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The Evolution of Extortion in Iterated Prisoner's Dilemma Games
Christian Hilbe, Martin A. Nowak, Karl Sigmund
TL;DR
The paper asks how zero-determinant and extortion strategies perform evolutionarily in repeated cooperation games. It analyzes their evolution within populations and between populations with differing evolutionary rates, finding that extortioners catalyze cooperation but gain lasting advantage mainly in small or differentially evolving populations. The slower-evolving population benefits in two-population arms races, producing a Red-King effect.
Problem
The paper examines whether newly proposed zero-determinant strategies can arise, invade, and remain dominant in evolving populations.
Method
The paper investigates ZD strategies in evolutionary contests within well-mixed populations and between two populations evolving at different rates.
Results
ZD strategies catalyze cooperation within populations but prevail only when populations are small; between populations, extortion can emerge when evolutionary rates differ.
Takeaways & Limitations
When populations evolve on different time scales, extortion may evolve in endosymbiotic relationships through the Red-King effect.
Abstract
from arXiv · showhide
Iterated games are a fundamental component of economic and evolutionary game theory. They describe situations where two players interact repeatedly and have the possibility to use conditional strategies that depend on the outcome of previous interactions. In the context of evolution of cooperation, repeated games represent the mechanism of reciprocation. Recently a new class of strategies has been proposed, so called 'zero determinant strategies'. These strategies enforce a fixed linear relationship between one's own payoff and that of the other player. A subset of those strategies are 'extortioners' which ensure that any increase in the own payoff exceeds that of the other player by a fixed percentage. Here we analyze the evolutionary performance of this new class of strategies. We show that in reasonably large populations they can act as catalysts for the evolution of cooperation, similar to tit-for-tat, but they are not the stable outcome of natural selection. In very small populations, however, relative payoff differences between two players in a contest matter, and extortioners hold their ground. Extortion strategies do particularly well in co-evolutionary arms races between two distinct populations: significantly, they benefit the population which evolves at the slower rate - an instance of the so-called Red King effect. This may affect the evolution of interactions between host species and their endosymbionts.
Introduction
The paper examines whether zero-determinant strategies can arise and persist evolutionarily, finding that their effects depend on population structure and evolutionary rates. Extortioners catalyze cooperation within populations but gain lasting advantage when distinct populations evolve at different speeds.
- Zero-determinant strategies: Zero-determinant strategies let players unilaterally enforce a linear relationship between their own and the co-player’s payoffs.Equalizers fix the co-player’s score independently of the co-player’s strategy, while extortion strategies form another subset.
- Research question: The study asks whether ZD strategies can arise by mutation, invade populations, and remain evolutionarily dominant.The question is motivated by the possibility that extortioners receive no surplus when matched with one another.
- Within-population evolution: In well-mixed populations, ZD strategies can catalyze cooperation but do not become the long-term evolutionary outcome.The IPD models repeated interactions whose conditional strategies depend on previous-round outcomes.
- Two-population evolution: In distinct populations evolving on different time scales, extortion strategies can prevail in the slower-evolving population and enslave the faster-evolving one.This is identified as an example of the Red-King effect.
- Game setting: The Prisoner’s Dilemma is defined by T > R > P > S, with cooperation represented by C and defection by D.In the donation game, cooperation costs c and provides benefit b, where 0 < c < b.
- Approach: The analysis investigates ZD strategies in evolutionary contests involving important memory-one strategies and, separately, all memory-one strategies.The paper also considers interactions between members of two distinct populations.
Methods and Results
The paper analyzes zero-determinant strategies through stochastic evolutionary models, pairwise comparisons, and co-evolutionary interactions between distinct populations. Extortioners can catalyze cooperation but are generally not favored in large single populations, whereas slower-evolving populations can gain an advantage in two-population arms races.
- Methods and Results: ZD strategies are memory-one strategies satisfying a linear payoff relation, with equalizers fixing the co-player’s payoff and extortioners enforcing a surplus ratio χ > 1.Extortioners guarantee that player I’s surplus over P is χ times the co-player’s surplus.
- Methods and Results: The analysis combines pairwise comparisons with finite-population selection-mutation dynamics and simulations over memory-one strategy spaces.Mutations ensure all population states are reachable, producing a steady-state distribution of strategies.
- Methods and Results: WSLS dominates extortioners in pairwise comparison, while extortioners have no associated Nash equilibrium and can coexist with AllC under specified proportions.In a direct competition, WSLS has a higher payoff than Eχ when M > 1 + χ.
- Methods and Results: In reasonably large populations, extortioners, equalizers, and ZD strategies are not favored by evolution, but extortioners and TFT can catalyze transitions from AllD toward WSLS.Extortioners and TFT can subvert an AllD population through neutral drift; WSLS then prevails when populations are sufficiently large.
- Methods and Results: Very small populations promote extortion-related behaviors because relative payoff differences determine strategy fate, while larger populations favor WSLS-like strategies and yield higher average payoffs.The reported qualitative pattern remains robust to changes in benefits, costs, and selection strength.
- Methods and Results: In two-population evolutionary arms races, the slower-evolving population can adopt extortion, force cooperation, and receive more than 90% of the surplus under the illustrated parameters.When hosts and symbionts evolve at similar rates, neither population extorts the other; faster symbiont adaptation can precede long-term host extortion.
Discussion
The paper finds that ZD strategies are not generally favored in large populations, although some strategies perform well, while extortion can emerge between populations evolving at different rates. Its analysis also characterizes ZD, equalizer, and extortion strategies mathematically and identifies broader settings where the results may apply.
- Evolutionary outcomes: Extortion strategies can catalyze cooperation within one population but prevail only when the population is small.The paper reports this as a main result of its evolutionary analysis.
- Evolutionary outcomes: Within large populations, ZD strategies are not favored overall, although Generous TFT and compliant strategies perform well.Compliant strategies, which seek a larger share of the loss relative to mutual cooperation, perform as well as WSLS.
- Relation to prior work: The paper’s evolutionary conclusions are supported by prior work showing that ZD strategies do not prevail in large populations and that extortion remains promising in very small populations.It also notes that separate evolving populations can reproduce features of classical two-player game theory.
- Two-population evolution: In endosymbiotic interactions, extortion may evolve when host and symbiont populations evolve on different time scales, with the slower-evolving species gaining a disproportionate share of benefits.The authors describe this as a Red-King effect and explain that extortioner hosts can make symbiont cooperation more profitable for hosts than symbionts.
- Strategy characterization: The central ZD relation is proved for any 2 × 2 game, although feasible probability solutions may not exist in many cases.The proof derives the relation from the strategy dynamics and shows it holds independently of the second player’s strategy.
- Strategy characterization: Within the four-dimensional unit cube of memory-one strategies, ZD strategies form a three-dimensional subset containing two-dimensional equalizer and extortion subsets.For the donation game, all reactive strategies are ZD strategies; reactive equalizers satisfy p − q = c/b, while reactive χ-extortioners satisfy q = 0 and p = (b + χc)/(c + χb).