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On the Evolution of Continuous Types Under Replicator and Gradient Dynamics: Two Examples

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Abstract

This paper illustrates techniques for assessing the dynamic stability of games where a continuum of types might be present by re-analyzing two models under incomplete information, the Lohman et al. (Unpublished manuscript, 2001) public goods game and the Kopel et al. (J Econ Dyn Control 48:394–409, 2014) Cournot duopoly game. The evolution of continuous types follows either replicator dynamics Oechssler and Riedel (Econ Theory 17:141–162, 2001; J Econ Theory 107:223–252 2002) or gradient dynamics Friedman and Ostrov (J Math Econ 46:691–707, 2010; J Econ Theory 148:743–777, 2013). The techniques rely on a system of partial differential equations. Numerical solutions obtained through replicator and gradient dynamics highlight the differences and the similarities that arise under both approaches. In the public goods game, the dynamic system affects the stationary distribution of types while in the Cournot duopoly model, the types evolve to a single mass point regardless of the dynamics used. Lastly, these techniques allow us to endogenize the distribution of player types.

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Notes

  1. Similar ideas have been applied to the evolution of preferences in a discrete case in [21].

  2. Related to my work, [7] apply replicator dynamics to the evolution of continuous types. Unfortunately, I cannot apply gradient dynamics to [7] since the gradient term is invariant to the types.

  3. Königstein and Müller [13] also study a variant of the standard Cournot duopoly model. In their model, there is a continuum of types whose subjective profits are dependent on both the monetary profits and consumer welfare. The firms have perfect information about the counterparty type. Using static analysis, they find that firms that consider consumer welfare are evolutionary stable.

  4. In a complete information environment, [12] assume that a manager can decide whether the firm should be profit maximizer, or should consider consumer welfare. Their conclusions are similar to [13], showing that socially responsible firms can do better than purely profit maximizing firms.

  5. Ritzberger and Weibull [22] use the term sign-preserving dynamics when the growth rate of strategy i is positive (negative) and the payoff of strategy i is greater than (less than) the average payoff in the population. Rabanal and Friedman [21] point out that gradient dynamics follow the aforementioned sign-preserving dynamics when there are only two possible choices. In this case, given a particular type c, the dynamics become a replicator.

  6. Use of conservation of probability mass; see [3, 4].

  7. This method is also known as semidiscretization, since only one dimension (space) is discretized. For example, see [10].

  8. This scheme is called upwind, because it follows the direction that gradient term flows. For a description of the scheme applied to a continuous action space, see Appendix B of [4].

  9. This assumption simplifies the presentation of the model. KLS14 analyze exhaustively both complements and substitutes in their static analysis, but make similar assumptions to ease the exposition of their dynamics.

  10. As mentioned earlier, I adopt gradient dynamics since they are a familiar specification that are a reasonable approximation for dynamics arising from a wide variety of evolutionary processes. KLS14 work with best-replies, which are equivalent to gradient dynamics at \(\hbox {d}\pi _i/\hbox {d}q_{i}=0\).

  11. Defining the parameters \(c=0,\) \(\theta _{A}=\frac{3-a}{2}-a^{2}+a\sqrt{a^{2}+a-2}\), and \(\theta _{B}=\frac{1}{2}(\sqrt{9a^{2}-6a-3}-3a+3)\), they find the following

    • When \(a>1\) and \(0<\theta <\theta _{A}\), the model admits two equilibria: (i) all firms are type S, producing \(a/(3-2\theta )\), which is locally stable, and (ii) all firms are type P, producing 1 / 3, which is a saddle point.

    • When \(a>1\) and \(\theta _{A}<\theta <\theta _{B}\), the model admits two corner equilibria, and an inner equilibrium. Both corners equilibria are saddle points, and the inner equilibrium is defined as coexistence of both types. The type S produces \((a(\theta (2\theta -3)+2)-(\theta -3)\theta -2-2\sqrt{\psi })/\theta ^{2}\), where \(\psi =(a-1)(\theta -1)^{2}(a((\theta -1)\theta +1)+\theta -1).\) The type P responds as \(h(q)=(2-\theta )q/2(1-\theta )+(1-a-\theta )/(2(1-\theta )).\)

    • When \(a>1\) and \(\theta _{B}<\theta \le 1\), or when \(a=1\) and \(0<\theta \le 1\), the model admits two corner equilibria. All firms are type P, which is locally stable, or all firms are type S, which is a saddle point.

  12. To ease the exposition of the paper, I do not rewrite most of the equations presented by KLS14 that describe firm production and firm types share.

  13. The gradient term for the type S firm evaluated at the optimal share and firm’s best reply is \(\frac{4 \theta (15 \theta ^3 - 7 \theta ^4 - 48 \sqrt{\theta ^2 + \theta ^4} + \theta (48 + 2 \sqrt{\theta ^2 + \theta ^4}) + \theta ^2 (-6 + 7 \sqrt{\theta ^2 + \theta ^4}))}{5 (\theta - \theta ^2 + \sqrt{\theta ^2 + \theta ^4})^3}\).

  14. I confirm their results by replacing \(\theta ^*\) and \(a=1+\theta ^*\) in the expressions that prove their Proposition 4. Out of the three families of equilibria described in footnote 11, only the first family survives in the long run.

  15. See footnote 8.

  16. In this numerical solution, I use a larger number of iterations compared to the public goods game since the initial distribution of types (uniform) is quite different compared to the stationary distribution.

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Acknowledgments

I am grateful to two anonymous referees of this journal, Mark Cheung, Dan Friedman, Michael Kopel and Olga Rabanal for their comments and suggestions.

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  1. Bates College, Pettengill Hall, Lewiston, ME, 04240, USA

    Jean Paul Rabanal

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Rabanal, J.P. On the Evolution of Continuous Types Under Replicator and Gradient Dynamics: Two Examples. Dyn Games Appl 7, 76–92 (2017). https://doi.org/10.1007/s13235-015-0164-0

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