Can zero-determinant strategies control payoffs in continuous games?
Zero-determinant strategies in repeated continuously-relaxed games
This paper explores "zero-determinant" (ZD) strategies within repeated games where players' actions are probabilistic (mixed strategies). ZD strategies allow a player to unilaterally enforce a specific linear relationship between their own and other players' payoffs. The research extends ZD strategies to scenarios where action choices are continuous rather than discrete, finding that this "continuous relaxation" makes more ZD strategies possible. A special case, "one-point" ZD strategies, where a player repeats a single mixed strategy, is shown to connect to properties of Nash equilibria in classic game theory examples like Matching Pennies and the Battle of the Sexes.
For LLM-based multi-agent systems, this research suggests ways to design agents that can control payoff relationships in complex, dynamic interactions. The continuous relaxation of action spaces may be relevant to LLMs outputting probability distributions over possible actions. One-point ZD strategies could offer a simple yet effective mechanism for an LLM-agent to influence the overall outcome of a multi-agent interaction by strategically choosing a single mixed strategy distribution.