A. The General Framework

Consider a large population of individual traders who have two strategies (or phenotypes) available: rational strategy (type-1) and nonrational strategy (type-2).

At any time t, let Mi (t) be the number of individual traders who adopt strategy (type) i ε {1,2}. The associated population profile is therefore defined as the ordered pair x(t) = (x1(t); x2(t)), where xi (t) D Mi (t)=(M1(t) C M2(t)) is the population share of type-i traders. The population state x(t) is thus identified with a mixedstrategy in the associated strategy simplex Δ such that x(t) ε Δ= {x(t) ε R²+ | x1(t)+x2(t)=1}. Each strategy induces a payoff for the individual trader who adopts it, given the strategy profile of the rest of the population. Let the payoff to any pure strategy i ε {1,2}, given the population state x(t), be denoted as u(i, x(t)). The average payoff to an individual trader drawn at random from the population is thus given by

Following the approach in evolutionary games, let current payoffs be the determining factor for the relative fitness of different strategies and, as a result, drive the evolution of their corresponding population shares. In other words, current payoffs from trading activities represent the incremental effect on the fitness of different types, measured as the number of each type of trader. All other factors that are independent of the current payoffs affect only the absolute fitness of each type, but not the relative fitness of different strategies. Let the net birthrate · at any time t represent these other factors as background fitness independent of the current payoffs and let the population of traders evolve continuously over time. This setup results in the following population dynamic for type-i traders,

(1)

where ˙M i (t) is the instantaneous rate of the change in the population of type-i traders at time t. The corresponding dynamic for the population share of type-i traders xi (t) is straightforward to calculate and obtained as follows:

(2)

This dynamic implies that the type of traders associated with better-than-average payoffs increases, while the type associated with worse-than-average payoffs decreases in the process of evolution. As expected, this dynamic is independent of the common background fitness measure, i.e., net birthrate ·. It is worth nothing that the dynamic exhibits the same form of the usual replicator dynamic (Taylor and Jonker (1978)) in evolutionary games.

So far, we derive a general population dynamic without specifying a particular payoff function. Clearly, the dynamic depends on the choice of the payoff function, which in turn depends on the asset market in question. In Section B, we show that in a large economy the investment return of each strategy (type) emerges as the strategy’s (type’s) payoff function for the population dynamic in the process of wealth accumulation.

Prof. F. Albert Wang

Next: Population Dynamic in a Large Economy

Summary: Index