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Evolutionary Game Theory

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Handbook of Dynamic Game Theory

Abstract

Evolutionary game theory developed as a means to predict the expected distribution of individual behaviors in a biological system with a single species that evolves under natural selection. It has long since expanded beyond its biological roots and its initial emphasis on models based on symmetric games with a finite set of pure strategies where payoffs result from random one-time interactions between pairs of individuals (i.e., on matrix games). The theory has been extended in many directions (including nonrandom, multiplayer, or asymmetric interactions and games with continuous strategy (or trait) spaces) and has become increasingly important for analyzing human and/or social behavior as well. This chapter initially summarizes features of matrix games before showing how the theory changes when the two-player game has a continuum of traits or interactions become asymmetric. Its focus is on the connection between static game-theoretic solution concepts (e.g., ESS, CSS, NIS) and stable evolutionary outcomes for deterministic evolutionary game dynamics (e.g., the replicator equation, adaptive dynamics).

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Notes

  1. 1.

    In the basic biological model for evolutionary games, individuals are assumed to engage in random pairwise interactions. Moreover, the population is assumed to be large enough that an individual’s fitness (i.e., reproductive success) \(\pi (p,\hat{p})\) is the expected payoff of p if the average strategy in the population is \(\hat{p}\). In these circumstances, it is often stated that the population is effectively infinite in that there are no effects due to finite population size. Such stochastic effects are discussed briefly in the final section.

  2. 2.

    Clearly, the unit interval [0, 1] is (forward) invariant under the dynamics (3) (i.e., if \(\varepsilon (t)\) is the unique solution of (3) with initial value \(\varepsilon (0) \in [0, 1]\), then \(\varepsilon (t) \in [0, 1]\) for all t ≥ 0). The rest point \(\varepsilon = 0\) is (Lyapunov) stable if, for every neighborhood U of 0 relative to [0, 1], there exists a neighborhood V of 0 such that \(\varepsilon (t) \in U\) for all t ≥ 0 if \(\varepsilon (0) \in V \cap [0, 1]\). It is attracting if, for some neighborhood U of 0 relative to [0, 1], \(\varepsilon (t)\) converges to 0 whenever \(\varepsilon (0) \in U\). It is (locally) asymptotically stable if it is both stable and attracting. Throughout the chapter, dynamic stability is equated to local asymptotic stability.

  3. 3.

    The approach of Taylor and Jonker (1978) also relies on the population being large enough (or effectively infinite) so that n i and p i are considered to be continuous variables.

  4. 4.

    With \(N \equiv \sum _{j=1}^{m}n_{j}\) the total population size,

    $$\displaystyle\begin{array}{rcl} \dot{p}_{i}& =& \frac{\dot{n}_{i}N - n_{i}\sum _{j=1}^{m}\dot{n}_{j}} {N^{2}} {}\\ & =& \frac{n_{i}\pi (e_{i},p) - p_{i}\sum _{j=1}^{m}n_{j}\pi (e_{j},p)} {N} {}\\ & =& p_{i}\pi (e_{i},p) - p_{i}\sum _{j=1}^{m}p_{ j}\pi (e_{j},p) {}\\ & =& p_{i}\left (\pi (e_{i},p) -\pi (p,p)\right ) {}\\ \end{array}$$

    for i = 1, …, m. This is the replicator equation (4) in the main text. Since \(\dot{p}_{i} = 0\) when p i  = 0 and \(\sum _{1=1}^{m}\dot{p}_{i} =\pi (p,p) -\pi (p,p) = 0\) when \(p \in \Delta ^{m}\), the interior of \(\Delta ^{m}\) is invariant as well as all its (sub)faces under (4). Since \(\Delta ^{m}\) is compact, there is a unique solution of (4) for all t ≥ 0 for a given initial population state \(p(0) \in \Delta ^{m}\). That is, \(\Delta ^{m}\) is forward invariant under (4).

  5. 5.

