Advantages of Evolutionary Game Theory Applications

Field of Economics Neoclassical economics is based on the theory of atomism and mechanics, which assumes that the participants are completely rational and consistent in their preferences. Participants can get an optimal solution under the given conditions, for example, producers can find a production solution to get the maximum profit under the condition of certain technology and resources, and consumers can get a consumption solution to get the maximum utility under the condition of given budget, and so on. Game theory adds the interaction between actors on the basis of neoclassical economics, which makes the theory closer to reality, but in general, game theory still has not jumped out of the framework of neoclassical economics. As a result, when using game theory to build models, the assumptions made about various relationships are often unrealistic, and therefore, the decisions made based on such models are often far from reality, which can easily lead to mistakes.

Evolutionary game theory abandons the assumption of complete rationality, and takes Darwin's theory of biological evolution and Lamarck's theory of genetic inheritance as the ideological basis, and from the system theory, regards the adjustment process of group behavior as a dynamic system, in which the behavior of each individual and its relationship with the group are individually portrayed. Game theory assumes that the actors have perfect rational thinking, that is, the actors always aim at their own maximum interests, have the judgment and decision-making ability of pursuing their own interests maximization in various environments, have the perfect judgment and prediction ability in the game environment where interaction exists, and will not make mistakes, will not be impulsive, and have no irrationality. In addition, one of the most important assumptions in game theory is the assumption of "*** the same knowledge" of the two actors in the game, i.e., all participants are rational, all participants know that all participants are rational, and so on to infinity. This is an inconceivably infinite process of reasoning, and is a very strict assumption as far as the actor's ability to know the real world is concerned. Obviously, real world such assumptions are usually not guaranteed.

Evolutionary game theory adopts the assumption of limited rationality for the actors, therefore, these individuals do not have the "omniscience and omnipotence" of the actors in the game theory, and they cannot obtain the optimal results instantly in the economic activities. Game theory focuses on the study of equilibrium state and ignores the process of reaching equilibrium. In game theory, the actors can immediately make perfect judgment on the external environment and reach the equilibrium state. Game theory ignores the problem of time, emphasizes the equilibrium of the actors instantaneously, and even if the problem of time is considered, time is regarded as symmetrical or reversible.

In evolutionary game theory, time occupies a very important position. Actors constantly modify and improve their behavior during the evolutionary process, imitating successful strategies and so on. The actors in traditional game theory are completely rational, usually, under the assumption of complete rationality, if the Nash equilibrium exists, then the two sides of the game can directly reach the Nash equilibrium by playing the game once. This result does not depend on the initial state of the market, so there is no need for any dynamic adjustment process. On the other hand, according to evolutionary game theory, the Nash equilibrium should be reached after many games, which requires a dynamic adjustment process, and the equilibrium is dependent on the initial state, which is path-dependent.

In addition, in the case of multiple Nash equilibria, if a certain Nash equilibrium will definitely be adopted, there must exist some mechanism that can lead to the emergence of a certain equilibrium expected by each party to the game. However, the notion of Nash equilibrium in game theory does not itself have such a mechanism. Therefore, when there are multiple Nash equilibria in a game, it is impossible to predict what the outcome of the game will be, even assuming that the parties to the game are perfectly rational, and it is even more difficult to predict the outcome of the game if the parties to the game are only finitely rational. Of course, in game theory, when there are multiple Nash equilibria, backward induction can be used to achieve the refinement of the Nash equilibrium, but the precondition of this method is that the participants need to satisfy a stronger rationality assumption than complete rationality - sequential rationality. This cannot be achieved in reality. In evolutionary game theory, on the other hand, the refinement of equilibrium is achieved by forward induction, i.e., participants choose their future behavioral strategies based on the history of the game, which is a dynamic selection and adjustment process. Therefore, although the participants are all finitely rational, the dynamic selection mechanism will make it possible to reach one of the Nash equilibria in a situation where multiple Nash equilibria exist, realizing the refinement of Nash equilibrium.

The most common dynamic equations of selection mechanism have three types: the first type is positive payment dynamic equation, in this type of dynamic equation, all the pure strategies that obtain the payment greater than the average payment of the group have positive growth rate, and all the pure strategies that obtain the payment less than the average payment of the group have negative growth rate; the second type is monotonic dynamic equation, in this type of dynamic equation, if a pure strategy or mixed strategy obtains a payment greater than that obtained by another pure strategy, then the growth rate of the former is greater than that of the latter; and the third category is weakly positive payment dynamic equations, in which at least some pure strategies that obtain a payment higher than the average payment of the population (if they exist) have a positive growth rate. Clearly, the weakly positive payment dynamic equation contains both the positive payment dynamic equation and the monotonic dynamic equation.

The most widely used dynamic equation for choice mechanisms in evolutionary game theory is the replicator dynamic equation proposed by Taylor and Jonker (Taylor&Jonker, 1978), who at that time only studied symmetric two-player games. Subsequently, Taylor (Taylor, 1979) generalized the symmetric case to the asymmetric case. In the replicator dynamics equation, the growth rate of a pure strategy is proportional to the relative payoff or fitness (the difference between the payoff received by the pure strategy and the average payoff of the population). Obviously, the replicator dynamic equation is included in the first three types of dynamic equations of choice mechanisms. Replicator dynamic equations are most widely used in the economic field, and scholars have used replicator dynamic equations to successfully study a series of socio-economic problems such as social customs, institutions, and behavioral norms.

So, how to relate the basic concept of evolutionary game theory, the evolutionary stable strategy, to the dynamic equation of choice mechanism? Is it true that the refinement of equilibrium obtained through the choice mechanism is the evolutionary stable strategy? Intuitively, evolutionary stabilization strategies seem to guarantee that the equilibrium is stable. However, the formal definition of stability addresses dynamic systems, not the game's payoff or fitness function, and the evolutionary stabilization strategy can only describe the local dynamic nature of the system; it is not capable of expressing the relationship between the equilibrium and the dynamic selection process. Therefore, the evolutionary stabilization strategy is not necessarily the same concept as the dynamic equilibrium achieved by the dynamic equations of the choice mechanism. Therefore, in order to better describe the dynamic evolutionary process and unify the static concepts in evolutionary game theory with the dynamic process, Hirshleifer (Hirshleifer, 1982) proposed the concept of evolutionary equilibrium. According to Hirshleifer's concept, if the trajectory from any small neighborhood that makes a certain equilibrium point of the dynamic system eventually converge to the equilibrium point, the equilibrium point is said to be locally and asymptotically stable, such a dynamically stable equilibrium is the evolutionary equilibrium (Evolutionary Equilibrium).

It is well known that the evolutionary stabilization strategy is a refinement of the Nash equilibrium. Then, what is the relationship between Evolutionary Equilibrium and Evolutionary Stability Strategy and Nash Equilibrium? Friedman (Friedman, 1998) pointed out:

(1) Every Nash equilibrium is an equilibrium of a dynamic system;

(2) Evolutionary equilibrium is necessarily a Nash equilibrium;

(3) Evolutionary stabilization strategy is not necessarily an evolutionary equilibrium.

The replicator's dynamic equation guarantees that the evolutionary stable strategy is an evolutionary equilibrium, but in the general dynamic equation the evolutionary stable strategy is neither sufficient nor necessary for an evolutionary equilibrium. Friedman also argues that the most useful and widely used notion of equilibrium in evolutionary game theory is not that of an evolutionary stable strategy, but that of an evolutionary equilibrium. This is because the assumption that behavior changes over time according to some dynamic is sensible.