Learning, Mutation and the Economics of Convention
A Selected Review of Evolutionary Game Theory
1. Introduction
The idea of interaction, just as two players play chess, had fundamentally changed the traditional view of optimization after the publication of the masterpiece of Theory of Games and Economic Behavior in 1944 by von Neumann and Morgenstern， although pieces of interaction thoughts can be traced back into a few classical books , such as Researches on the Mathematic Principles of the Theory of Wealth by Cournot (1838). Maximization the utility given the budget constraints is the core in Economics; however, the general framework of analyzing the noncooperative strategic behaviors among different agents with different maximization objects did not appear until Nash (1951). Although the full powerfulness of the concept of Nash Equilibrium (NE) did not discover until the early 1970s, the formulation of the strategic equilibrium concept has replace to a certain extent the traditional Walrasian equilibrium, and become the frame reference of modern microeconomics.
Given all the other players’ strategies, that no one can increase his own payoff by bilateral deviation of his given strategy is the critical intuition which upon based the concept of NE. When the expectation of omnipotence of NE was hold by everyone, however, two difficulties had been haunting every game theorist: how and which NE can be uniquely predicted. Typically, games have many NEs, not only pure NEs, but also mix NEs. Which NEs will be the final outcome and how to justify mix NEs? In addition, how all the players know that one NE will be played?
The first attempt made by almost all the game theorists is focused on various kinds of refinements of the concept of NE and try to make the unique prediction for one game, thus multifarious definitions of equilibrium have appeared which are in fact resulted from different definitions of rationality. As a consequence, almost any NE could be justified in terms of other’s refinement and the available set of refinements of NE became embarrassingly large. Therefore, inspired by the evolutionary thoughts in biology, much attention has shifted to dynamic formation of the equilibrium. In fact, this kind of evolutionary explanation of Nash Equilibrium has been originated in Nash’s unpublished Ph.D. dissertation (1950) in which Nash suggests a population-statistical interpretation of the concept of Nash Equilibrium. He suggested that “some of the players are randomly selected from a large population and play to each other repeatedly, they are assumed to have full knowledge of the total structure of the game, or the abi
lity and inclination to go through any complex reasoning process” (Weibull, 1995).
With relaxation of the traditional super-rationality assumption made on the player, the evolutionary game theorists try to justify NE outside the game itself and base their explanation on other factors outside the game. The pioneer work along this direction was carried out by Schelling (1960) who applied the convention, referred to as foci point in his own term then, to justify the only prediction in coordination game in which there are multi-equilibriums. Although Schelling has successfully applied his foci concept to explain many economic, sociologic and politic issues, the origin of those conventions remained largely still a puzzle. So during the 1980s and 1990s many game theorists had a great deal of attempts to discover the underlying dynamic mechanism of the evolutionary process of convention. Just as in the revolutionary work of the origin of species by Darwin (1882) who contributed the evolution of biological world to natural selection , game theorists has to specify a general dynamic mechanism underlying the social evolutionary process and to explain and predict the uniqueness of convention in some regions (local conformity) or diversity of conventions in other regions (global diversity effect) in social world.
The first motivation of evolutionary game theory was given by Smith and Price (1973) in which work the notation of an evolutionary stable strategy (ESS) had been advanced. In the following works by Taylor and Jonker (1978), Smith (1981), and Fudenberg and Maskin (1990), Nelson and Winter (1982), the importance of evolutionary ideas has been emphasized in explaining economic changes and the formation of evolution. With respect to the concept of ESS, the learning ability which based on the trial-and-error experiments was treated as the key step to explain a convention emerges. While ESS concept has successfully specified the dynamics of the convention evolution, this evolutionary process is largely treated as a deterministic process which contradicts the economic and social reality. During the same period, Harsanyi and Selten (1998) has developed the idea of risk dominance strategy equilibrium which taken the exogenous shock into the game and then derive the stable strategy equilibrium in the static game theory.
The first attempt to combine the idea of evolutionary stable strategy and risk dominant strategy was made by Foster and Young (1990), in which they extended the risk dominance idea to the repeated game situation and allowed a series of continuous exogenous shocks (mutations) to disturb the evolutionary process. Then they derived the so called stochastically table equilibrium (SSE) which is identical to risk dominance equilibrium in the case while different in other cases. Then in the following work by Fudenberg, Drew and Harris (1992), Kandori, Mailath and Rob (1993), the focus of evolutionary game theory has laid on the specification of dynamics of mutation. In contrasted with the generation of exogenous disturbance at the aggregate level in Foster and Young (1990) and Fudenberg, Drew and Harris (1992), in which a Brownian motion reflects aggregate randomness in the population, Kandori, Mailath and Rob (1993) made the generation of disturbance at the individual level and then made the independent individual mutation process. Then in Young (1993), idea of independent individual mutation process has been generalized into the SSE and extended Kandori, Mailath and Rob (1993) discrete process to continuous case and built the basic framework for evolutionary game theory. Therefore, the basic model of this review will be built on Young (1993) .
The remaining parts of this review are organized as follows. Before the present of basic model, one example is given in section 2 to illustrate the economic intuition behind the SSE. The general model, including the three hypotheses, adaptive play setting and basic results is laid out in section 3. Sections 4 study the asymptotic behavior of this model. Section 5 and 6 illustrated the procedure to compute the SSE in 2×2 and3×3 cases. This review concludes with a comment.