    The proof of this equivalence relies on the compactness of \(\Delta ^{m}\) and the bilinearity of the payoff function π(p, q) as shown by Hofbauer and Sigmund (1998).

  6. 6.

    Under the replicator equation, \(\dot{V }(p) =\sum _{ i=1}^{m}p_{i}^{{\ast}}p_{i}^{p_{i}^{{\ast}}-1}\dot{p}_{i}\prod _{\{j\mid j\neq i,p_{ j}^{{\ast}}\neq 0\}}p_{j}^{p_{j}^{{\ast}}} =\sum _{ i=1}^{m}p_{i}^{{\ast}}\prod _{j}p_{j}^{p_{j}^{{\ast}}}(\pi (e_{i},p)-\pi (p,p)) = V (p)(\pi (p^{{\ast}},p)-\pi (p,p)) > 0\) for all \(p \in \Delta ^{m}\) sufficiently close but not equal to an ESS p ∗. Since V (p) is a strict local Lyapunov function, p ∗ is locally asymptotically stable. Global stability (i.e., in addition to local asymptotic stability, all interior trajectories of (4) converge to p ∗) in part (c) follows from global superiority (i.e., π(p ∗, p) > π(p, p) for all p ≠ p ∗) in this case.

  7. 7.

    This intuition is correct for small constants c greater than 1. However, for large c, the discrete-time trajectories approach the continuous-time ones and so \(p^{{\ast}} = (1/3, 1/3, 1/3)\) will be asymptotically stable under (6) when \(\varepsilon > 0\).

  8. 8.

    On the other hand, an ESS remains locally asymptotically stable for all selection dynamics that are uniformly monotone according to Cressman (2003) (see also Sandholm 2010).

  9. 9.

    Since the best response dynamics is a differential inclusion, it is sometimes written as \(\dot{p} \in BR(p) - p\), and there may be more than one solution to an initial value problem (Hofbauer and Sigmund 2003). Due to this, it is difficult to provide an explicit formula for the vector field corresponding to a particular solution of (8) when BR(p) is multivalued. Since such complications are beyond the scope of this chapter, the vector field is only given when BR(p) is a single point for the examples in this section (see, e.g., the formula in (10)).

  10. 10.

    Broom and Krivan (chapter “Biology (Application of Evolutionary Game Theory),” this volume) give more details of this result and use it to produce analytic expressions for the IFD in several important biological models. They also generalize the IFD concept when the assumptions underlying the analysis of Fretwell and Lucas (1969) are altered. Here, we concentrate on the dynamic stability properties of the IFD in its original setting.

  11. 11.

    The results summarized in this example do not depend on linearity as shown in Krivan et al. (2008) (see also Cressman and Tao 2014).

  12. 12.

    To see that the habitat selection game is a potential game, take \(F(p) \equiv \sum _{i=1}^{H}\int _{0}^{p_{i}}\pi (e_{i},u_{i})du_{i}\). Then \(\frac{\partial F(p)} {\partial p_{i}} =\pi (e_{i},p_{i})\). If patch payoff decreases as a function of patch density, the habitat selection game is a strictly stable game (i.e., \(\sum \left (p_{i} - q_{i}\right )\left (\pi (e_{i},p) -\pi (e_{i},q)\right ) < 0\) for all p ≠ q in \(\Delta ^{H}\)). This follows from the fact that F(p) is strictly concave since \(\frac{\partial ^{2}F(p)} {\partial p_{i}\partial p_{j}} = \left \{\begin{array}{cc} \frac{\partial \pi (e_{i},p_{i})} {\partial p_{i}} &\ \text{ if }i = j \\ 0 & \ \text{ if }i\neq j \end{array} \right.\) and \(\frac{\partial \pi (e_{i},p_{i})} {\partial p_{i}} < 0\). Global asymptotic stability of p ∗ for any dynamics (9) that satisfies the conditions of Theorem 3 follows from the fact that \(W(p) \equiv \max _{1\leq i\leq H}\pi (e_{i},p)\) is a (decreasing) Lyapunov function (Krivan et al. 2008).

