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A Unified Model of Direct and Indirect Reciprocity in Multichannel Games
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Proposes a unified game-theoretic model combining direct and indirect reciprocity in linked two-channel repeated prisoner's dilemma games, with ternary reputations, and characterizes partner and semi-partner equilibria via analysis and evolutionary simulations.
Problem
How cooperation is sustained when players interact across multiple linked repeated games and both direct and indirect reciprocity can be used, given that reputation evaluation across interdependent channels is complex.
Approach
Players hold private ternary reputations (good, neutral, bad) for co-players, updated from both direct interactions and (with probability λ) observed third-party interactions across two linked PD channels; equilibrium analysis identifies partner (full cooperation) and semi-partner (partial cooperation) strategies and shows linking channels enlarges the full-cooperation region.
Key results
- Partner strategy (full cooperation) existence conditions: for direct reciprocity delta >= 2c/(b1+b2); four parameter regions delineated by error rate and pairwise continuation probability.
- Linking two channels expands the full-cooperation equilibrium region compared with two unlinked games (Figure 2).
- Evolutionary experiments (n=50) show populations converge to defection or cooperation strategies consistent with equilibrium predictions, and rely on direct, indirect, or both information depending on rounds and error rate.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| link two repeated games instead of treating them as separate, unlinked games | multichannel reciprocity framework · two-channel games | full-cooperation equilibrium range | Helped |
| use direct rather than indirect reciprocity | partner strategy · multichannel games with moderate noise and numerous interactions | reciprocity mechanism supporting full cooperation | Helped |
| use indirect rather than direct reciprocity | partner strategy · multichannel games with minimal noise and frequent interactions | reciprocity mechanism supporting full cooperation | Helped |
| cooperate in the higher-payoff channel when encountering a neutral player | semi-partner strategy · multichannel games | partial-cooperation equilibrium range | Helped |
| adjust the cooperation probabilities for neutral players across channels | semi-partner strategy · multichannel games | partial-cooperation equilibria | Mixed |
| evolve strategies through social learning | players using direct information · evolutionary experiments | strategy adoption | Mixed |
Surprising findings
- Using both direct and indirect information simultaneously (0<λ<1) gives no advantage for sustaining full cooperation equilibria; full cooperation exists only at λ=0 or λ=1 endpoints.
Limitations
- Restricted to first-order norms; higher-order norms not analyzed.
- Reputation mechanism is susceptible to misclassification from temporary behavior fluctuations.
- Analysis limited to two-channel linked PD with symmetric costs and a fixed population size.
Experimental setup
Metrics: cooperation rate Nash equilibrium conditions payoff of deviating player
Builds on: Donahue et al. 2020 multichannel games Ohtsuki and Iwasa norm analysis repeated prisoner's dilemma Kessinger, Tarnita, Plotkin 2023 indirect reciprocity
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