Optimal dynamic incentive scheduling for Hawk-Dove evolutionary games

The Hawk-Dove evolutionary game offers a paradigm of the trade-offs associated with aggressive and passive behaviors. When two (or more) populations of players compete, their success or failure is measured by their frequency in the population, and the system is governed by the replicator dynamics. We develop a time-dependent optimal-adaptive control theory for this dynamical system in which the entries of the payoff matrix are dynamically altered to produce control schedules that minimize and maximize the aggressive population through a finite-time cycle. These schedules provide upper and lower bounds on the outcomes for all possible strategies since they represent two extremizers of the cost function. We then adaptively extend the optimal control schedules over multiple cycles to produce absolute maximizers and minimizers for the system.