Phase diagram of Symmetric Iterated Prisoners Dilemma of Two-Companies with Partial Imitation Rule

The problem of two companies of agents with one-step memory playing game is investigated in the context of the Iterated Prisoner’s Dilemma under the partial imitation rule, where a player can imitate only those moves that he has observed in his games with his opponent. We limit our study to the special case where the players in the two groups enjoy the same conditions on a fully connected network, so that there are only two payoff matrices required: one for players playing games with members of the same company, and the other one for players playing games with members from a different company. We show that this symmetric case of two companies of players can be reduced to the one-company case with an effective payoff matrix, from which a phase diagram for the players using the two dominant strategies, Pavlov and Grim Trigger can be constructed. The phase diagram is computed by numerical integration of the approximate mean value equations. The results are in good agreement with simulations of the two-company model. The phase diagram leads to an interesting conclusion that a player will more likely become a Grim Trigger, regardless of their affiliated company, when the noise level increases so that he is more irrational, or when the intra-group temptation to defect increases.

💡 Research Summary

The paper investigates evolutionary dynamics in a two‑company (or two‑population) setting of the Iterated Prisoner’s Dilemma (IPD) when agents possess one‑step memory and adopt a partial imitation rule (pIR). Under pIR an agent can only copy those moves of an opponent that have actually been observed during their encounters, in contrast to the traditional full‑imitation rule (tIR) which assumes perfect knowledge of the opponent’s entire meta‑strategy.

The authors consider a fully connected network of agents split equally between two companies, A and B. Interactions within a company are governed by an intra‑company payoff matrix (M_{\text{intra}}) and interactions across companies by an inter‑company matrix (M_{\text{inter}}). Both matrices share the standard Prisoner’s Dilemma structure with reward (R), temptation (T), punishment (P) and sucker’s payoff (S); only the temptation parameter (b) (with (T= b), (R=1), (P=0), (S=0)) is allowed to vary. The intra‑company temptation is denoted (b_{\text{intra}}) and the inter‑company temptation (b_{\text{inter}}) (with (b_{\text{inter}}\ge b_{\text{intra}}) to model “hostility toward strangers”).

With one‑step memory each agent’s strategy can be encoded by five bits: an initial move and a response to each of the four possible previous outcomes (DD, DC, CD, CC). This yields a strategy space of (2^{5}=32) possible strategies. Among them, Pavlov (win‑stay‑lose‑shift) and Grim Trigger (cooperate until a single defection, then defect forever) dominate large regions of the parameter space and are the focus of the analysis.

The dynamics proceeds in discrete Monte‑Carlo (MC) steps. In each step a pair of agents is randomly selected (either from the same company or from different companies) and one of them may imitate the other. The probability that agent (i) imitates agent (j) follows a Fermi‑type function

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