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1. Introduction

The term ‘stochastic resonance’ has been one of the hottest topics in the field of statistical physics. It is now broadly applied to describe any phenomenon where the presence of noise in a nonlinear system is better for output signal quality than its absence. In the usual context, noise is recognized as what is unwanted to gain and what should be removed. But for recent decades, quite a few physicists have found an interesting phenomenon that is called stochastic resonance effect, where adding some appropriate noise to a non-linear system featured with some cyclic nature enhances the rhythm laying behind, which brings somehow useful and preferable effects for us. Incidentally, for past decades, evolutionary game theory (EGT) has attracted much attention from various fields; not only theoretical biology, statistical physics, information science but also economics as well as other social sciences. It is because EGT may give a breakthrough to solve one of the most challenging questions of why many animal species including human being show lots of proofs indicating that mutual cooperation has evolved among egocentric individuals even in an environment of selfish behavior being beneficial than altruistic one [2,3]. As a commonly shared template to discuss this mysterious puzzle, prisoner’s dilemma (PD), one of the four classes of 2-player & 2-strategy (2 × 2) games where cooperation (C) never be able to survive in defection (D) in a well-mixed and infinite population, has well accepted. Quite rich stock by many previous studies theoretically, numerically as well as experimentally elucidates that a mechanism to decrease anonymity among players can bring an enhanced possibility of cooperation surviving, which is called reciprocity mechanism. Finite population is simplest example, although its effect is not strong as compared with other tangible mechanisms [4] such as direct reciprocity, indirect reciprocity, reciprocity supported by multi-level selection etc. Among those mechanisms, perhaps, most heavily concerned one is what-is-called network reciprocity. Since 1992, when the first study of the spatial prisoner’s dilemma (SPD) was conducted by Nowak and May [5], the number of papers dealing with network reciprocity have climbed up perhaps thousands. One reason why so many people have attracted in SPD is that network reciprocity may explain the evolution of cooperation even among primitive organisms without any sophisticated intelligence. Network reciprocity relies on two effects. The first is limiting the number of game opponents (that is meant “depressing anonymity” as opposed to the situation assumed by an infinite and well-mixed population), and the second is a local adaptation mechanism, in which an agent copies a strategy from a neighbor linked by a network. These explain how cooperators survive in a system with asocial dilemma, even though it requires agents to use only the simplest strategy—either cooperation (C) or defection (D).

Meanwhile in SPD with assumption of a finite population, from the system dynamics point of view, a demographic fluctuation can be observed intrinsically. In view of statistical physics, such dynamical system may show a stochastic resonance if it would be exposed to an appropriate noise that is extrinsically given through an additional mechanism, where cooperation, originally vanishing, can survive or even can be surged as opposed to defection [6]. Along with this context, there have been many works concerned on “what noise (and by how) additionally imposed to SPD model can effectively enhance network reciprocity”.

Schematic view while each presumed error setting taking place in the flow of SPD games.

In sum, referring to the previous works above-mentioned, we should explore action error, copy error, observation error, random agents and zealous agents when we say noise effect in SPD games. This paper is organized as follows. Section 2 gives a model description, and precisely describes what noise mechanisms are presumed. Section 3 presents and discusses simulation results, and Section 4 draws conclusions.

2. Model setup 

We presume standard SPD game setting. Agents of N = 104 are placed in each of vertices in an assumed underlying network explained below. Each of agents plays a PD with all his neighbors and accumulates payoffs resulting from all games with his neighbors. In a game, a player receives a reward (R) for mutual cooperation and a punishment (P) for mutual defection. If one player chooses cooperation (C) and the other chooses defection (D), the latter obtains a temptation payoff (T), and the former the sucker’s payoff (S). We assume a spatial game with R = 1 and P = 0, parametrized as R S T P = 1 −Dr 1 + Dg 0 ,where Dg = T − R and Dr = P − S imply a chicken-type dilemma and stag-hunt dilemma, respectively [34,35]. We limit the PD game class by assuming 0 ≤ Dg ≤ 1 and 0 ≤ Dr ≤ 1.

We vary strategy updating rule either Imitation Max (IM) or Pair-wise Fermi (PW-Fermi). IM is the most well-accepted deterministic update rule where a focal player copies the strategy of the neighbor or himself who getting the largest payoff in the current time step. Also, we presumed PW-Fermi as the most representative stochastic update rule where a player compares his accumulated payoff (i) with that of a randomly selected neighbor (j) and copies the neighbor’s strategy according to Pi←j copy = 1 1+exp[(i−j)/κ] . Here, κ indicates noise coefficient, which is presumed 0.1 throughout the study.

As population structure, we assume a 2D lattice (hereafter, Lattice) with degree of 8 (k = 8), i.e. a Moore neighborhood and scalefree network by Barabasi–Albert algorithm [36] (hereafter, BA-SF). 

In each simulation setting, we explored how each of different error settings presumed below influences on network reciprocity by varying whether deterministic or stochastic updating; {IM, PWFermi} and whether homogeneous or heterogeneous underlying network; {Lattice, BA-SF} are presumed.
 
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