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Games and the Treatment Convexity of Cancer

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Abstract

Evolutionary game theory has been highly valuable in studying frequency-dependent selection and growth between competing cancer phenotypes. We study the connection between the type of competition as defined by the properties of the game, and the convexity of the treatment response. Convexity is predictive of differences in the tumor’s response to treatments with identical cumulative doses delivered with different variances. We rely on a classification of \(2\times 2\) games based on the signs of “gains of switching,” containing information about the kind of selection through the game’s equilibrium structure. With the disease starting in one game class, we map the type of effects treatment may have on the game depending on dosage and the implications of treatment convexity. Treatment response is a linear function of dose if the game is a Prisoner’s Dilemma, Coordination, or Harmony game and does not change game class, but may be convex or concave for Anti-Coordination games. If the game changes class, there is a rich variety in response types including convex–concave and concave–convex responses for transitions involving Anti-Coordination games, response discontinuity in case of a transition out of Coordination games, and hysteresis in case of a transition through Coordination games.

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Notes

  1. Each of the four games corresponds to an archetypal psychological pressure that the authors dubbed “exploiter” (a change of strategy affects self positively, has negative effects on other), “leader” (positive for both, higher for self), “hero” (positive for both, higher for other), and “martyr” (negative for self, positive for other, coinciding with the Prisoner’s Dilemma).

  2. Strict means that indifferences are ruled out, ordinal means that games are considered equivalent up to monotone transformations of the payoffs.

  3. Note that while the classifications are based on the same logic of payoff differences, the gain functions of [23] and the dilemma strengths of [37] have the opposite sign; the former measuring the payoff advantage of “cooperation,” the latter that of “defection.” Our \(\varDelta _i\)s measure the payoff advantage of matching the opponent’s strategy and are thus insensitive of the labels of the strategies.

  4. In non-cooperative game theory, a player i’s strategy, say \(s_i\), dominates another strategy, \(s'_i\) of the player if playing \(s_i\) always gives a higher payoff to i independently of what the opposition plays. A strategy \(s_i\) is dominant if it dominates every other strategy of player i and dominated if there exists another strategy of i that dominates it.

  5. The authors of [9] do not name the name classes, whereas [16] calls an Anti-Coordination game a “Leader game,” following [26]. We argue that this classification is not necessarily a good fit for evolutionary games. In behavioral applications of non-cooperative games, games may be considered equivalent up to a relabeling of one or both players’ strategies. Coordination games and Anti-Coordination games are equivalent up to a relabeling of only one player’s strategies; [26] calls their “Leader game” a variant of the “Battle of the Sexes” game, an asymmetric Coordination game. In evolutionary games with no relabeling allowed, “Leader games” and “Hero games” are always Anti-Coordination games, whereas symmetric Coordination games are not covered in this classification, as [26] considers them to be trivial. Citing the nomenclature of [8, 16] correctly calls a harmony game “Deadlock,” but this topological classification does not distinguish between “Prisoner’s Dilemma” and “Deadlock.”

  6. Note that because the labeling of the quadrants is made in terms of the growth game \(\Pi \), interaction games in H or PD quadrant may not be Harmony or Prisoner’s Dilemma games. For example, a game \(\Gamma (c)\) where type 1 is dominated is in the PD quadrant as type 1 is growth-optimal without treatment, but \(\gamma _{11}(c)<\gamma _{22}(c)\) may obtain, in which case \(\Gamma (c)\) is not itself be a true Prisoner’s Dilemma game. Similarly, a game \(\Gamma (c)\) in the H quadrant is not a true Harmony game itself unless \(\gamma _{11}(c)>\gamma _{22}(c)\) holds. Interaction games in the AC or the C quadrant, however, are always indeed Anti-Coordination and Coordination games, respectively.

  7. Note: linearity (or convexity/concavity) serves as only an approximation of outcomes due to the fact that tumors may be initially far from equilibrium growth dynamics (see Sect. 4 for more).

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Acknowledgements

Péter Bayer acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (Investissements d’Avenir, grant ANR-17-EURE-0010) and from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 789111—ERC EvolvingEconomics to Ingela Alger). Jeffrey West acknowledges funding from the Center of Excellence for Evolutionary Therapy at Moffitt Cancer Center.

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  1. Toulouse School of Economics and Institute for Advance Study in Toulouse, 1 Esplanade de l’Université, 31080, Toulouse, France

    Péter Bayer

  2. Department of Integrated Mathematical Oncology, Moffitt Cancer Center, 12902 USF Magnolia Drive, Tampa, FL, 33612, USA

    Jeffrey West

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This article is part of the topical collection “Evolutionary Games and Applications” edited by Christian Hilbe, Maria Kleshnina and Kateřina Staňková.

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Bayer, P., West, J. Games and the Treatment Convexity of Cancer. Dyn Games Appl 13, 1088–1105 (2023). https://doi.org/10.1007/s13235-023-00520-z

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