Different perceptions of social dilemmas: Evolutionary multigames in structured populations

I. INTRODUCTION

• Social dilemmas pit individual interests against societal interests, and the prisoner’s dilemma captures the tension between mutual cooperation’s collective reward and a defector’s advantage against a cooperator.Research on cooperative resolution has examined mechanisms including kin selection, reciprocity, network reciprocity, and group selection.Source: §I · p. 1
• In a multigame, different players adopt different payoff matrices, modeling how they may perceive the same dilemma differently.The paper motivates this with drivers facing different losses if a collision occurs.Source: §I · p. 1
• Because multigames had been studied in well-mixed systems, the paper examines them in structured populations, where stable outcomes can differ and network reciprocity can sustain cooperation.The study uses square lattices and scale-free networks, with the weak prisoner’s dilemma as its core game.Source: §I · p. 1
• Greater heterogeneity in adopted payoff matrices promotes cooperation, supporting heterogeneity-enhanced network reciprocity.The authors present this as evidence that differing perceptions can contribute to resolving social dilemmas.Source: §I · p. 2

II. EVOLUTIONARY MULTIGAMES

• Players on square lattices or Barabási–Albert networks use the weak prisoner’s dilemma, while a fraction ρ receives either S = +∆ or S = −∆.The two variants are equally represented so the average payoff matrix remains the weak prisoner’s dilemma; some simulations also vary payoff matrices over time.Source: §II · p. 2
• Each update compares the payoffs of a randomly selected player and a neighbor, then uses a Fermi probability for the selected player to adopt the neighbor’s strategy.Each player’s payoff is calculated using that player’s own payoff matrix, allowing asymmetric payoff allocation.Source: §II · p. 2
• With K = 0.1, better-performing players’ strategies are readily adopted, though copying a worse-performing player remains possible.This choice represents imperfect information, opponent-evaluation errors, and other unpredictable factors.Source: §II · p. 2
• Simulations typically use N = 10^4 players and average final results over 200 independent realizations.Each realization includes network generation; results are measured after relaxation of up to 2 · 10^5 MCS, with larger systems used near phase transitions.Source: §II · p. 2

• In well-mixed populations, cooperators go extinct for all T > 1, whereas structured populations support coexistence over a wide range of parameters.In the well-mixed case, extinction proceeds first among S = −∆ players, then S = 0, then S = +∆ players.Source: §III · p. 2
• Larger ∆ lets cooperators survive at larger T, and at ∆ = 0.8 cooperation persists across the whole T interval; larger fC also accompanies higher average payoffs.The cooperation increase strengthens with both ρ and ∆, even when higher S values are balanced by equally high negative S values.Source: §III · pp. 2–3, Figure 1 · p. 3
• Increasing ρ and ∆ monotonically raises stationary cooperation, supporting the conclusion that greater payoff-matrix heterogeneity strengthens cooperation.This pattern holds irrespective of T and includes populations with equal numbers of players using positive and negative S values.Source: §III · p. 3, Figure 2 · p. 3
• Cooperators using S = −∆ still exceed the homogeneous weak prisoner’s dilemma baseline, as positive-S cooperators act as leaders whose strategies spread through asymmetric imitation.The “−” subpopulation adopts strategies from the “+” subpopulation more often, and this feedback supports higher social welfare.Source: §III · pp. 3–4, Figure 3 · p. 4
• On scale-free networks with degree-normalized payoffs, assigning S = +∆ to only 2% of hubs promotes cooperation; at ∆ = 0.8, cooperators and defectors coexist across almost the whole T interval.Qualitatively identical cooperation-supporting effects also appear on random regular and Bender–Canfield configurational networks.Source: §III · pp. 4–5, Figure 4 · p. 4
• When players switch between S = +∆ and S = −∆ with equal probability, cooperation is still promoted beyond network reciprocity, though less strongly than with fixed game assignments.The positive effect remains on heterogeneous interaction networks; influential cooperative players emerge only temporarily in this setting.Source: §III · p. 5, Figure 5 · p. 5

IV. DISCUSSION

• Greater payoff-matrix heterogeneity promotes cooperation across network structures and whether game assignments remain fixed or vary over time.The authors identify asymmetric strategy imitation from players using positive sucker’s payoffs toward those using negative sucker’s payoffs; the higher cooperation in the positive-payoff group raises cooperation in the other groups.Source: §IV · p. 5
• Unlike structured populations, well-mixed populations recover the baseline weak prisoner’s dilemma regardless of multigame parametrization.The discussion notes that well-mixed models have limited applicability for realistic systems when interactions are better described by networks.Source: §IV · pp. 5–6
• The results add evidence for heterogeneity-enhanced network reciprocity and identify heterogeneity as a feature that can raise cooperation beyond traditional network reciprocity’s boundaries.The authors connect different perceptions of the same conflict to a mechanism for resolving social dilemmas rooted in principles shared with other heterogeneity-based cooperation mechanisms.Source: §IV · p. 6