  13. 13.

    Specifically, \(\pi (v, 0) = av^{2}\, < 0 =\pi (0, 0)\) for all v ≠ 0 if and only if a < 0.

  14. 14.

    Much of the literature on evolutionary games for continuous trait space uses the term ESS to denote a strategy that is uninvadable in this sense. However, this usage is not universal. Since ESS has in fact several possible connotations for games with continuous trait space (Apaloo et al. 2009), we prefer to use the more neutral game-theoretic term of strict NE in these circumstances when the game has a continuous trait space.

  15. 15.

    Typically, δ > 0 depends on u (e.g., δ < ∣u − u ∗∣). Sometimes the assumption that u ∗ is a strict NE is relaxed to the condition of being a neighborhood (or local) strict NE (i.e., for some \(\varepsilon > 0\), π(v, u) < π(u, u) for all \(0 < \mid v - u\mid <\varepsilon\)).

  16. 16.

    In particular, adaptive dynamics is not applied to examples such as the War of Attrition, the original example of a symmetric evolutionary game with a continuous trait space (Maynard Smith 1974, 1982; Broom and Krivan, (chapter “Biology (Application of Evolutionary Game Theory),” this volume), which have discontinuous payoff functions. In fact, by allowing invading strategies to be far away or individuals to play mixed strategies, it is shown in these references that the evolutionary outcome for the War of Attrition is a continuous distribution over the interval S. Distributions also play a central role in the following section. Note that, in Sect. 3, subscripts on π denote partial derivatives. For instance, the derivative of π with respect to the first argument is denoted by π 1 in (13). For the asymmetric games of Sect. 4, π 1 and π 2 denote the payoffs to player one and to player two, respectively.

  17. 17.

    This concept was first called m-stability by Taylor (1989) and then convergence stability by Christiansen (1991), the latter becoming standard usage. It is straightforward to show that the original definition is equivalent to Definition 2 (c).

  18. 18.

    This general characterization of a CSS ignores threshold cases where π 11(u ∗, u ∗) = 0 or \(\pi _{11}(u^{{\ast}},u^{{\ast}}) +\pi _{12}(u^{{\ast}},u^{{\ast}}) = 0\). We assume throughout Sect. 3 that these degenerate situations do not arise for our payoff functions π(v, u).

  19. 19.

    We particularly object to this phrase since it causes great confusion with the ESS concept. We prefer calling these evolutionary branching points.

  20. 20.

    Here, stability means that \(\delta _{u^{{\ast}}}\) is neighborhood attracting (i.e., for any initial distribution P 0 with support sufficiently close to u ∗ and with P 0(u ∗) > 0, P t converges to \(\delta _{u^{{\ast}}}\) in the weak topology). As explained in Cressman (2011) (see also Cressman et al. 2006), one cannot assert that \(\delta _{u^{{\ast}}}\) is locally asymptotically stable under the replicator equation with respect to the weak topology or consider initial distributions with P 0(u ∗) = 0. The support of P is the closed set given by \(\{u \in S\mid P(\{y: \mid y - u\mid <\varepsilon \}) > 0\) for all \(\varepsilon > 0\}\).

  21. 21.

    This result is often stated as ESS + NIS implies CSS (e.g., Apaloo 1997; Apaloo et al. 2009). Furthermore, an ESS + NIS is often denoted ESNIS in this literature.

  22. 22.

    In fact, \(P^{{\ast}} = -\dfrac{(a + b)\alpha + a\beta } {b(\beta -\alpha )} \delta _{\beta } + \dfrac{(a + b)\beta + a\alpha } {b(\beta -\alpha )} \delta _{\alpha }\) since this dimorphism satisfies π(P ∗, Q) > π(Q, Q) for all distributions Q not equal to P ∗ (i.e., P ∗ is globally superior by the natural extension of Definition 3 (b) to non-monomorphic P ∗ as developed by Cressman and Hofbauer 2005).

  23. 23.

