Positive and Negative Determinant Strategies in Repeated Games with Behavior-Value Inconsistency
For researchers in evolutionary game theory and multi-agent systems, this work addresses a realistic gap in payoff control models by accounting for internal inconsistency costs, though the results are incremental extensions of ZD strategies.
This paper introduces a game-theoretic framework incorporating internal costs from behavior-value inconsistency, proving that zero-determinant strategies fail under such conditions. Instead, it identifies positive/negative determinant strategies that enable unilateral payoff control, outperforming ZD strategies in control ability.
Direct reciprocity, based on the repeated interactions, is a fundamental mechanism to promote cooperation. Zero-determinant (ZD) strategies have opened an avenue for unilateral payoff control. However, previous studies neglect internal costs provided what agents do differ from what agents think, which is crucial for decision making of intelligent agents. Motivated by this, we establish a game theoretical framework by assuming that an individual pays the internal cost if the behavior is inconsistent with the internal thought. We prove that ZD strategy does not exist if the cost via behavior-value inconsistency is present. Instead, we find a new class of repeated strategies that enforce a unilateral payoff control, which is termed as positive/negative determinant strategy. The found strategy allows an individual to enforce an affine combination of two individuals' average payoffs above/below zero. Consequently, a focal individual is able to unilaterally control the opponent's payoff below a given value via negative determinant strategy, and a focal individual is able to get more payoff than the opponent via positive determinant strategy. We also find that the control ability of positive/negative determinant strategies is better off than that of ZD strategies. Our work highlights the importance of inconsistency between the behavior and value on payoff control, which is typically absent in classic ZD strategies.