“Survival of Dominated Strategies Under Evolutionary Dynamics,” J. Hofbauer & W. Sandholm (2011)

There is a really interesting tension in a lot of economic rhetoric. On the one hand, we have results that derive from optimal behavior by agents with rational foresight: “price equals marginal cost in competitive markets because of profit-maximizing behavior” or “Policy A improves welfare in a dynamic general equilibrium setting with utility-maximizers”. Alternatively, though, we have explanations that rely on dynamic consequences to even non-maximizing agents: “price equals marginal cost in competitive markets because firms who price about MC are driven out by competition” or “Policy A improves welfare in a dynamic general equilibrium, and the dynamic equilibrium is sensible because firms adjust myopically as if in a tatonnement process.”

These two types of explanation, without further proof, are not necessarily the same. Profit-maximizing firms versus firms disciplined by competition give completely different welfare results under monopoly, since the non profit-maximizing monopolist can be very wasteful and yet still make positive profits. In a dynamic context, firms adjust myopically to excess demand in some markets, rather than profit-maximizing according to rational expectations, will not necessarily converge to equilibrium (a friend mentioned that Lucas made precisely this point in a paper in the 1970s).

How can we square the circle? At least in static games, there has been a lot of work here. Nash and other strategic equilibrium concepts are well known. There is also a branch of game theory going back to the 1950s, evolutionary games, where rather than choosing strategically, a probability vector lists what portion of the players are playing a given strategy at a given time, resulting in some payoffs. A revision rule, perhaps stochastic to allow for “mutations” as in biology, then tells us how the vector of strategies updates conditional on payoffs in the previous round. Fudenberg and Kreps’ learning model from the 1980s is a special case.

Amazingly, it is true for almost all sensible revision rules that the set of rest points of the dynamic includes every Nash equilibrium of the underlying static game, and further that for many revision rules the dynamic rest points are exactly equivalent to the set of Nash equilibria. We have one problem, however: dynamic systems needn’t converge to points at all, but rather may converge to cycles or other outcomes.

Hofbauer and Sandholm – Sandholm being both a graduate of my institution and probably the top economist in the world today on evolutionary games – show that for any revision rule satisfying a handful of common properties, we can construct a game where strictly dominated strategies are played with positive probability. This includes any dynamic meeting the following four properties: the population law of motion is continuous in payoffs and the current population vector, there is positive correlation between strategy growth rates and current payoffs, the dynamic is at rest iff the strategy vector is a Nash equilibrium of the underlying static game, and if an unplayed strategy has sufficiently high rewards, then with positive probability some agents begin using it. These criterion are satisfied by “excess payoff dynamics” like BNN where strategies with higher than average payoffs have higher than average growth rates, and by “pairwise comparison dynamics” where agents switch with positive probability to strategies which have higher payoff than their own current payoff. A myopic best response is not continuous, and indeed, myopic best response has been shown to eliminate strictly dominated strategies.

The proof involves a quite difficult topological construction which I don’t discuss here, but it’s worth discussing the consequence of this result. In strategic situations where we may think agents lack full rationality or rational foresight, and where we observe cycle or other non-rest behavior over time, we should be hesitant to ignore strictly dominated actions (particularly ones that are only dominated by a small amount) in our analysis of the situation. There is also scope for policy improvements: if agents are learning using a dynamic which does not rule out strictly dominated strategies, we may be able to provide information which coerces an alternative dynamic which will rule out such strategies.

Final version in Theoretical Economics 6 (IDEAS version). Another big thumbs up to the journal Theoretical Economics, the Lake Wobegon of econ, where the submissions are free, the turnaround time is fast, and all the articles are totally ungated.

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