    Although Darwinian dynamics can also be based solely on changing strategy frequency with population size fixed (Vincent and Brown 2005), the theory developed here considers changing population size combined with strategy evolution.

  24. 24.

    As in Sect. 3.1, we ignore threshold cases. Here, we assume that a and 2a + b are both nonzero.

  25. 25.

    Technically, at this rest point, \(\frac{du_{1}} {dt} = 2ka\beta > 0\) and \(\frac{du_{2}} {dt} = -2ka\beta < 0\) are not 0. However, their sign (positive and negative, respectively) means that the dimorphism strategies would evolve past the endpoints of S, which is impossible given the constraint on the strategy space.

    These signs mean that local asymptotic stability follows from the linearization of the ecological dynamics at the rest point. It is straightforward to confirm this 2 × 2 Jacobian matrix has negative trace and positive determinant (since a > 0 and b < 0), implying both eigenvalues have negative real part.

    The method can be generalized to show that, if S = [α, β] with α < 0 < β, the stable evolutionary outcome predicted by Darwinian dynamics is now \(u_{1}^{{\ast}} =\beta,u_{2}^{{\ast}} =\alpha\) with \(n_{1}^{{\ast}} = (a\alpha \beta - 1)\frac{(a+b)\alpha +a\beta } {b(\beta -\alpha )},n_{2}^{{\ast}} = (1 - a\alpha \beta )\frac{(a+b)\beta +a\alpha } {b(\beta -\alpha )}\) both positive under our assumption that a > 0 and 2a + b < 0. In fact, this is the same stable dimorphism (up to the population size factor 1 − a α β) given by the replicator equation of Sect. 3.2 (see Remark 2).

  26. 26.

    Many others (e.g., Barabás and Meszéna 2009; Barabás et al. 2012, 2013; Bulmer 1974; D’Andrea et al. 2013; Gyllenberg and Meszéna 2005; Meszéna et al. 2006; Parvinen and Meszéna 2009; Sasaki 1997; Sasaki and Ellner 1995; Szabó and Meszéna 2006) have examined the general LV competition model.

  27. 27.

    Specifically, the Gaussian distribution is given by

    $$\displaystyle{P^{{\ast}}(u) = \frac{K_{m}\sigma _{k}} {\sigma _{a}\sqrt{2\pi (\sigma _{k }^{2 } -\sigma _{ a }^{2 })}}\exp (-u^{2}/(2(\sigma _{ k}^{2} -\sigma _{a}^{2}))).}$$
  28. 28.

    Cressman et al. (2016) also examined what happens when there is a baseline competition between all individuals no matter how distant their trait values are. This leads to a stable evolutionary outcome supported on finitely many strategies as well. That is, modifications of the basic LV competition model tend to break up its game-theoretic solution P ∗(u) with full support to a stable evolutionary outcome supported on finitely many traits, a result consistent with the general theory developed by Barabás et al. (2012) (see also Gyllenberg and Meszéna 2005).

  29. 29.

    Covariance matrices C 1 are assumed to be positive definite (i.e., for all nonzero u  ∈ R n, u ⋅ C 1 u > 0) and symmetric. Similarly, a matrix A is negative definite if, for all nonzero u ∈ R n, u ⋅ Au < 0.

  30. 30.

    A similar covariance approach was applied by Hines (1980) (see also Cressman and Hines 1984) for matrix games to show that p ∗ ∈ int\((\Delta ^{m})\) is an ESS if and only if it is locally asymptotically stable with respect to the replicator equation (4) adjusted to include an arbitrary mutation process.

  31. 31.

    These two games are described more fully in Broom and Krivan’s (chapter “Biology (Application of Evolutionary Game Theory),” this volume).

  32. 32.

    To see this result first proven by Selten (1980), take (p, q) = (p, q ∗). Then (25) implies p ⋅ Aq ∗ < p ∗⋅ Aq ∗ or q ∗⋅ Bp ∗ < q ∗⋅ Bp ∗ for all p ≠ p ∗. Thus, p ⋅ Aq ∗ < p ∗⋅ Aq ∗ for all p ≠ p ∗. The same method can now be applied to (p, q) = (p ∗, q).

