Abstract
Do evolutionary processes lead economic or biological agents to behave as if they were rational? To test this idea, many authors examined whether evolutionary game dynamics eliminate strictly dominated strategies. We survey, unify, and fill some gaps in this literature in the case of monotonic dynamics: a class of selection dynamics in which the growth rates of the pure strategies are ordered in accordance with their payoffs. We also survey results for other dynamics.
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Notes
Regular selection dynamics are often defined (e.g., Ritzberger and Weibull 1995) as dynamics of type \(\dot{x}_i(t)=x_i(t) \phi _i(\mathbf {x}(t),\mathbf {y}(t))\) with \(\sum _{k \in I} x_k \phi _k(\mathbf {x},\mathbf {y})=0\) for all \((\mathbf {x}, \mathbf {y})\). This is equivalent to (1): the above dynamics are of type (1) with \(g_i= \phi _i\); and dynamics (1) are of the above type with \(\phi _i=g_i - \sum _k x_k g_k\).
The quantity \(e^{-H_{\mathbf {q}}(\mathbf {x})}\), which is proportional to \(\prod _{i \in I} x_i^{q_i}\), may be seen as an upper bound on the “frequency” of strategy \(\mathbf {q}\) in the following sense: assume that players may use mixed strategies and that a proportion \(\alpha _{\mathbf {q}}(t)\) uses strategy \(\mathbf {q}\) at time \(t\); let \(\mathbf {p}(t)\) be the mean strategy in the remainder of the population, so that the mean strategy in the whole population is \(\mathbf {x}=\alpha _q \mathbf {q}+ (1-\alpha _q) \mathbf {p}\). It is easily shown that \(e^{-H_{\mathbf {q}}(\mathbf {x})} \ge \alpha _{\mathbf {q}}\), with equality if and only if the supports of \(\mathbf {q}\) and \(\mathbf {p}\) are disjoint. Thus, if strategy \(\mathbf {q}\) is eliminated, then the subpopulation playing strategy \(\mathbf {q}\) gets extinct: \(\alpha _{\mathbf {q}}(t) \rightarrow 0\) as \(t \rightarrow +\infty \). Moreover, if one observes only \(\mathbf {x}(t)\) and does not know which subpopulations are initially present (that is, which mixed strategies are played), then one cannot be sure that \(\alpha _{\mathbf {q}}(t) \rightarrow 0\) unless \(\mathbf {q}\) is eliminated.
Samuelson and Zhang (1992) showed that in bimatrix games, for any aggregate monotonic dynamics, there exists a positive speed function \(\lambda \) such that \(\dot{\mathbf {x}}(t)=\lambda (\mathbf {x}(t),\mathbf {y}(t)){\dot{\mathbf {x}}}^{\textit{REP}}\), where \({\dot{x}}^{\textit{REP}}_i=x_i^{{\textit{REP}}}(U_{i}(\mathbf {y})-\sum _{k} x_{k}^{{\textit{REP}}} U_k(\mathbf {y}))\). For single-population dynamics (\(\mathbf {y}(t)=\mathbf {x}(t)\) \(\forall t\)), this implies that all aggregate monotonic dynamics have the same orbits as the replicator dynamics; for multi-population dynamics, this need not be so, because the speed function is population specific.
Sandholm’s imitative dynamics are defined by positive functions \(r_{ik}\) that represent the rate at which an agent playing strategy \(k\) switches to strategy \(i\) upon meeting an agent playing strategy \(i\). In our framework, they take the form:
$$\begin{aligned} \dot{x}_i= x_i \left[ \sum _{k \in I} x_k\left( r_{ik}(\mathbf {x}, \mathbf {y}) - r_{ki}(\mathbf {x}, \mathbf {y})\right) \right] \end{aligned}$$with \(U_i(\mathbf {y}) \ge U_j(\mathbf {y}) \Leftrightarrow r_{ik}(\mathbf {x}, \mathbf {y}) - r_{ki}(\mathbf {x}, \mathbf {y}) \ge r_{jk}(\mathbf {x}, \mathbf {y}) - r_{kj}(\mathbf {x}, \mathbf {y})\) for all pure strategies \(i\), \(j\), \(k\). These dynamics are easily seen to be monotonic. Conversely, monotonic dynamics whose growth rate functions \(g_i\) are positive—which we may always assume—may be written as imitative dynamics with switching rate functions \(r_{ik}= g_i\) for all \(k\). Note that the above definition of “imitative dynamics” is somewhat misleading in that dynamics based on imitation need not satisfy it; see, e.g., footnote 26.
