Restricted Non-Cooperative Game Theory

In conventional non-cooperative game theory, each player can see and can instantaneously select any element of its strategy set in response to the other players' strategy selections. In real settings, however, the strategies available to a player at any given time will often be a function of the strategy it selected at a prior time. For example, changing only one aspect of a strategy at a time may be
possible. Sometimes these constraints on the dynamics of strategy selection may be the result of external circumstances or cognitive limitations on the part of the player; other times they may be deliberately engineered by the player itself. Either way, however, the result is to overlay the strategies with a network connecting them (a topology) in which some strategies are connected and others are not. The top left control in this Demonstration permits you to select among a 25-member random sample of the 2,562,890,625 possible strategy topology pairs and visualizes the selection. The top right section of the Demonstration shows the resulting connections among the strategy combinations, with blue edges denoting possible moves among strategies 1-4 by player 1 and pink edges denoting possible moves among strategies A-D by player 2.

As in conventional non-cooperative game theory, the players in the game each receive a payoff based on the strategy combination that results from their chosen strategies. The control on the bottom left of this Demonstration lets you select among 25 different payoff arrangements. By requiring the players to behave in a greedy fashion and without benefit of any memory--such that they move to the strategy that, given the complementary strategies of the other players, gives them the immediate highest payoff--you can pare the network shown at the top right down to the "game network" shown on the bottom right. In such a "game network" each strategy combination has only two edges leading out of it ( edges in an -player game).

If you then assume that the players take a random walk on the game network starting from some random strategy combination, each strategy combination has some stationary probability of being selected. The network shown in the bottom right of this Demonstration reflects these stationary probabilities with the colors of the nodes. Darker blues represent improbable strategy combinations. Lighter colors represent more probable strategy combinations. Finally, if you divide (a) the expected payoffs to each player weighted according to these stationary probabilities of being at each strategy combination by (b) the expected payoffs to each player with the strategy combinations weighted equally, you obtain the "normalized score" for each player for each payoff/topology combination. Scores are thus a measure of how much better (or worse) than pure chance the players fare by pursuing this greedy, restricted approach. A graphic in the center of the Demonstration shows these normalized scores. The blue rectangle represents the score for player 1. The pink rectangle represents the score for player 2.