Extortion under uncertainty: Zero-determinant strategies in noisy games

Abstract

Repeated game theory has been one of the most prevailing tools for understanding the long-run relationships, which are footstones in building human society. Recent works have revealed a new set of "zero-determinant (ZD)" strategies, which is an important advance in repeated games. A ZD strategy player can exert a unilaterally control on two players' payoffs. In particular he can deterministically set the opponent's payoff, or enforce an unfair linear relationship between the players' payoffs, thereby always seizing an advantageous share of payoffs. One of the limitations of the original ZD strategy, however, is that it does not capture the notion of robustness when the game is subjected to stochastic errors. In this paper, we propose a general model of ZD strategies for noisy repeated games, and find that ZD strategies have high robustness against errors. We further derive the pinning strategy under noise, by which the ZD strategy player coercively set the opponent's expected payoff to his desired level, although his payoff control ability declines with the increase of noise strength. Due to the uncertainty caused by noise, the ZD strategy player cannot secure his payoff to be higher than the opponent's, which implies strong extortions do not exist even under low noise. While we show that the ZD strategy player can still establish a novel kind of extortions, named weak extortions, where any increase of his own payoff always exceeds that of the opponent's by a fixed percentage, and the conditions under which the weak extortions can be realized are more stringent as the noise becomes stronger.

Strategies that enforce linear payoff relationships under observation errors in Repeated Prisoner’s Dilemma game

“…In other words, if D(p, q, αS X +βS Y +γ1) = 0 is satisfied, there is a linear payoff relationship between the two players' payoffs. Press and Dyson (without error) [9] and Hao et al (with error) [49] only searched for the case that the second and fourth columns take the same value. This makes the determinant become zero.…”

“…However, in some regions near (P E , P E ), X's payoff is lower than Y 's payoff. We mathematically restate the difference between dominant and contingent based on Hao et al's formalism [49]. We transform α = φs , β = −φ, γ = φ(1 − s )l in Eq.…”

“…Thus, the horizontal axis is the payoff of Equalizer (player X) and the vertical axis is the payoff of player Y . As Hao et al already suggested [49], there exist Equalizer strategies even if errors are incorporated. When + ξ = 0 (no error), if we set (α, β, γ) = (0, −2/3, 1/3) in Eq.…”

“…Thus, the effect of errors has been considered in the literature of the RPD game [40][41][42][43][44][45][46][47][48]. However, except for [49], the effect of errors has not been considered for strategies that enforce a linear payoff relationship. There are typically two types of errors: perception errors [42] and implementation errors [43].…”

Linear algebraic structure of zero-determinant strategies in repeated games

“…Although ZD strategy was originally formulated in two-player two-action (iterated prisoner's dilemma) games, ZD strategy was ex-tended to multi-player two-action (iterated social dilemma) games (12,13), two-player multi-action games (14,15), and multi-player multi-action games (16). In addition, ZD strategy was extended to two-player two-action noisy games (17), which is one example of the repeated incomplete-information games. Furthermore, besides these fundamental theoretical studies, ZD strategies are also applied to resource sharing in wireless networks (18,19).…”