics above can be generated (all agents simultaneously update rather than individually update) which similarly “ranges” from something like the myopic replicator to something like a best response dynamic as beta grows. Note that strictly dominated strategies are not played for any beta in both the continuous and discrete time i-logic dynamics.

Now consider a simple two strategy game with the following payoffs:

Left Right
Left   (1,1) (a,2)
Right (2,a) (1,1)

The unique Nash equilibrium is X=1/A. Let, say, A=3. When beta is very low (say, beta=1), and players are “relatively myopic”, and the initial condition is X=.1, the discrete time i-logic dynamic converges to X=1/A. But if beta gets higher, say beta=5, then players are “more rational” yet the dynamic does not converge or cycle at all: indeed, whether the population plays left or right follows a chaotic system! This property can be generated for many initial points X and A.

The dynamic here doesn’t seem crazy, and making agents “more rational” in a particular sense makes convergence properties worse, not better. And since play is chaotic, a player hoping to infer what the population will play next is required to know the initial conditions with certainty. Nash or correlated equilibria may have some nice dynamic properties for wide classes of reasonable learning rules, but the point that some care is needed concerning what “reasonable learning rules” might look like is well taken.

Final 2013 preprint. Big thumbs up to Wagner for putting all of his papers on his website, a real rarity among philosophers. Actually, a number of his papers look quite interesting: Do cooperate and fair bargaining evolve in tandem? How do small world networks help the evolution of meaning in Lewis-style sender-receiver games? How do cooperative “stag hunt” equilibria evolve when 2-player stag hunts have such terrible evolutionary properties? I think this guy, though a recent philosophy PhD in a standard philosophy department, would be a very good fit in many quite good economic theory programs…