If the payoff matrix of player 1 is \(\left( \begin{array}{ll} 1 &{} 1 \\ 1 &{} 0 \end{array}\right) \), then the dynamics \(\dot{x}_1= x_1 x_2; \dot{x}_2= - x_2 x_1\) are monotonic—even aggregate monotonic—in the sense of (5), but no longer monotonic if the implication in (5) is replaced by an equivalence; but for dynamics defined for all games and depending continuously on the game, both definitions coincide. Note that Samuelson and Zhang (1992) define monotonicity with an equivalence but aggregate monotonicity with an implication. This might be a typo since the above dynamics would then be aggregate monotonic but not monotonic!
The analogy with attitudes towards risk is as follows: convex utility functions favor risky lotteries over sure ones in that an expected gain advantage of a risky lottery over a sure one always results in the risky lottery being preferred. Similarly, concave utility functions favor sure lotteries. Linear utility functions are a limit case where a lottery with a higher expected gain is always preferred.
Formally, let \(I_0=\tilde{I}_0=I\) and \(S_0=\tilde{S}_0=\Delta (I)\). Similarly, let \(J_0=\tilde{J}_0=J\) and \(T_0=\tilde{T}_0=\Delta (J)\). Inductively, let \(I_{k+1}\) (resp. \(\tilde{I}_{k+1}\), \(S_{k+1}\), \(\tilde{S}_{k+1}\)) denote the set of strategies in \(I_k\) (resp. \(\tilde{I}_{k}\), \(S_{k}\), \(\tilde{S}_k\)) that are not dominated by any strategy in \(I_k\) (resp. \(\Delta (\tilde{I}_{k})\), \(I \cap S_{k}\), \(\tilde{S}_k\)) when player 2 chooses strategies in \(J_k\) (resp. \(\tilde{J}_k\), \(T_k\), \(\tilde{T}_k\)). Similar definitions apply to player 2.
Solutions of single-population dynamics may be seen as particular cases of solutions of two-population dynamics: the case of a symmetric game with symmetric initial conditions. Thus, as already noted by Hofbauer and Weibull (1996), proving that some single-population dynamics need not eliminate dominated strategies a fortiori implies the same result for two-population dynamics (and also for \(n\)-population dynamics: just add dummy players).
These two mild additional conditions are related to the proof technique, but can probably be dispensed with.
This is a simplification. A more precise argument is that close to a vertex of the simplex corresponding to one of the pure strategies \(R\), \(P\) and \(S\), the average unnormalized growth rate in the population is close to \(f(a)\), which is less than \(f(m + \beta )\) for \(\beta \) small enough, so the share of strategy \(B\) increases. Moreover, within the face \(x_4=0\), the RPS cycle is only locally, not globally attracting, so more subtle arguments are needed. See Hofbauer and Weibull (1996) for details.
This is where we use that \(f'\) is strictly positive, and actually only that \(f'(m)>0\).
To obtain this expression of \(\dot{w}\), note that \((\mathbf {A}\mathbf {x})_i = m + (\mathbf {A}\mathbf {h})_i\), thus \(f((\mathbf {A}\mathbf {x})_i)=f(m) + f'(m) (\mathbf {A}\mathbf {h})_i + o((\mathbf {A}\mathbf {h})_i)\). Using this and \(\sum _{i} h_i=0\), we get: \(\dot{w}= - f'(m) \mathbf {h}\cdot \mathbf {A}\mathbf {h}+ o( || \mathbf {h}||^2)\). A standard computation shows that for the Euclidean norm \(\mathbf {h}\cdot \mathbf {A}\mathbf {h}=[a-(b+c)/2] || \mathbf {h}||^2\), hence the result. This is a variant of Exercise 8.1.1 in Hofbauer and Sigmund (1998).
Björnerstedt et al. (1996) consider an overlapping-generations dynamics which has as limit cases the discrete-time and the continuous-time replicator dynamics, for respectively no overlap and full overlap. For a fixed game, if the degree of overlap is large enough, strictly dominated strategies are eliminated.
The background fitness \(C(t)\) does not affect Eq. (20), and Imhof takes \(C(t)=0\).
