Fig 1.
Memory length m required for each of currently known friendly-rivalry strategies in the n-person PG game [18, 19, 21].
The dashed blue line depicts a theoretical lower bound m = n for friendly rivalry [19], and the strategy proposed in this work, called CAPRI-n, has m = 2n − 1.
Fig 2.
Schematic diagram of the transition between states of CAPRI-n.
The five rules of the strategy can be identified with the player’s internal states [26], each of which is represented as a node in this diagram. An exception is state I, which corresponds to two nodes to clarify the following point: When t* ≥ t − m, the state may have outgoing connections to A and P. When t* < t − m, on the other hand, the only possible next state is R. The player has to choose c at a blue node and d at a red node. We have omitted error-caused transitions for the sake of simplicity.
Fig 3.
Distribution of long-term payoffs when a CAPRI-n player meets co-players whose pμν’s are randomly sampled from the unit interval.
Darker shades toward blue indicate higher frequency of occurrence. The multiplication factors for n = 2, 3, and 4 are 1.5, 2, and 3, respectively, and the solid lines indicate the region of feasible payoffs. In each case, the filled circle means the long-term payoffs when CAPRI-n is adopted by all the players, whereas the cross shows those of TFT players as a reference point. In each panel, we have drawn a dotted line along the diagonal as a simple check for defensibility. For n = 3 or 4, the parallelogram surrounding the blue area indicates the set of feasible payoffs when the focal player is AllD, which indicates that the behavior of CAPRI-n is similar to AllD against most of the memory-one players.
Fig 4.
Abundance of strategies for n = 2 as the benefit-to-cost ratio b and the population size N vary.
The default values were b = 3 and N = 30 unless otherwise specified. The strength of selection and the error probability were set to be σ = 1 and ϵ = 10−4, respectively. (A) Simulation result with 16 memory-one deterministic strategies, classified into three categories, i.e., efficient, defensible, and the other strategies. (B) Effect of CAPRI-2 when it was added to the available set of strategies.
Fig 5.
Abundance of strategies for n = 3 as the benefit-to-cost ratio b and the population size N vary.
The default values were b = 3 and N = 30 unless otherwise specified. The strength of selection and the error probability were set to be σ = 1 and ϵ = 10−4, respectively. (A) Simulation result with 64 memory-one deterministic strategies, classified into three categories, i.e., efficient, defensible, and the other strategies. (B) Effect of CAPRI-3 when it was added to the available set of strategies.
Fig 6.
Normalized distribution of fixation probabilities of mutants, which were randomly sampled from the set of deterministic strategies with the same memory length as CAPRI-n’s.
When we simulated the two-person game with taking CAPRI-2 as the resident strategy, 109 mutants were sampled. In case of the three-person game in which CAPRI-3 was the resident strategy, the number of sampled mutants was 5 × 106. In either case, no mutant had higher fixation probability than 1/N (the vertical dashed line). On the other hand, when the resident was randomly drawn from the same strategy set, mutants frequently achieved fixation with probability higher than 1/N. For each random sample of the resident strategy, 102 mutants were tested, and this process was repeated 107 and 105 times when n = 2 and 3, respectively. Throughout this calculation, we used N = 10 as the population size, ϵ = 10−4 as the error probability, σ = 1 as the selection strength, and b = 2 as the benefit of cooperation.