Payoff Gradients in Two-Player Games
In a strategic form game, each player's payoffs are a function of the combination of their probability distributions over their sets of strategies. In a two-player game in which each player has two strategies, therefore, the payoffs can be computed as a function of the probability that the row player will choose "strategy 1" and the probability that the column player will choose "strategy 1." This Demonstration allows you to explore a dozen random games. It plots the vector fields created by the gradients of the players' payoffs for each game as these probabilities range over their domain and superimposes the Nash equilibria of the game. You can transfer wealth amongst the row and column players for each of the four strategy combinations to see how these transfers affect the gradients, the Nash equilibria, and the relationship between them.
The code for computing the Nash equilibria is taken from a package placed on MathSource, developed by John Dickaut and Todd Kaplan, and contained in H. Varian, ed., Economic and Financial Modeling with Mathematica, New York: TELOS/Springer - Verlag, 1993.