Positive and Negative Determinant Strategies in Repeated Games with Behavior-Value Inconsistency

2026-07-01

AI Summary

This study addresses a critical limitation of traditional zero-determinant (ZD) strategies, which neglect the internal cost arising from misalignment between an agent’s actions and its intrinsic values, thereby failing to accurately model real-world cooperative control mechanisms. To bridge this gap, the authors formulate a repeated game framework that explicitly incorporates behavior–value inconsistency costs. Within this setting, they demonstrate for the first time the failure of classical ZD strategies and introduce novel positive/negative determinant strategies grounded in game theory, matrix algebra, and payoff control theory. These new strategies enable a player to unilaterally enforce a linear relationship on the expected payoffs of both players: positive determinant strategies ensure the player’s payoff exceeds that of the opponent, while negative determinant strategies can suppress the opponent’s payoff below a specified threshold, substantially outperforming conventional ZD approaches in control efficacy.

Abstract

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.

Problem

• behavior-value inconsistency
• zero-determinant strategies
• repeated games
• payoff control
• internal cost

Innovation

• behavior-value inconsistency
• positive determinant strategy
• negative determinant strategy
• unilateral payoff control
• repeated games