A Third Strategy Incorporation Into N-person Snowdrift Game And Explorations About The Effects

The study of cooperative behavior in a competitive population has become an urgent and important question across several disciplines. Up till today,game theory has offered the most effective way to study that topic. Among those typical models mentioned in classical game theory, prisoner dilemma (PD) draws much attention, yet snowdrift game (SG) seems to be less attractive. In this paper, based on previous studies and by introducing a third strategy,the author makes an innovative attempt to generalize the prototypical SG model. The paper applies two different ways to study the generalized models,not only the system's dynamical equation is derived,but a specific numerical simulation is given as well, the results performed by both methods are in perfect match. The study finds out that unlike public goods game (PGG), which was the N-person generalized model of PD, NSG reveals different dynamical evolution.While as system evolves, PGG keeps changing from one phase to another, unable to reach a steady state, NSG however, evolves into one of two or three phases with different properties, once a specific phase is reached, it is a steady state. These findings provide new clues to further study into cooperation behavior evolution.The introduction of costly punishment was never used in NSG in previous studies,in Chapter 2, the author generalizes the N-person evolutionary snowdrift game to incorporate the effects of costly punishment in the well-mixed population. A set of dynamical equations that account for the evolution of the frequencies of the three strategies under replicator dynamics is formulated. At long time, the system evolves into one of two phases with different properties consisting only of two strategies,and three-strategy coexistence is not allowed. Small cost-to-benefit ratio, big competing group size, and severe punishment tend to suppress non-cooperators, and lead to a cooperative system with a mixture of cooperators and punishers. The resulting composition depends on the initial conditions as the dynamics is frozen once non-cooperators extinct. Large cost-to-benefit ratio, small competing group size, and light punishment tend to be self-destructive for the punishers, and lead to a mixture of cooperators and non-cooperators with composition independent of initial conditions and continual dynamics. The frozen phase and dynamical phases correspond to a line of fixed points and a single fixed point on different axes in the phase space, respectively.A simulation algorithm that mimics the replicator dynamics exactly is proposed. Results of the dynamical equations and numerical simulations are found to be in exact agreement.Then the author explores to incorporate another third strategy to generalize the prototypical NSG model in Chapter 3. In addition to the cooperative C and non-cooperative D strategies,a strategy L representing a loner behavior is introduced.Dynamical equations governing the time evolution of the frequencies of the strategies in the well-mixed population are derived. Detailed studies on how a system evolves indicated that the steady state could be an All L, All C, or C+D state, depending on the parameters r, L , and group size N. The strategy L plays two roles. It leads to an Al1L phase and helps give an Al1C phase. An algorithm for simulating the model numerically is described and validated. The algorithm will be useful in studying our model in various structured populations.In Chapter 4, based on the optional NSG, the author adds a threshold to cooperation,makes it harder to cooperate, but somehow gives some explanation to things abound in real life. The evolutionary dynamics is given in the context,also a critical value of r*that splits the system into two different evolutionary stable states, i.e. the C, D coexistence and the all-L state, by a sudden shift. However, as the threshold T gets bigger, cooperation is suppressed and it is more tempting to take the Loner strategy for the players. Further more the author investigates the special form of the model when N=T. In this special situation, defectors no longer get the chance to free-ride on cooperators,making the D strategy a dominated one,rather than dominating other strategies in our previous models.In our fourth model in Chapter 5, the author adds another strategy, i.e. punishment,to the protypical optional NSG model. Punishment is a common way to regulate behavior. Models of costly punishment have been introduced in which some players,i.e., the punishers, are willing to pay an extra cost so as to punisher the non-cooperators in the form of a deduction in their payoff. It was found that a high level of cooperation can be achieved by imposing punishment. The author first gives out the replicator dynamics of the model, then the author looks further into the influences that the parameters can cause after both Loner and Punisher are introduced to the NSG model.It's not surprised to find there's also a critical value of r* that splits the system into two different evolutionary stable states, i.e. the C, D and P coexistence or C and P state and the all-L state, by a sudden shift. As for the parameters N, ? and L. An increase in ? will cause the critical shift r* earlier to come. An increase of ? has little influency in the critical value of r* to the corresbonding set of parameters, yet it suppresses defectors and makes C and P coexistence more possible. And with bigger N,cooperation is suppressed and thus making a final C and D coexisting state consist of more Ds and less Cs.