  33. 33.

    This game is also considered briefly by Broom and Krivan (chapter “Biology (Application of Evolutionary Game Theory),” this volume). There the model parameters are given biological interpretations (e.g., M is the fixed total population size of species one and K 1 is its carrying capacity in patch one, etc.). Linearity then corresponds to Lotka-Volterra type interactions. As in Example 1 of Sect. 2.4, our analysis again concentrates on the dynamic stability of the evolutionary outcomes.

  34. 34.

    When the outcome is a single node, this is understood by saying the outcome is the payoff pair at this node.

  35. 35.

    This is clear for the replicator equation (27). For this example with two strategies for each player, it continues to hold for all other game dynamics that satisfy the basic assumption that the frequency of one strategy increases if and only if its payoff is higher than that of the player’s other strategy.

  36. 36.

    For Example 3, this is either (2, 2) or (1, 4).

  37. 37.

    This is the obvious extension to bimatrix games of the best response dynamics (8) for symmetric (matrix) games.

  38. 38.

    An extensive form game that is not of perfect information has at least one player “information set” containing more than one decision point of this player. This player must take the same action at all these decision points. Matrix games then correspond to symmetric extensive form games (Selten 1983) where there is a bijection from the information sets of player 1 to those of player 2. Bimatrix games can also be represented in extensive form.

  39. 39.

    These equivalences are also shown by Leimar (2009) who called the concept strong convergence stability.

  40. 40.

    In (30), we assume payoff linearity in the distributions P and Q. For example, the expected payoff to \(u^{{\prime}}\) in a random interaction is \(\pi (u^{{\prime}}; P,Q) \equiv \int _{S}\int _{T}\pi _{1}(u^{{\prime}}; u,v)Q(dv)P(du)\) where P (Q) is the probability measure on S (T) corresponding to the current distribution of the population one’s (two’s) strategies. Furthermore, \(\pi (P; P,Q) \equiv \int _{S}\pi (u^{{\prime}}; P,Q)P(du^{{\prime}})\), etc.

  41. 41.

    Note that (u∗ ,v ∗ ) is neighborhood attracting if (P t ,Q t ) converges to \((\delta _{u^{{\ast}}},\delta _{v^{{\ast}}})\) in the weak topology whenever the support of (P 0 ,Q 0 ) is sufficiently close to (u ∗ ,v ∗ ) and \((P_{0},Q_{0}) \in \Delta (S) \times \Delta (T)\) satisfies P 0 ({u ∗ })Q 0 ({v ∗ }) > 0.

  42. 42.

    These deterministic dynamics all rely on the assumption that the population size is large enough (sometimes stated as “effectively infinite”) so that changes in strategy frequency can be given through the payoff function (i.e., through the strategy’s expected payoff in a random interaction).

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Acknowledgements

The authors thank Abdel Halloway for his assistance with Fig. 4. This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 690817.

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Authors and Affiliations

  1. Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, Canada

    Ross Cressman

  2. Department of Mathematics, Statistics and Computer Science, St Francis Xavier University, Antigonish, NS, Canada

    Joe Apaloo

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  1. Ross Cressman
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  2. Joe Apaloo
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Correspondence to Ross Cressman .

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  1. Center for Advanced Study, University of Illinois Urbana-Champaign, Urbana, Illinois, USA

    Tamer Basar

  2. GERAD, HEC Montréal, Montreal, Québec, Canada

    Georges Zaccour

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  1. Coordinated Science Laboratory, University of Illinois, 1308 West Main, 61801, Urbana, Illinois, USA

    Tamer Basar

  2. Département de sciences de la décision, HEC Montreal, Montreal, Canada

    Georges Zaccour

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Cressman, R., Apaloo, J. (2016). Evolutionary Game Theory. In: Basar, T., Zaccour, G. (eds) Handbook of Dynamic Game Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-27335-8_6-1

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