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An Introduction to Dynamo: Diagrams for Evolutionary Game Dynamics

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Abstract

Dynamo: Diagrams for Evolutionary Game Dynamics is free, open-source software used to create phase diagrams and other images related to dynamical systems from evolutionary game theory. We describe how to use the software’s default settings to generate phase diagrams quickly and easily. We then explain how to take advantage of the software’s intermediate and advanced features to create diagrams that highlight the key properties of the dynamical system under study. Sample code and output are provided to help demonstrate the software’s capabilities.

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Notes

  1. See Helbing (1992), Björnerstedt and Weibull (1996), Weibull (1995), Hofbauer (1995a), and Schlag (1998).

  2. For a list of articles containing Dynamo output, see http://www.ssc.wisc.edu/~whs/dynamo/papers.html.

  3. For shorter and less technical introductions to evolutionary game theory, see Sandholm (2009, 2012).

  4. In principle, we could create a fifth notebook for two-population games with three strategies for one population and two for the other, for which the state space is a triangular prism.

  5. Many commands from Mathematica version 5 were eliminated entirely from version 6, so notebooks written on either side of this divide are typically incompatible with the application on the other side.

  6. The full location is Choice of game → Specification of payoff parameters → Specification of normal form game .

  7. This example is due to Arneodo et al. (1980), who introduce it in the context of the Lotka–Volterra equations; also see Skyrms (1992). The attractor here is known as a Shilnikov attractor; see Hirsch et al. (2004, Chap. 16).

  8. See http://www.ssc.wisc.edu/~whs/dynamo/3D.html. These notebooks can be opened in Mathematica 6–8, or using Wolfram’s free CDF Player; see http://www.wolfram.com/products/player.

  9. Its location is Specification of graphical output → Specifications for phase diagram → Solution trajectories.

  10. The pdcontourplot option is located in Specification of graphical output → Specifications for phase diagram .

  11. Reversing the order of these times causes Dynamo to interpret {.4, .3, .3} as the terminal condition at time 30, and to compute the solution backward until time 0. Working from the terminal condition is useful when trying to locate unstable cycles, or to find or trajectories that approach unstable rest points along their stable manifolds.

  12. Here is a detailed explanation of what this line of code does. Dynamo’s DEsol calls upon Mathematica’s NDSolve to a the numerical solution to the differential equation. The output of NDSolve is presented as a list of so-called transformation rules. In the present case, the list contains a single transformation rule, corresponding to the lone solution of the differential equation from the initial condition provided. The transformation rule refers to the first two components of the solution using the functions {Xt1, Xt2} , with the third component of the solution being defined implicitly by the fact that solutions live on the simplex.

    The code /., which is the short form of Mathematica’s ReplaceAll command, has Mathematica specify the values of the expression that precedes it by applying the transformation rule that follows it. Here {Xt1[30], Xt2[30], 1-Xt1[30]-Xt2[30]}/.DEsol[{.4, .3, .3}, 0, 30] uses /. to obtain for the time 30 position of the each solution found by DEsol . The result is a list containing a single position vector; the final [[1]] extracts the lone element from this list. It is worth noting that if this code is run in a separate cell after initial run of Dynamo (but without the final ; that suppresses the output), Mathematica immediately reports the point that lies very close to the closed orbit.

  13. The options that control these diagrams are located in Specification of graphical output → Specifications for other diagrams .

  14. By projected payoff, we mean the orthogonal projection of the payoff vector onto the tangent space of the simplex. It is obtained by subtracting the unweighted average of all strategies’ payoffs from each strategy’s payoff—see Sandholm (2010c, Sect. 2.3).

  15. This transformation is useful for studying the replicator dynamic in potential games, since under this transformation, the replicator dynamic is the gradient system defined by the game’s potential function. See Akin (1979) and Sandholm et al. (2008).

  16. See Hofbauer and Weibull (1996), as well as Viossat (2011), from which this example is taken. It follows that one could have obtained the same differential equation by defining the appropriate payoff functional imitative dynamic in Dynamo, and then applying it to F(x) = Ax. Defining new dynamics is not difficult, but takes a few steps to accomplish. The definitions of the built-in dynamics are presented in Choice of dynamic → Definitions of dynamics ; these can be used as a template for defining a new dynamic under the heading Other .

  17. This definition works because of a quirk in Mathematica’s syntax: when a bulit-in function that takes a scalar argument is applied to a vector, the function is evaluated separately on each component of the vector.

  18. The relevant parameters can be found in Specification of graphical output → Specifications for phase diagram → Face shading and viewpoint .

  19. These parameters are located in Specification of graphical output → Specifications for phase diagram → Drawing dots at rest points and Nash equilibria .

  20. An excellent source of information on working with Mathematica graphics, including the creation of files for publication, is a website maintained by Jens Nöckel: see pages.uoregon.edu/noeckel/MathematicaGraphics.html .

  21. This option is found in Specification of graphical output → Specifications for phase diagram .

  22. See Sandholm (2010c, Fig. 5.7).

  23. Mac users who work in TeXShop will find that bitmap figures appearing in pdfs look ugly in TeXShop’s pdf viewer. Fortunately, the figures will look just fine when the pdf is opened in any standard viewer.

  24. See, for example, Press et al. (2007).

  25. See reference.wolfram.com/mathematica/tutorial/NDSolveOverview.html .

  26. See the Dynamo notebooks for documentation. Dynamo takes advantage of this property when finding solutions to the projection dynamic (Nagurney and Zhang 1996; Lahkar and Sandholm 2008), which seems to proceed most smoothly using the Runge–Kutta–Fehlberg method of orders 5 and 4.

  27. See Sandholm (2010c, Chaps. 4–6).

  28. By default, FindRoot employs Newton’s method, but other methods are available and can be selected manually. See reference.wolfram.com/mathematica/ref/FindRoot.html.

  29. For more on linearization of game dynamics, see Sandholm (2010c, Sects. 8.5, 8.6, and 8.C).

  30. For more on local stability of ESSs, see Hofbauer and Sigmund (1988), Cressman (1997), Sandholm (2010a, c, Sects. 8.3 and 8.4).

  31. If a certain class of examples systematically causes problems, one can tune the parameters of the test to obtain better performance. The relevant parameters are located in Specification of graphical output → Specifications for phase diagram → Tuning stability tests .

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Acknowledgments

Financial support from NSF Grant SES-1155135 is gratefully acknowledged.

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  1. Department of Economics, University of Wisconsin, 1180 Observatory Drive, Madison, WI, 53706, USA

    Francisco Franchetti & William H. Sandholm

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  1. Francisco Franchetti
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  2. William H. Sandholm
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Correspondence to William H. Sandholm.

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Franchetti, F., Sandholm, W.H. An Introduction to Dynamo: Diagrams for Evolutionary Game Dynamics . Biol Theory 8, 167–178 (2013). https://doi.org/10.1007/s13752-013-0109-z

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