The Evolution of Extortion in Iterated Prisoner's Dilemma Games

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.Comment: contains 4 figure

Strong Amplifiers of Natural Selection: Proofs

We consider the modified Moran process on graphs to study the spread of genetic and cultural mutations on structured populations. An initial mutant arises either spontaneously (aka \emph{uniform initialization}), or during reproduction (aka \emph{temperature initialization}) in a population of $n$ individuals, and has a fixed fitness advantage $r>1$ over the residents of the population. The fixation probability is the probability that the mutant takes over the entire population. Graphs that ensure fixation probability of~1 in the limit of infinite populations are called \emph{strong amplifiers}. Previously, only a few examples of strong amplifiers were known for uniform initialization, whereas no strong amplifiers were known for temperature initialization. In this work, we study necessary and sufficient conditions for strong amplification, and prove negative and positive results. We show that for temperature initialization, graphs that are unweighted and/or self-loop-free have fixation probability upper-bounded by $1-1/f(r)$, where $f(r)$ is a function linear in $r$. Similarly, we show that for uniform initialization, bounded-degree graphs that are unweighted and/or self-loop-free have fixation probability upper-bounded by $1-1/g(r,c)$, where $c$ is the degree bound and $g(r,c)$ a function linear in $r$. Our main positive result complements these negative results, and is as follows: every family of undirected graphs with (i)~self loops and (ii)~diameter bounded by $n^{1-\epsilon}$, for some fixed $\epsilon>0$, can be assigned weights that makes it a strong amplifier, both for uniform and temperature initialization