Social dilemmas occur when incentives for individuals are misaligned with group interests1–7. According to the ‘tragedy of the commons’, these misalignments can lead to overexploitation and collapse of public resources. The resulting behaviours can be analysed with the tools of game theory8. The theory of direct reciprocity9–15 suggests that repeated interactions can alleviate such dilemmas, but previous work has assumed that the public resource remains constant over time. Here we introduce the idea that the public resource is instead changeable and depends on the strategic choices of individuals. An intuitive scenario is that cooperation increases the public resource, whereas defection decreases it. Thus, cooperation allows the possibility of playing a more valuable game with higher payoffs, whereas defection leads to a less valuable game. We analyse this idea using the theory of stochastic games16–19 and evolutionary game theory. We find that the dependence of the public resource on previous interactions can greatly enhance the propensity for cooperation. For these results, the interaction between reciprocity and payoff feedback is crucial: neither repeated interactions in a constant environment nor single interactions in a changing environment yield similar cooperation rates. Our framework shows which feedbacks between exploitation and environment—either naturally occurring or designed—help to overcome social dilemmas.Cooperation is more likely to evolve in a public-goods-distribution game when payoffs can change between rounds so that the stakes increase when players cooperate and decrease when players defect.

中文翻译：

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随机博弈中合作的演变

当个人激励与群体利益不一致时，就会出现社会困境1-7。根据“公地悲剧”，这些错位会导致公共资源的过度开发和崩溃。由此产生的行为可以用博弈论的工具进行分析。直接互惠理论 9-15 表明重复互动可以缓解这种困境，但之前的工作假设公共资源随着时间的推移保持不变。在这里，我们引入了公共资源是可变的，取决于个人的战略选择的想法。一个直观的场景是合作增加了公共资源，而叛逃则减少了它。因此，合作允许玩具有更高回报的更有价值的游戏的可能性，而叛逃会导致价值较低的游戏。我们使用随机博弈理论 16-19 和进化博弈论来分析这个想法。我们发现公共资源对先前互动的依赖可以大大增强合作的倾向。对于这些结果，互惠和回报反馈之间的相互作用至关重要：在恒定环境中的重复交互和在变化环境中的单一交互都不会产生相似的合作率。我们的框架显示了开发和环境之间的哪些反馈——无论是自然发生的还是设计的——有助于克服社会困境。当收益在轮次之间变化时，合作更有可能在公共物品分配游戏中发展，这样当玩家合作时赌注就会增加并在玩家叛逃时减少。