Research has shown that the addition of abstention as an option transforms social dilemmas to rock-paper-scissor type games, where defectors dominate cooperators, cooperators dominate abstainers (loners), and abstainers (loners), in turn, dominate defectors. In this way, abstention can sustain cooperation even under adverse conditions, although defection also persists due to cyclic dominance. However, to abstain or to act as a loner has, to date, always been considered as an independent, third strategy to complement traditional cooperation and defection. Here we consider probabilistic abstention, where each player is assigned a probability to abstain in a particular instance of the game. In the two limiting cases, the studied game reverts to the prisoner's dilemma game without loners or to the optional prisoner's dilemma game. For intermediate probabilities, we have a new hybrid game, which turns out to be most favorable for the successful evolution of cooperation. We hope this novel hybrid game provides a more realistic view of the dilemma of optional/voluntary participation.
is also the literature on games with an exit strategy. For example, research has been done on the dictator game with an exit strategy 58,59 .
However, we believe that in many situations involving voluntary participation, such as in human interactions, the use of abstention as a pure strategy may not be ideal to capture the social dilemma. In reality, depending on the context and the type of social relationships we are modelling, abstention can also mean laziness, shyness or lack of proactivity, and all those emotions, feelings or characteristics may exist within a certain range. Thus, we propose that in a round of interactions, some agents might be interested in interacting with all of its neighbours (i.e., never abstain), while others may be willing to interact with only a few of them and abstain from interacting with others. To give another example, in the context of a poll of a number of individuals, there might be some who vote and others who do not. In the latter case, considering all the non-voters as abstainers might be too simplistic. In reality, there might be some who abstain because they do not have a view at all and those who occasionally abstain from convenience, lack of interest or because of some external event. In this way, we believe that abstention should be seen and explored as an extra attribute of each agent, and not as a pure strategy.
Given this motivation, in this paper, we introduce a prisoner’s dilemma with probabilistic abstention (PDPA), which is a hybrid of two well-known games in evolutionary game theory: the PD and the OPD game (also known as the PD game with voluntary participation). As occurs in the PD game, in the hybrid game each agent can choose either to cooperate or defect. The only difference is that in the PDPA game, in addition to the game strategy, each agent is defined by a value α = [0, 1] to denote a probability of abstaining from any interaction.
This work aims to investigate the differences between the PDPA game and the classic PD and OPD games. A number of Monte Carlo simulations are performed to investigate the effects of α in the evolution of cooperation. In order to have a more complete analysis of the evolutionary dynamics, both synchronous and asynchronous updating rules [60][61][62] are explored.
In order to increase the understanding of the outcomes associated with the hybrid game proposed in this paper (i.e., the prisoner’s dilemma game with probabilistic abstention -PDPA), in the following experiments we adopt ε = (1s)(1 -α) to denote the effective cooperation rate of an agent, where s = {0, 1} and α = [0, 1] correspond to the agent’s strategy and its probability of abstaining from a game interaction respectively. Note that here s = 0 means cooperator, s = 1 means defector, α = 0 indicates that the agent never abstains and α = 1 indicates that the agent always abstains. In this way, we can have two types of agents for each strategy: the pure-cooperators and the pure-defectors (i.e., the agents who always play the game, α = 0); and the agents who sporadically play the game (i.e., the sporadic-cooperators and sporadic-defectors, 0 < α < 1). Thus, the value of ε is very important to easily distinguish between a cooperator who always abstains (i.e., {s = 0, α = 1} ⇒ ε = 0), from the sporadic-cooperators (i.e., {s = 0, α = (0, 1)} ⇒ ε > 0), and the pure-cooperators (i.e., {s = 0, α = 0} ⇒ ε = 1).
We start by comparing the outcomes of the PDPA game with those obtained for the classic prisoner’s dilemma (PD) and optional prisoner’s dilemma (OPD) games for both synchronous and asynchronous updating rules (Fig. 1). We test a number of randomly initialized populations of agents playing the PDPA game with three different setups:
• α = 0 for all agents (equivalent to the PD game); • α is either 0 or 1 with equal probability (equivalent to the OPD game);
For all setups, we investigate the relationship between the fraction of effective cooperation ε and the probability of abstaining α for different values of the temptation to defect T and the loner’s payoff L.
As shown in Fig. 1, for the synchronous rule, it is possible to observe that the PDPA sustains higher levels of cooperation even for large values of the temptation to defect T. The difference between the outcomes of the synchronous and asynchronous versions in the classic games occur as expected: cooperation has more chance of surviving when the updating rules are synchronous, with less stochasticity and more awareness of the neighbourhood’s behaviour, i.e., the agent knows who is the best player in its neighbourhood. Surprisingly, results indicate that when the PDPA is considered, this enhancement also holds for the asynchronous updating model, which is a well-known adverse scenario for both classic games 62 . In general, it is clear that irrespective of the updating rule, the PDPA game is most beneficial for the evolution of cooperation. Moreover, when comparing the OPD with the PDPA game, we see a
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