New 'Determinant Strategies' Outperform Zero-Determinant in Games with Internal Costs

A new paper on arXiv (2607.00625) from researchers Yuan Liu, Yakun Wang, and Bin Wu tackles a blind spot in repeated game theory: what happens when agents' actions don't align with their internal values? Classic zero-determinant (ZD) strategies give players unilateral control over payoffs in repeated Prisoner's Dilemma games, enabling cooperation without reciprocity. But prior models ignored the internal cost of inconsistency—the 'cognitive dissonance' of doing one thing while thinking another. The authors show that introducing a small cost for behavior-value mismatch destroys ZD strategies entirely. No player can enforce a linear payoff relationship when such inconsistency is penalized.

Instead, the team identifies a new class of strategies they call 'positive and negative determinant strategies.' These allow a player to enforce an affine (linear plus constant) combination of both players' average payoffs above or below zero. A positive determinant strategy lets the focal player guarantee they earn more than their opponent over repeated rounds. A negative determinant strategy lets them cap the opponent's payoff below a set value. Crucially, these strategies offer stronger control than ZD strategies—they can still enforce payoff relationships even with internal costs. This work highlights the importance of modeling agents with 'behavior-value inconsistency,' a realistic feature for AI systems, human-machine interactions, and any intelligent agent that acts against its values.