In many applications, it is important to model continuing changes in the relative popularity of "strategies" used by a large population of "players". If these strategies are inherited (as in distinct sub-populations of animals with different phenotypes), changes in relative abundance are driven by different reproductive rates, which might also be changing with the composition of the entire population. Perhaps surprisingly, the popularity of patterns in human (non-inherited) behavior may also evolve in a similar fashion if the strategy-switching is primarily driven by (occasional) emulation of more successful players. Evolutionary Game Theory (EGT) provides a formal framework for describing the dynamics of such processes by ordinary (or stochastic) differential equations. But in some cases, passive observation is not enough: we might want to change the rules of the game dynamically, thus altering that population's trajectory and driving it to some target outcome. Mathematical control theory provides useful computational tools to accomplish this rule-changing "optimally".

We will first illustrate the EGT by attempting to answer a few classical questions: Why aren't animals fighting to death more often? How do many people manage to stay community-minded while others around them remain selfish? Are aggressive, defensive, or sneaky lizards more successful in producing offspring?

We will then illustrate the ideas of "controlled EGT" in the context of cancer modeling and optimization of adaptive drug therapies. In a joint project with Mark Gluzman and Jake Scott, we showed that the competition among sub-populations of cancer cells can be exploited to find the best time and duration for treatment, thus decreasing the total amount of drugs used and the total time to recovery. This was accomplished by solving a Hamilton-Jacobi-Bellman PDE numerically to obtain the optimal treatment policy in the feedback form. In a more recent project with MingYi Wang and Jake Scott, we extend this model to account for environmental stochasticity in cancer evolution. To deal with the possibility of failure due to random perturbations, we switch to a different optimization goal and compute treatment policies that maximize the probability of desirable outcomes (e.g., curing a patient without exceeding the prescribed time and drugs "budget"). These underlying models are certainly too simple to make them directly relevant for oncological practice. But our control-theoretic approach is general, and the qualitative insights gained from the EGT will, hopefully, inform the design of future clinical trials.