有限总体中制度激励的成本效率
Cost efficiency of institutional incentives in finite populations
摘要：
在合作行为不常见的人群中，机构可以提供激励措施，以增加这种行为。这一过程成本高昂，因此优化总体支出非常重要。这个问题可以用数学公式表示为一个多目标最优化问题，在这个问题中，人们希望最小化提供激励的成本，同时确保群体内期望的合作水平。在本文中，我们对此问题进行了严格的分析。我们研究了两两合作的困境（捐赠游戏）和多玩家（公共产品游戏）设置。我们证明了（总激励）费用函数的正则性，刻画了它的渐近极限（无穷总体，弱选择和大选择），并准确地显示当奖励或惩罚更有效。我们证明了当选择强度变化时，费用函数呈现相变现象。我们计算了相变的临界阈值，并研究了选择强度在临界值之下和之上时的最优化问题。它使我们能够提供一个精确的计算最优成本的激励，对于一个给定的选择强度。最后，我们提供了数值模拟来证明分析结果。总的来说，我们的分析首次提供了一个选择依赖计算的最佳成本的制度激励（奖励和惩罚），保证了最低限度的合作。这是至关重要的现实世界中的制度激励的应用，因为强度的选择是特定于一个给定的人口和潜在的游戏回报结构。
英文摘要：
Institutions can provide incentives to increase cooperation behaviour in a population where this behaviour is infrequent. This process is costly, and it is thus important to optimize the overall spending. This problem can be mathematically formulated as a multi-objective optimization problem where one wishes to minimize the cost of providing incentives while ensuring a desired level of cooperation within the population. In this paper, we provide a rigorous analysis for this problem. We study cooperation dilemmas in both the pairwise (the Donation game) and multi-player (the Public Goods game) settings. We prove the regularity of the (total incentive) cost function, characterize its asymptotic limits (infinite population, weak selection and large selection) and show exactly when reward or punishment is more efficient. We prove that the cost function exhibits a phase transition phenomena when the intensity of selection varies. We calculate the critical threshold in regards to the phase transition and study the optimization problem when the intensity of selection is under and above the critical value. It allows us to provide an exact calculation for the optimal cost of incentive, for a given intensity of selection. Finally, we provide numerical simulations to demonstrate the analytical results. Overall, our analysis provides for the first time a selection-dependent calculation of the optimal cost of institutional incentives (for both reward and punishment) that guarantees a minimum amount of cooperation. It is of crucial importance for real-world applications of institutional incentives since intensity of selection is specific to a given population and the underlying game payoff structure.
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