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Error-Awareness and Equilibria in Normal Form Games Peter Kriss ’07 Swarthmore College, Department of Mathematics & Statistics 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 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. E I [Si ] aij p j j1 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 higherror best response. Then, the deceptive player might play a traditional best response to this strategy and thereby arrive at a higher payoff outcome. E I[Si ] ( p j (1II ) II /n)aij j1 E II [T j ] bijqi That is, the payoff depends on the opponent’s probability distribution. 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). n n i1 Deceptive Error Rates. Error-Adjusted Expected Payoff Functions With errors incorporated into our expected payoff functions, we find that the expected payoff to Player I of playing Si and to Player II of playing Tj are now: Expected Payoff Functions As can be inferred from the n x n game matrix, the expected payoff to Player I of playing Si and to Player II of playing Tj are as follows: n Incorporating Errors n E II[T j ] (qi (1I ) I /n)bij i1 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? represented as a matrix like the one below. High-Error Best Response The a- and b-values, S’s and T’s, and q’s and 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. p’s are the payoffs, strategies, and probabilities of playing those strategies of Players I and II, respectively. What do these two situations have in common? 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? 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. 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.