Imhof’s Eq. (1.4) corresponds to Cabrales’ Eq. (3) with a single population, a number of Brownian motions equal to the number of pure strategies, and the \(i\)th Brownian motion affecting only player \(i\). That is, with Cabrales’ notation, \(N=1\), \(d=n_i\) and \(\sigma ^i_{\alpha l}= 0\) if \(\alpha \ne l\).
Cabrales also considers a model with an additional mutation term. He shows (Prop. 1B, p. 459) that when the mutation rates are small and symmetric enough (not orders of magnitude apart), then after a long time, the frequency of iteratively dominated strategies is negligible with high probability. We recommend the discussion following Prop. 1B.
In the Associate Editor’s view, the literature on dynamics in games for which the set of pure strategies are continua may be misleading because (quoting): “often these continua are not arbitrary, but subsets of an Euclidean space (with its natural topology). Yet, when strategies are derived from an extensive form, as they ought to be (...), they are typically functions spaces that do not have a “natural” topology.” While we agree that the choice of topology may be an issue (though more for convergence to equilibria than elimination of dominated strategies), it seems to us that some interactions, in particular in biology, are naturally modeled as normal-form games with a pure strategy set that is an interval of \(\mathbb {R}\), and that the papers we mention make sense.
Matsui (1992) original definition is more restrictive: it only allows for piecewise-linear solutions, and such that the times at which the direction of movement changes do not accumulate.
Balkenborg et al. (2013) ( Theorem 3) extend this result as follows: define (p. 173) generalized best-reply correspondences based on pure strategies as correspondences \(\tau \) with the same structural and topological properties as the usual, independent best-reply correspondence \(\beta \) and such that the image of any mixed strategy profile \(x\) contains a best reply to \(x\), that is, \(\tau (x) \cap \beta (x) \ne \emptyset \). An example is the better-reply correspondence used by, e.g., Ritzberger and Weibull (1995). Balkenborg et al. show that for any such correspondence \(\tau \), the generalized best-reply dynamics \(\dot{x} \in \tau (x)- x\) eliminates all pure strategies that are not \(\tau \)-rationalizable, where \(\tau \)-rationalizability is the notion obtained by replacing the best-reply correspondence by \(\tau \) in the definition of rationalizability. For some correspondences \(\tau \), dominated strategies may be \(\tau \)-rationalizable, so this is not a result on dominated strategies per-se, but on generalizations of rationalizability. The proof is similar to the proof of Proposition 3.
Hopkins (1999, p. 92) proves a similar and similarly obvious result on a version of fictitious play.
The game used by Hofbauer and Sandholm is not the mixed extension of a finite game. They argue convincingly that this is a technical trick, that allows to prove survival of a pure strategy dominated by another pure strategy for a large class of dynamics using a single game, but that for any dynamics in this class, survival of pure strategies dominated by a pure strategy should also occur in the mixed extension of a finite game adapted to these dynamics. We share this view.
Survival of pure strategies strictly dominated by other pure strategies also arises with the (innovative) projection dynamics from transportation science (Nagurney and Zhang 1996; Sandholm et al. 2008), but other arguments are needed since they do not satisfy Continuity. Besides, Björnerstedt et al. (1996) give an example of a game and of dynamics such that a pure strategy dominated by all other pure strategies survives for most initial conditions.
In many areas of France, children play a variant of Rock-Paper-Scissors with an additional strategy, “Well”, that beats Rock and Scissors (as they fall into the well), but is beaten by Paper (as it covers the well). In this four strategy game, Rock is weakly dominated by Well, but anecdotical evidence shows that all strategies are still played at high frequencies. Similarly, before writing the payoff matrix, third-year university students in my game theory class typically fail to recognize that Rock is a “bad” strategy. Though this is a story about weakly, not strictly dominated strategies, and maybe learning rather than evolutionary dynamics, this suggests dynamics more akin to Hofbauer and Sandholm’s innovative dynamics than to the replicator dynamics.
Consider dynamics derived from a revision protocol (see Sandholm 2010) such that a revising agent: first, randomly meets \(n\ge 3\) agents from the population; second, chooses one of the \(m \le n\) strategies played by these agents, with equal probability \(1/m\) (and not \(1/n\): this is what advantages rare strategies); third, if this strategy earns more than the one they currently use, adopt it with probability proportional to the payoff difference, as in revision protocols leading to the replicator dynamics. Preliminary results suggest that in game (24), for \(\varepsilon \) small enough, the strictly dominated strategy T survives under these dynamics. Note that, though based on imitation, these dynamics are not imitative in the sense of footnote 6, because the probability according to which a revising agent considers switching to strategy \(i\) is not exactly \(x_i\).
