Often players in a scenario that can be modeled as a strategic form game engage in repeated interactions with each other. In such instances, their selection of a strategy on a given "iteration" may depend on their history of previous interactions. By way of example, a player might start by playing strategy 1 and continue playing it unless the history of interactions were such that the opposing player had played 0 on his/her last two turns. This Demonstration permits a selection for each player from the 2,097,152 (2^21) strategies that depend on the prior two interactions of the players. It shows the resulting "directed graph" of histories the players could observe, the steady-state cycle of histories that will be observed, the payoffs from the selected strategic form game, and the mean payoffs received by the players in the steady state.
Histories that are part of the steadyÃ¢ÂÂstate cycle are shown as framed.
There are 2^42 (≈ 4.4 trillion) possible strategy combinations that the players can employ in this game and that can be selected using the two top sliders.
A "tit-for-tat" strategy can be implemented by having player 1 use strategy 1398101 and by having player 2 use strategy 1973785.