Control And Optimization Of Incentives For Evolution Of Collective Cooperation

Cooperative behaviors are common in nature and human society.In recent decades,understanding the emergence and maintenance of cooperation among self-interested individuals has been a major scientific puzzle.Evolutionary game theory provides a powerful mathematical framework to investigate the problem of cooperation,and the prisoner’s dilemma game and public goods game have been widely adopted as representative paradigms for studying cooperation among selfish individuals.Incentive is an effective measure to promote the evolution of cooperation.However,thus far few studies have investigated the control and optimization of incentives.In this thesis,we apply evolutionary game theory to deeply study the control and optimization of incentives for cooperation from four aspects,and the details are as follows:First,we study the evolutionary dynamics of tax-based pure punishment and reward strategies.In an infinite well-mixed population,we propose tax-based pure punishment and reward strategies,which are respectively introduced to the public goods game.By means of the replicator equation,we respectively investigate the evolutionary dynamics of the strategies above-mentioned and derive the theoretical condition in which the equilibria exist and are stable.By comparison,we find that tax-based pure punishment(taxbased pure reward)has an evolutionary advantage over pure punishment(pure reward)in sustaining cooperation.Besides,tax-based pure reward can lead to a higher level of cooperation than tax-based pure punishment.Next,we study the optimal incentive protocols with the lowest cost in well-mixed populations.Since providing incentives is costly,we formulate an index function for quantifying the cumulative cost during the process of incentive implementation in a wellmixed population,and respectively explore the optimal positive and negative incentive protocols in the public goods game.Using the Hamilton-Jacobi-Bellman equation,we theoretically obtain the optimal positive and negative incentive protocols with minimal cost,respectively.Additionally,we provide numerical examples to verify that compared with other given incentive schemes,the obtained optimal incentive protocol allows the dynamical system to reach the desired destination at the lowest cumulative cost.Furthermore,we find that applying punishment can induce a lower cumulative cost than the usage of reward if the initial cooperation level is larger.Otherwise,applying reward requires a lower cost.Furthermore,we study the optimal incentive protocol with the lowest cost under different update rules in structured populations.By establishing an index function for quantifying the cumulative cost during the process of incentive implementation,we theoretically derive the optimal positive and negative incentive protocols for cooperation on regular networks.For each update rule,we find that both types of optimal incentive protocols are identical and time-invariant.Moreover,we compare the optimal rewarding and punishing protocols concerning implementation cost,and find that if the initial cooperation level is lower,the optimal rewarding scheme requires a lower cumulative cost than the punishing scheme.Otherwise,the usage of punishment is more efficient.We perform computer simulations confirming that the obtained theoretical results are valid in a broad class of network structures.Lastly,we respectively study the optimally combined incentive protocol with the lowest cost in well-mixed and structured populations.Combined incentives,integrating reward for cooperators and punishment for defectors,are effective tools to promote the evolution of cooperation.In the prisoner’s dilemma game,we investigate the optimally combined incentive schemes on complete and regular networks,respectively.By devising an index function for quantifying the cost of incentives,we yield the optimally combined incentive protocol by using optimal control theory.By means of theoretical analysis,we theoretically identify a mathematical condition,under which the optimally combined incentive scheme requires the lowest cost than the isolated usage of punishment and reward.In addition to numerical calculations,we further perform simulations to verify that the obtained theoretical results are valid in a broad class of network structures.