References
Akin, E.: Domination or equilibrium. Math. Biosci. 50, 239–250 (1980)
Alchian, A.: Uncertainty, evolution and economic theory. J. Polit. Econ. 58, 211–222 (1950)
Balkenborg, D., Hofbauer, J., Kuzmics, C.: Refined best reply correspondence and dynamics. Theor. Econ. 8, 165–192 (2013)
Berger, U., Hofbauer, J.: Irrational behavior in the Brown-von Neumann-Nash dynamics. Games Econ. Behav. 56, 1–6 (2006)
Bernheim, D.G.: Rationalizable strategic behavior. Econometrica 52, 1007–1028 (1984)
Björnerstedt, J.: Experimentation, Imitation and Evolutionary Dynamics. Department of Economics, Stockholm University, Stockholm (1995)
Björnerstedt, J., Dufwenberg, M., Norman, P., Weibull, J.: Evolutionary selection dynamics and irrational survivors. In: Albers, W., Güth, W., Hammerstein, P., Moldovanu, B., van Damme, E. (eds.) Understanding Strategic Interaction: Essays in Honor of Reinhard Selten, pp. 128–148. Springer, Berlin (1996)
Cabrales, A.: Stochastic replicator dynamics. Int. Econ. Rev. 41, 451–481 (2000)
Cabrales, A., Sobel, J.: On the limit points of discrete selection dynamics. J. Econ. Theory 57, 407–420 (1992)
Cressman, R.: Stability of the replicator equation with continuous strategy space. Math. Soc. Sci. 50, 127–147 (2005)
Cressman, R., Hofbauer, J.: Measure dynamics on a one- dimensional continuous trait space: theoretical foundations for adaptive dynamics. Theor. Popul. Biol. 67, 47–59 (2005)
Cressman, R., Hofbauer, J., Riedel, F.: Stability of the replicator equation for a single species with a multi-dimensional continuous trait space. J. Theor. Biol. 239, 273–288 (2006)
Dekel, E., Scotchmer, S.: On the evolution of optimizing behavior. J. Econ. Theory 57, 392–406 (1992)
Foster, D., Young, H.P.: Stochastic evolutionary game dynamics. Theor. Popul. Biol. 38, 219–232 (1990)
Franchetti, F., Sandholm, W.H.: An Introduction to Dynamo: diagrams for evolutionary game dynamics. Biol. Theor. 8, 167–178 (2013)
Friedman, D.: Evolutionary games in economics. Econometrica 59, 637–666 (1991)
Friedman, M.: The methodology of positive economics. In: Friedman, M. (ed.) Essays in Positive Economics. University of Chicago Press, Chicago (1953)
Fudenberg, D., Harris, C.: Evolutionary dynamics with aggregate shocks. J. Econ. Theory 57, 420–441 (1992)
Gaunersdorfer, A.: Time averages for heteroclinic attractors. SIAM J. Appl. Math. 52, 1476–1489 (1992)
Gaunersdorfer, A., Hofbauer, J.: Fictitious play, Shapley polygons, and the replicator equation. Games Econ. Behav. 11, 279–303 (1995)
Gilboa, I., Matsui, A.: Social stability and equilibrium. Econometrica 59, 859–867 (1991)
Heifetz, A., Shannon, C., Spiegel, T.: The dynamic evolution of preferences. Econ. Theory 32, 251–286 (2007a)
Heifetz, A., Shannon, C., Spiegel, Y.: What to maximize if you must. J. Econ. Theory 133, 31–57 (2007b)
Hofbauer, J.: A difference equation model for the hypercycle. SIAM J. Appl. Math. 44, 772–792 (1984)
Hofbauer, J., Imhof, L.A.: Time averages, recurrence and transience in the stochastic replicator dynamics. Ann. Appl. Probab. 19, 1347–1368 (2009)
Hofbauer, J., Schlag, K.H.: Sophisticated imitation in cyclic games. J. Evol. Econ. 10, 523–543 (2000)
Hofbauer, J., Sandholm, W.H.: Survival of dominated strategies under evolutionary dynamics. Theor. Econ. 6, 341–377 (2011)
Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics. Cambridge Univ. Press, Cambridge (1998)
Hofbauer, J., Sorin, S., Viossat, Y.: Time average replicator and best-reply dynamics. Math. Oper. Res. 34, 263–269 (2009)
Hofbauer, J., Weibull, J.W.: Evolutionary selection against dominated strategies. J. Econ. Theory 71, 558–573 (1996)
