Memory- n strategies of direct reciprocity

Abstract

Humans routinely use conditionally cooperative strategies when interacting in repeated social dilemmas. They are more likely to cooperate if others cooperated before, and are ready to retaliate if others defected. To capture the emergence of reciprocity, most previous models consider subjects who can only choose from a restricted set of representative strategies, or who react to the outcome of the very last round only. As players memorize more rounds, the dimension of the strategy space increases exponentially. This increasing computational complexity renders simulations for individuals with higher cognitive abilities infeasible, especially if multiplayer interactions are taken into account. Here, we take an axiomatic approach instead. We propose several properties that a robust cooperative strategy for a repeated multiplayer dilemma should have. These properties naturally lead to a unique class of cooperative strategies, which contains the classical Win-Stay Lose-Shift rule as a special case. A comprehensive numerical analysis for the prisoner's dilemma and for the public goods game suggests that strategies of this class readily evolve across various memory- spaces. Our results reveal that successful strategies depend not only on how cooperative others were in the past but also on the respective context of cooperation.

Emergence and suppression of cooperation by action visibility in transparent games

“…Although frequently observed in humans and animals (see, for instance, [5,13], these strategies have up to now remained beyond the scope of game-theoretical studies, but naturally emerge in our transparent games framework. Note that here we focused on memory-one strategies for the reasons of better tractability, results for strategies with longer memory can differ considerably [37].…”

Symmetric Decomposition of Asymmetric Games

“…As pointed out by Veller and Hayward [36], many real-world situations, in which one aims to study evolutionary or learning dynamics of several interacting agents, are better modelled by asymmetric games. As such these theoretical findings can facilitate deeper analysis of equilibrium structures in evolutionary asymmetric games relevant to various topics including economic theory, evolutionary biology, empirical game theory, the evolution of cooperation, evolutionary language games and artificial intelligence [11,12,[37][38][39][40]. Finally, the results of this paper also nicely underpin what is said in H. Gintis' book on the evolutionary dynamics of asymmetric games, i.e., 'although the static game pits the row player against the column player, the evolutionary dynamic pits row players against themselves and column players against themselves' [32] (chapter 12, p.292).…”

Friendly-rivalry solution to the iteratedn-person public-goods game

“…Another important solution concept to the n-person dilemma can be derived from a different set of criteria: By requiring mutual cooperation, error correction, and retaliation with a time scale of k rounds, one can characterize the all-or-none (AON-k) strategy, which is defined as prescribing c only when everyone cooperated or no one did in each of the previous k rounds [30,34,35]. For example, WSLS= (1, 0, 0, 1) is equivalent to AON-1.…”