Svoboda Jakub, Chatterjee Krishnendu
Institute of Science and Technology Austria, Klosterneuburg 3400, Austria.
Proc Natl Acad Sci U S A. 2024 Dec 10;121(50):e2405605121. doi: 10.1073/pnas.2405605121. Epub 2024 Dec 6.
Spatial games provide a simple and elegant mathematical model to study the evolution of cooperation in networks. In spatial games, individuals reside in vertices, adopt simple strategies, and interact with neighbors to receive a payoff. Depending on their own and neighbors' payoffs, individuals can change their strategy. The payoff is determined by the Prisoners' Dilemma, a classical matrix game, where players cooperate or defect. While cooperation is the desired behavior, defection provides a higher payoff for a selfish individual. There are many theoretical and empirical studies related to the role of the network in the evolution of cooperation. However, the fundamental question of whether there exist networks that for low initial cooperation rate ensure a high chance of fixation, i.e., cooperation spreads across the whole population, has remained elusive for spatial games with strong selection. In this work, we answer this fundamental question in the affirmative by presenting network structures that ensure high fixation probability for cooperators in the strong selection regime. Besides, our structures have many desirable properties: (a) they ensure the spread of cooperation even for a low initial density of cooperation and high temptation of defection, (b) they have constant degrees, and (c) the number of steps, until cooperation spreads, is at most quadratic in the size of the network.
空间博弈为研究网络中合作的演化提供了一个简单而优雅的数学模型。在空间博弈中，个体位于顶点，采用简单策略，并与邻居互动以获得收益。个体根据自己和邻居的收益来改变策略。收益由囚徒困境这一经典矩阵博弈决定，在该博弈中玩家可以选择合作或背叛。虽然合作是期望的行为，但背叛对于自私的个体来说能带来更高的收益。有许多关于网络在合作演化中作用的理论和实证研究。然而，对于强选择的空间博弈而言，是否存在这样的网络，即对于低初始合作率能确保高固定概率（也就是合作在整个人口中传播）这一基本问题，仍然难以捉摸。在这项工作中，我们通过给出在强选择 regime 中确保合作者具有高固定概率的网络结构，肯定地回答了这个基本问题。此外，我们的结构具有许多理想的特性：（a）即使对于低初始合作密度和高背叛诱惑，它们也能确保合作的传播；（b）它们具有恒定的度；（c）直到合作传播所需的步数最多是网络规模的二次方。