Hopkins, E.: Learning, matching, and aggregation. Games Econ. Behav. 26, 79–110 (1999)
Imhof, L.A.: The long-run behavior of the stochastic replicator dynamics. Ann. Appl. Probab. 15, 1019–1045 (2005)
Jouini, E., Napp, C., Viossat, Y.: Evolutionary beliefs and financial markets. Rev. Financ. 17, 727–766 (2013)
Laraki, R., Mertikopoulos, P.: Higher order game dynamics. J. Econ. Theory. 148, 2666–2695 (2013)
Matsui, A.: Best-response dynamics and socially stable strategies. J. Econ. Theory 57, 343–362 (1992)
Mertikopoulos, P., Moustakas, A.L.: The emergence of rational behavior in the presence of stochastic perturbations. Ann. Appl. Probab. 20, 1359–1388 (2010)
Nachbar, J.: Evolutionary selection dynamics in games: convergence and limit properties. Int. J. Game Theory 19, 59–89 (1990)
Nagurney, A., Zhang, D.: Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer, Dordrecht (1996)
Pearce, D.G.: Rationalizable strategic behavior and the problem of perfection. Econometrica 52, 1029–1050 (1984)
Ritzberger, K., Weibull, J.W.: Evolutionary selection in normal form games. Econometrica 63, 1371–1399 (1995)
Samuelson, L., Zhang, J.: Evolutionary stability in asymmetric games. J. Econ. Theory 57, 363–391 (1992)
Sandholm, W.H.: Population Games and Evolutionary Dynamics. MIT Press, Cambridge (2010)
Sandholm, W.H., Dokumaci, E., Lahkar, R.: The projection dynamic and the replicator dynamic. Games Econ. Behav. 64, 666–683 (2008)
Sandholm, W.H., Dokumaci, E., Franchetti, F.: Dynamo: Diagrams for Evolutionary Game Dynamics, Software. http://www.ssc.wisc.edu/~whs/dynamo (2012). Accessed 16 Nov 2014
Smith, M.J.: The stability of a dynamic model of traffic assignment—an application of a method of Lyapunov. Transp. Sci. 18, 245–252 (1984)
Taylor, P.D., Jonker, L.: Evolutionary stable strategies and game dynamics. Math. Biosci. 40, 145–156 (1978)
Viossat, Y.: The replicator dynamics does not lead to correlated equilibria. Games Econ. Behav. 59, 397–407 (2007)
Viossat, Y.: Evolutionary dynamics may eliminate all strategies used in correlated equilibria. Math. Soc. Sci. 56, 27–43 (2008)
Viossat, Y.: Deterministic monotone dynamics and dominated strategies, working paper, hal-00636620. http://hal.archives-ouvertes.fr/hal-00636620/en/ (2011). Accessed 16 Nov 2014
Weibull, J.: The “as-if” approach to game theory: three positive results and four obstacles. Eur. Econ. Rev. 38, 868–881 (1994)
Weissing, F.: Evolutionary stability and dynamic stability in a class of evolutionary normal form games. In: Selten, R. (ed.) Game Equilibrium Models: I. Evolution and Game Dynamics, pp. 29–97. Springer, Berlin (1991)
Acknowledgments
Previous version: 2011 (under the name “Deterministic monotone dynamics and dominated strategies”). I am deeply indebted to Larry Samuelson, and also to Francisco Franchetti, Josef Hofbauer, Panayotis Mertikopoulos, Bill Sandholm, Sylvain Sorin, Jörgen Weibull, seminar audiences, the Associate Editor, and anonymous referees. All errors and shortcomings are mine. The support of the Agence Nationale de la Recherche (GAGA project: ANR-13-JS01-0004-01) is gratefully acknowledged.
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Viossat, Y. Evolutionary dynamics and dominated strategies. Econ Theory Bull 3, 91–113 (2015). https://doi.org/10.1007/s40505-014-0062-4
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DOI: https://doi.org/10.1007/s40505-014-0062-4
Keywords
- As-if rationality
- Evolutionary games
- Dominated strategies
- Replicator dynamics
- Monotonic dynamics
- Innovative dynamics