Optimal dynamic incentive scheduling for Hawk-Dove evolutionary games

Abstract

The Hawk-Dove mathematical game offers a paradigm of the trade-offs associated with aggressive and passive behaviors. When two (or more) populations of players (animals, insect populations, countries in military conflict, economic competitors, microbial communities, populations of co-evolving tumor cells, or reinforcement learners adopting different strategies) compete, their success or failure can be measured by their frequency in the population (successful behavior is reinforced, unsuccessful behavior is not), and the system is governed by the replicator dynamical system. We develop a time-dependent optimal-adaptive control theory for this nonlinear dynamical system in which the payoffs of the Hawk-Dove payoff matrix are dynamically altered (dynamic incentives) to produce (bang-bang) control schedules that (i) maximize the aggressive population at the end of time T, and (ii) minimize the aggressive population at the end of time T. These two distinct time-dependent strategies produce upper and lower bounds on the outcomes from all strategies since they represent two extremizers of the cost function using the Pontryagin maximum (minimum) principle. We extend the results forward to times nT (n = 1, ..., 5) in an adaptive way that uses the optimal value at the end of time nT to produce the new schedule for time (n+1)T. Two special schedules and initial conditions are identified that produce absolute maximizers and minimizers over an arbitrary number of cycles for 0 ≤ T ≤ 3. For T > 3, our optimum schedules can drive either population to extinction or fixation. The method described can be used to produce optimal dynamic incentive schedules for many different applications in which the 2 x 2 replicator dynamics is used as a governing model.