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Review
. 2023 May 8;378(1876):20210495.
doi: 10.1098/rstb.2021.0495. Epub 2023 Mar 20.

Stackelberg evolutionary game theory: how to manage evolving systems

Alexander Stein  1 , Monica Salvioli  2 , Hasti Garjani  3 , Johan Dubbeldam  3 , Yannick Viossat  4 , Joel S Brown  5 , Kateřina Staňková  2
Affiliations
Review

Stackelberg evolutionary game theory: how to manage evolving systems

Alexander Stein et al. Philos Trans R Soc Lond B Biol Sci. .

Abstract

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'.

Keywords: Darwinian dynamics; cancer evolution; evolutionary game theory; evolutionary rescue; fisheries management; optimization.

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Conflict of interest statement

We declare we have no competing interests.

Figures

Figure 1.
Figure 1.
Illustration of the Stackelberg evolutionary game. It combines two types of games: (i) the leader–follower (Stackelberg) game between the rational leader and evolutionary followers, and (ii) the evolutionary game between the followers. The evolutionary game is defined by the fitness-generating function G(v,u,x,m), which determines the eco-evolutionary dynamics of the followers (§2). In the leader–follower game, the rational leader chooses their strategy m, with the goal to optimize their objective function Q(m,u,x) (§3). The Stackelberg strategy of the leader anticipates the eco-evolutionary response (x, u), whereas the Nash strategy anticipates the ecological response x only. (Online version in colour.)
Figure 2.
Figure 2.
(a) The best response curve (ESS) of the fish (bold solid line) and the best response curve of the fisheries manager (bold dashed line). The Nash equilibrium lies at the intersection of the fish’s ESS curve and the best response curve of the manager, while the Stackelberg equilibrium lies on the ESS curve of the fish but not on the best response curve of the manager. This is because the latter one is obtained by maximizing the profit over the best response of the followers. With the Stackelberg approach, the manager adopts a lower harvesting effort, which leads to bigger fish size. (b) The effect of the harvesting effort on the profit for the ecologically enlightened strategy (Nash) and evolutionarily enlightened strategy (Stackelberg). The ecologically enlightened manager considers the size of fish at maturation as fixed (u = uN) and selects the harvesting rate that maximizes the profit with this in mind (grey dotted curve). The evolutionarily enlightened manager assumes that the size of fish at maturation is the ESS (u*(m)) and selects a harvesting rate that maximizes the profit accordingly (red curve). The evolutionarily enlightened approach leads to higher profits with a lower harvesting rate than the ecologically enlightened one. Adapted from [24]. (Online version in colour.)
Figure 3.
Figure 3.
Bifurcation of an ESS of the fish with respect to harvesting rate. (a) G-function at the eco-evolutionary equilibrium for fixed m. At m = 0.25 < mc, there is one ESS (u0∗), while for m = 0.75 > mc, there are two ESS (u±∗). (b) A pitchfork bifurcation in ESS at mc ≈ 0.45. The leader’s best response curve is depicted in dotted orange. Parameter values: r = 1.0, Kmax = 10 000, σK = 0.55, σH = 0.50 and c = 500. (Online version in colour.)
Figure 4.
Figure 4.
The outcomes of the maximum tolerable dose (MTD), ecologically enlightened (Nash) strategy and evolutionarily enlightened (Stackelberg) strategy of the physician, when playing an SEG against cancer: the yellow and red/cross-hatched areas represent tumour stabilization (0 < x* ≤ δK) and progression (x* > δK) regions, respectively [42]. (a) The Nash and Stackelberg outcomes differ when Q defined by (5.3) is an explicit function of u. (b) The Nash and Stackelberg outcomes coincide when c2 = 0. Parameterization: δ = 0.7, rmax = 0.45, g = 0.8, K = 10 000, d = 0.01, k = 2, b = 10, αSS = αRR = 1, αSR = 0.1, αRS = 0.9, σS = 0, σR = 1, Qmax = 1; (a) c1 = 0.54, c2 = 0.21, c3 = 0.25, (b) c1 = 0.68, c2 = 0, c3 = 0.32. (Online version in colour.)
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