Algorithm 8’s Approach to Countering the Highest Average Payoff Opponent

Authors: (1) Avrim Blum, Toyota Technological Institute at Chicago, IL, USA; (2) Melissa Dutz, Toyota Technological Institute at Chicago, IL, USA. Table of Links Abstract and 1 Introduction 2 Setting and 2.1 Models of behaviorally-biased opponents 3 Preliminaries and Intuition 4.1 Myopic Best Responder and 4.2 Gambler’s Fallacy Opponent 4.3 Win-Stay, Lose-Shift Opponent 4.4 Follow-the-Leader Opponent and 4.5 Highest Average Payoff Opponent 5 Generalizing 5.1 Other Behaviorally-Biased Strategies 5.2 Exploiting an Unknown Strategy from a Known Set of Strategies 6 Future Work and References A Appendix A.1 Win-Stay Lose-Shift Variant: Tie-Stay A.2 Follow-the-Leader Variant: Limited History A.3 Ellipsoid Mistake Bounds A.4 Highest Average Payoff Opponent A.4 Highest Average Payoff Opponent We assume the opponent initializes each action with an average payoff of 0. For this opponent, our high-level strategy will be to learn a best response to each action, and then use the well-known ellipsoid algorithm to predict the opponent’s actions while playing best responses to the predicted actions. The round preceding such a switch must have been a loss for the opponent: its net payoff must have gone down during that round for the average payoff to go from non-negative to negative. We showed that the opponent will proceed through each of the n actions in order when it switches during phase 1, so as long as we can reliably force the opponent to switch actions, we correctly record a best response to each of the first n − 1 actions in phase 1. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Avrim Blum, Toyota Technological Institute at Chicago, IL, USA; (2) Melissa Dutz, Toyota Technological Institute at Chicago, IL, USA. Authors: Authors: (1) Avrim Blum, Toyota Technological Institute at Chicago, IL, USA; (2) Melissa Dutz, Toyota Technological Institute at Chicago, IL, USA. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Setting and 2.1 Models of behaviorally-biased opponents 2 Setting and 2.1 Models of behaviorally-biased opponents 3 Preliminaries and Intuition 3 Preliminaries and Intuition 4.1 Myopic Best Responder and 4.2 Gambler’s Fallacy Opponent 4.1 Myopic Best Responder and 4.2 Gambler’s Fallacy Opponent 4.3 Win-Stay, Lose-Shift Opponent 4.3 Win-Stay, Lose-Shift Opponent 4.4 Follow-the-Leader Opponent and 4.5 Highest Average Payoff Opponent 4.4 Follow-the-Leader Opponent and 4.5 Highest Average Payoff Opponent 5 Generalizing 5.1 Other Behaviorally-Biased Strategies 5.1 Other Behaviorally-Biased Strategies 5.2 Exploiting an Unknown Strategy from a Known Set of Strategies 5.2 Exploiting an Unknown Strategy from a Known Set of Strategies 6 Future Work and References 6 Future Work and References A Appendix A.1 Win-Stay Lose-Shift Variant: Tie-Stay A.1 Win-Stay Lose-Shift Variant: Tie-Stay A.2 Follow-the-Leader Variant: Limited History A.2 Follow-the-Leader Variant: Limited History A.3 Ellipsoid Mistake Bounds A.3 Ellipsoid Mistake Bounds A.4 Highest Average Payoff Opponent A.4 Highest Average Payoff Opponent A.4 Highest Average Payoff Opponent We assume the opponent initializes each action with an average payoff of 0. For this opponent, our high-level strategy will be to learn a best response to each action, and then use the well-known ellipsoid algorithm to predict the opponent’s actions while playing best responses to the predicted actions. The round preceding such a switch must have been a loss for the opponent: its net payoff must have gone down during that round for the average payoff to go from non-negative to negative. We showed that the opponent will proceed through each of the n actions in order when it switches during phase 1, so as long as we can reliably force the opponent to switch actions, we correctly record a best response to each of the first n − 1 actions in phase 1. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv