Reciprocity phase in various 2 × 2 games by agents equipped with two-memory length strategy encouraged by grouping for interaction and adaptation

Abstract

This paper numerically investigates 2 × 2 games involving the Prisoner's Dilemma, Chicken, Hero, Leader, Stag Hunt, and Trivial Games in which agents have a strategy expressed by five-bit, two-memory length. Our motivation is to explore how grouping for game interaction and strategy adaptation influence ST reciprocity and R reciprocity (Tanimoto and Sagara, 2007a [Tanimoto, J., Sagara, H., 2007a. A study on emergence of coordinated alternating reciprocity in a 2 × 2 game with 2-memory length strategy. Biosystems 90(3), 728–737]. Enhanced R reciprocity is observed with the stronger grouping for game interaction when a relatively stronger grouping for strategy adaptation is assumed. On the other hand, enhanced ST reciprocity emerged with the stronger grouping for strategy adaptation when the relatively weaker grouping for game interaction is imposed. Our numerical experiment deals with those two groupings independently and dependently.

Introduction

The question of how cooperative behavior can emerge in the real world has attracted much attention. Diverse fields of research have applied the 2-player–2-strategies game (the 2 × 2 game) as an archetype for investigating how cooperation can emerge in populations. The 2 × 2 game, as defined by its payoff in Table 1, is categorized into several classes. Among them, the Prisoner's Dilemma (PD) game (T > R > P > S and 2R > T + S) is the most well known. Nowak (2006) identified five mechanisms that produce cooperation (C) instead of defection (D) in PD games, which we call R reciprocity (Tanimoto and Sagara (2007a)) (because R equals the Pareto optimum for both players). The five mechanisms are kin selection, direct reciprocity, indirect reciprocity, network reciprocity, and group selection. All these mechanisms can be somewhat related to the lessening of an opposing player's anonymity relative to the so-called well-mixed situation. The term “lessening anonymity” here should be rephrased by “spatio-temporal correlated” (Roca et al., 2009) or “assortative” (Fletcher and Doebeli, 2009) that are opposing to “well-mixed”.

Network reciprocity relies on localities for both game interaction and strategy adaptation with neighbors on a network that lessens anonymity to deviate from the well-mixed situation. In most previous studies, which dealt with Spatial PD (SPD) games, involved these two localities; thus, indicating that game interaction and network adaptation work on the same network topology. For example, a focal agent plays with his immediate neighbors on the network and copies strategy C or D from one of them. However, if we control the localities independently, the question arises as to what that locality should be.

In this regard, Seo and Cho (1999) based their model on multi-player PD games and not on 2 × 2 PD games. Their results were not precise, as they attempted to discuss the locality effects of gaming and strategy adaptation independently. Ifti et al. (2004) employed a simulation approach in their milestone work and concluded that a consistent graph in terms of both game interaction and strategy adaptation is preferable for maximizing R reciprocity, where players obtain a higher payoff by obtaining Rs than mutual Ps in SPD games. Toward that end, Ohtsuki and his colleagues (Ohtsuki et al., 2007a, Ohtsuki et al., 2007b) proved what Ifti insisting by their excellent deduction with several premises. But Ohtsuki's deduction is invalid when selection pressure is assumed to be relatively large. Concerning this point, Wu and Wang (2007) and Suzuki and Arita (2009) found that a graph for strategy adaptation that has a range wider than a graph for game interaction is more appropriate for supporting cooperation than a consistent-ranging graph for both adaptation and interaction. That is, an SPD model wherein an agent copies his strategy from an immediate or proximate neighbor, but plays games with only with immediate neighbors, can support more robust cooperation.

In other words, so long as we are dealing with PD games that require R reciprocity to solve the dilemma, equal locality for interaction and adaptation is preferable under weak selection, while a stronger locality for interaction than for adaptation is better under strong selection.

Meanwhile, other interesting 2 × 2 game classes as a social metaphor for dilemma situations are Leader (T > S > R > P) and Hero (S > T > R > P) games that require ST reciprocity (Tanimoto and Sagara, 2007a), where players obtain a higher payoff by sharing S and T than mutual Rs (since S + T > 2R is valid for both Leader and Hero games). As mentioned, it is more profitable in both Leader and Hero games to offer C against an opponent's D (obtaining S) followed by offering D against a C opponent (obtaining T) than to offer C constantly (obtaining Rs). This situation encourages alternating coordinating strategy among agents, which is more sophisticated than offering only C, since there must be a specified concept concerning time sequence or role playing among agents. In some social contexts, ST reciprocity seems more important for evolution than R reciprocity. Tanimoto (2008) claims that ST reciprocity seems a crucially important metaphor in explaining why animal communication, including human language, can evolve in their biological perspective.

With respect to Leader and Hero games, however, no previous studies have explored effects of localities for game interaction and strategy adaptation. This paper intends to shed light on this issue through a simulation approach. In this paper, locality does not imply a network among agents but bears a more general meaning. In our terminology, “locality” is a grouping of agents. Hereafter, we use “grouping” instead of “locality”. Our model considers two independent groupings. The first grouping is a Gaming Grouping that regulates where agents’ groups can play. The second grouping is an Updating Grouping that regulates another kind of agents’ group where agents adapt their strategy in the evolutionary process.

Our motivation independently dealing with two groups of interaction and strategy evolution has been encouraged by the previous studies, mentioned-above, how the interaction and adaptation localities should be defined to maximize the network reciprocity in case of PD games. This kind of basic question seems meaningful, because the contemporary society is able to control human interactions in various ways by means of internet technology, for example, which brings huge spatial gaps between interaction and adaptation processes.