Zero-Determinant strategies in repeated multiplayer social dilemmas with discounted payoffs

TL;DR

This paper extends Zero-Determinant strategies to multiplayer social dilemmas with discounted payoffs, deriving conditions for enforceable payoff relations and analyzing how discounting influences equilibrium outcomes.

Contribution

It provides necessary and sufficient conditions for ZD strategies in multiplayer settings with discounting, revealing thresholds for enforceability based on group size and initial cooperation.

Findings

Existence of thresholds for discount factors enabling payoff enforcement.

Identification of extortionate, generous, and equalizer ZD strategies in multiplayer games.

Analysis of how discounting impacts ZD Nash equilibria.

Abstract

In two-player repeated games, Zero-Determinant (ZD) strategies enable a player to unilaterally enforce a linear payoff relation between her own and her opponent's payoff irrespective of the opponent's strategy. This manipulative nature of the ZD strategies attracted significant attention from researchers due to its close connection to controlling distributively the outcome of evolutionary games in large populations. In this paper, necessary and sufficient conditions are derived for a payoff relation to be enforceable in multiplayer social dilemmas with a finite expected number of rounds that is determined by a fixed and common discount factor. Thresholds exist for such a discount factor above which desired payoff relations can be enforced. Our results show that depending on the group size and the ZD-strategist's initial probability to cooperate there exist extortionate, generous and equalizer ZD-strategies. The threshold discount factors rely on the desired payoff relation and the variation in the single-round payoffs. To show the utility of our results, we apply them to multiplayer social dilemmas, and show how discounting affects ZD Nash equilibria.