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The time scales of the aggregate learning and sorting in market entry games with large number of players

We consider the dynamics of player's strategies in repeated market games, where the selection of strategies is determined by a learning model. Prior theoretical analysis and experimental data show that after large number of plays the average number of agents who decide to enter, per round of the game, approaches the market capacity and, after a longer wait, agents are being sorted into two groups: the agents in one group rarely enter the market, and in the other, the agents enter almost all the time. In this paper we obtain estimates of the characteristic times it takes for both patterns to emerge in the repeated plays of the game. The estimates are given in terms of the parameters of the game, assuming that the number of agents is large, the number of rounds of the game per unit of time is large, and the characteristic change of the propensity per game is small. Our approach is based on the analysis of the partial differential equation for the function $f(t,q)$ that describes the distribution of agents according to their level of propensity to enter the market, $q,$ at time $t.$