Stackelberg EvolutionaryGame (SEG) theory frames interactions between a rational leader and evolving followers, who play an evolutionary game among themselves. This framework has applications in managing evolving populations, including fisheries management, pest control, and cancer treatment. To manage an evolving systemin question, in the standard practice, the leader usually adopts a constant aggressive strategy, with the aim to either preserve (e.g., fisheries management) or eradicate (e.g., pest control) the evolving system. However, adopting an aggressive constant strategy ignores the evolving nature of the population in question.
In this thesis, we identify leader’s Nash and Stackelberg strategies in the game, assuming that the evolutionary followers have reached their eco-evolutionary equilibrium. We show that the constant aggressive strategy yields the least favorable outcome for the leader compared to the Nash and Stackelberg strategies. Furthermore, we show that the Stackelberg strategy consistently provides equal or better outcomes for the leader compared to the Nash strategy, as measured by the value of the leader’s objective function. We further explore the SEG framework in cancer treatment, where the followers are treatment-sensitive and treatment-resistant cancer cell populations. The resistant population develops treatment-induced resistance as a quantitative trait. We investigate how a physician as the leader can optimize treatment strategies to maximize patient’s quality of life by anticipating the cancer cells’ treatment-induced response. Three treatment strategies are compared: maximum tolerable dose (MTD), which is commonly used as the standard of care; the Nash strategy; and the Stackelberg strategy. The physician uses the Nash strategy when they take the cancer cells’ ecological equilibrium point into account. However, the physician uses the Stackelberg strategy when they take the eco-evolutionary response of cancer cells into account. Our results demonstrate that the Stackelberg strategy achieves the best outcomes, including reduced treatment-induced resistance, lower drug dosage, and improved patient’s quality of life. We show that the quality of life achieved with the Stackelberg strategy is at least as high as that of the Nash strategy, which typically outperforms theMTD approach.
The best strategy for the leader will depend on our understanding of the underlying eco-evolutionary dynamics of the evolutionary followers. To understand what the best evolutionary game for modeling cancer under treatment is, we fit various models to non-small cell lung cancer (NSCLC) in-vitro data analyzed earlier by Kaznatcheev et al. and Soboleva et al.. These experiments measure cell counts of Alectinib-sensitive and Alectinib-resistant cancer cells in environments with and without Alectinib and the presence or absence of cancer-associated fibroblasts (CAFs). We compare logistic, Gompertz, and von Bertalanffy growth models, along with Norton-Simon, linear, and ratiodependent treatment efficacy terms. We also examine how Alectinib and CAFs influence model parameters and, subsequently, the interactions between cancer cells. For monoculture data, our results indicate that the logistic model with ratio-dependent treatment efficacy provides the best fit. We derive inter-type competition coefficients for co-culture data using growth rate and carrying capacity estimates from monoculture. Statistical tests reveal that growth rate and carrying capacity parameters remain largely unaffected by the presence of CAFs. However, cell interactions in co-cultures vary significantly across environments due to changes in competition coefficients and drug efficacy. Specifically, we show that CAFs enable the coexistence of sensitive and resistant cells, whereas Alectinib favors the outcompetition of sensitive cells by resistant ones. This PhD thesis furthers Stackelberg evolutionary games to frame interactions between a rational leader and evolutionary followers. We integrate SEG theory with empirical cancer growth modeling, highlighting the potential of game-theoretic approaches to enhance cancer treatment outcomes. We also discuss the challenges and future opportunities for applying this framework to other domains where managing evolving systems is essential.

We present a game-theoretic model of a polymorphic cancer cell population where the treatment-induced resistance is a quantitative evolving trait. When stabilization of the tumor burden is possible, we expand the model into a Stackelberg evolutionary game, where the physician is the leader and the cancer cells are followers. The physician chooses a treatment dose to maximize an objective function that is a proxy of the patient’s quality of life. In response, the cancer cells evolve a resistance level that maximizes their proliferation and survival. Assuming that cancer is in its ecological equilibrium, we compare the outcomes of three different treatment strategies: giving the maximum tolerable dose throughout, corresponding to the standard of care for most metastatic cancers, an ecologically enlightened therapy, where the physician anticipates the short-run, ecological response of cancer cells to their treatment, but not the evolution of resistance to treatment, and an evolutionarily enlightened therapy, where the physician anticipates both ecological and evolutionary consequences of the treatment. Of the three therapeutic strategies, the evolutionarily enlightened therapy leads to the highest values of the objective function, the lowest treatment dose, and the lowest treatment-induced resistance. Conversely, in our model, the maximum tolerable dose leads to the worst values of the objective function, the highest treatment dose, and the highest treatment-induced resistance.

Stackelberg evolutionary game (SEG) theory combines classical and evolutionary game theory to frame interactions between a rational leader and evolving followers. In some of these interactions, the leader wants to preserve the evolving system (e.g. fisheries management), while in others, they try to drive the system to extinction (e.g. pest control). Often the worst strategy for the leader is to adopt a constant aggressive strategy (e.g. overfishing in fisheries management or maximum tolerable dose in cancer treatment). Taking into account the ecological dynamics typically leads to better outcomes for the leader and corresponds to the Nash equilibria in game-theoretic terms. However, the leader's most profitable strategy is to anticipate and steer the eco-evolutionary dynamics, leading to the Stackelberg equilibrium of the game. We show how our results have the potential to help in fields where humans try to bring an evolutionary system into the desired outcome, such as, among others, fisheries management, pest management and cancer treatment. Finally, we discuss limitations and opportunities for applying SEGs to improve the management of evolving biological systems. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.