1. Abstract Though previous explorations of equilibria in game theory have incorporated the concept of error-making, most do not consider the possibility of anticipation of errors. Instead of treating them as inherently unpredictable, I allow the awareness of error-making to directly affect a player's choice of strategy before any errors actually occur. I explore the consequences of allowing players to be estimate their opponent's error rate and incorporate this information into an expected payoff function. I show that if both players are aware of a high error rate of their opponent, a new, stable, non-Nash equilibrium can be achieved. The General n × n Game Matrix Any normal form game can, by definition, be represented as a matrix like the one below. The a- and b-values, S’s and T’s, and q’s and p’s are the payoffs, strategies, and probabilities of playing those strategies of Players I and II, respectively. Background Information Nash Equilibrium The Nash Equilibrium (NE) is the equilibrium concept in game theory. It is defined as a set of strategies such that no player has an incentive to make a unilateral change of strategy. That is, each player is playing the best response to his opponent’s choice of strategy. Expected Payoff Functions As can be inferred from the n x n game matrix, the expected payoff to Player I of playing S i and to Player II of playing T j are as follows: That is, the payoff depends on the opponent’s probability distribution. Deceptive Error Rates. One important point to note is that these results depend only on the players' estimations of their opponent's error rate. If players are using error-aware best response, the game can move between equilibria without errors ever being made. Since a high estimate of the opponent's error rate can cause a player to change strategies, there can exist an incentive for a player to give the impression that his error rate is higher than it is. If successful, this tactic could force the opponent to play a high- error best response. Then, the deceptive player might play a traditional best response to this strategy and thereby arrive at a higher payoff outcome. What do these two situations have in common? Error-Awareness and Equilibria in Normal Form Games Peter Kriss ’07 Swarthmore College, Department of Mathematics & Statistics Reference Young, H. Peyton. 1998. Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton University Press, Princeton. Acknowledgements I would like to thank Dr. Robert Muncaster of the University of Illinois without whose introduction and guidance in evolutionary game theory, this project would not have been possible. Nash vs. High-Error Equilibria If we reexamine the Nash equilibrium concept, the fact that the HEE is not necessarily a NE will not surprise us. One formulation of the Nash equilibrium concept is a set of strategies such that there is no incentive for unilateral deviation. But for significant error rates, the players have good reason to suspect that a deviation would not be unilateral -- it would be bilateral. Thus, this error-making environment takes us out of the traditional Nash equilibrium context. Incorporating Errors The Error Rate We now introduce the error rate, . Let  represent the probability that a player's choice of strategy is not executed as such, but a strategy is instead chosen randomly (Young, 1998). Error-Adjusted Expected Payoff Functions With errors incorporated into our expected payoff functions, we find that the expected payoff to Player I of playing S i and to Player II of playing T j are now: High-Error Best Response In general, the strategies that maximize these functions depend on the p’s and q’s. But as  1, they do not. For some  and greater, a single strategy will always be the best response. The set of these High-Error Best Responses is the High-Error Equilibrium (HEE). An Example: The Big Risk Game We have three Nash Equilibria: (A,A), (B,B) and (C,C). Why? High-Error Best Response Imagine we start at (A,A). Now if Player II’s error rate increases past the threshold, Player I will switch to strategy B. (Why?) Then, if Player I’s error rate increases past the threshold, Player II will switch to strategy C, the High-Error Best Response. High-Error Equilibrium So at (B,C), we have the High-Error Equilibrium. But this High-Error Equilibrium not a Nash equilibrium of the original game! What happened to NE being the equilibrium concept?