A Tale of Two Equations Ludi Vitae or Motus Vita?

Abstract

In this paper, we investigate the meaning and significance of a remarkable fact: the equivalence of the core equations in two apparently disconnected areas in biology. The core equation in evolutionary game theory expresses the dynamic law known as the replicator equation, and the core equation of population dynamics expresses the dynamic law known as the Lotka–Volterra equation. They are proven equivalent by a simple transformation of variables. What does it mean? Does it mean that evolution via natural selection is better modeled as a process of game playing—ludi vitae—or that of mechanical motion—motus vita? We analyze the complexity of this question in this paper.

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Notes

  1. 1.

    See von Neumann and Morgenstern (1944) and Luce and Raiffa (1957), where the first begins the life of game theory in economics and the second is the book from which Dawkins (1998) actually learned game theory, though he said he never managed to finish the book.

  2. 2.

    The relationship between EGT explanations and those of Modern Synthesis is complex (how phenotype selections affect genotype selections, etc.). For lack of space, we leave it out in this paper.

  3. 3.

    For this simple model, we ignore the complications of sexual reproduction and treat everything as haploids.

  4. 4.

    It is perfectly feasible in EGT models to think of learning or brainwashing as mutations in the reproduction of memes.

  5. 5.

    For the formal definitions of NE and ESS in the general n-strategies cases, see Hofbauer and Sigmund (1998).

  6. 6.

    Note, the frequencies and fitness functions are all explicit functions of time, which is true in general regarding replicator dynamics.

  7. 7.

    For theorems that connect the stability of various strength between population states and strategies and for the extension to other game dynamics, see Cressman and Tao (2014), Weibull (1995) and Hofbauer and Sigmund (1998).

  8. 8.

    Volterre published his result as an application in a set of 11 lessons on differential equations, “Leçons sur l’intégration des Équations Différentielles aux Dérivées Partielles,” Professées a Stockholm sur l’Invitation de S. M. le Roi de Suéde, 1906. The same result was independently discovered earlier by an American scientist, Alfred Lotka, and therefore the equation was known in the literature as the Lotka–Volterra equation; see Hofbauer and Sigmund (1998).

  9. 9.

    The lack of terms such as \(x^2\) in the first and \(y^2\) in the second equation indicates that competitions among the same species are negligible; but they can easily be introduced to have a fuller representation.

  10. 10.

    A detailed comparative study of LV models and the Hamiltonian formulation of classical mechanics of many-particle systems is given in a separate forthcoming paper.

  11. 11.

    See “Appendix” for the proof.

  12. 12.

    For more interesting variations on this game such as models for food chains, see Freedman and Waltman (1985), Harrison (1979), Bomze (1983, 1995), Hofbauer and Sigmund (1998, pp. 79–82) and Nowak (2006, pp. 57–60).

  13. 13.

    The technical account of some of the material in that book can be found in Hofbauer and Sigmund (1998)

  14. 14.

    Apart from being intuitively obvious, it is not clear why the RE–LV equivalence is not one of this sort. For lack of space, the question is not pursued deeper here.

  15. 15.

    This sketch may be too brief; for better discussion, see Earman (1989).

  16. 16.

    Together with the previous case, we have covered special relativity and quantum physics.

  17. 17.

    For lack of space, we state the equivalence in a rough and ready manner; for a properly given account, see von Neumann (1955) and Schiff (1968).

  18. 18.

    For philosophical arguments for “wave-function realism,” see Ney and Albert (2013), esp. Introduction, pp. 1–51.

  19. 19.

    In a 1-person game, the player is playing against a MEJE set of situations.

  20. 20.

    In a 1-person game, the probabilities of situations add up to 1.

  21. 21.

    Warning: population dynamics is a fundamentally different area from population genetics, as we mentioned at the beginning.

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Acknowledgements

The project is partially supported by 2018-2019 Fudan Startup Research Fund for Incoming Talent, and I would like to thank my assistant, Su Wuji, for his help of bringing the paper to publication.

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Appendix: The Equivalence of the Replicator Equation and the LV Equation

Appendix: The Equivalence of the Replicator Equation and the LV Equation

Here is the proof in the completely spelled-out form such that one can see how the simple transformation of variables works to transform one equation to the other (Hofbauer 1981). To prove Theorem 1, we map all the trajectories in \(\{ {\mathbf{y}} \in {\mathbb{R}}^n_+ \mid y_n = 1 \}\) to those in the simplex \(\{ {\mathbf{x}} \in S_n \mid x_n > 0 \}\) with the following maps:

$$\begin{aligned} {\mathbf{y}} \mapsto {\mathbf{x}}: \;\; x_i= & {} \frac{y_i}{\sum _{j=1}^n y_j}, i = 1,\ldots , n, \\ {\mathbf{x}} \mapsto {\mathbf{y}}: \;\; y_i= & {} \frac{y_i}{y_n} = \frac{x_i}{x_n}. \end{aligned}$$

replicator equation\(\Rightarrow\)LV equation (Hofbauer and Sigmund 1998)

We begin with an easily proven lemma:

$$\begin{aligned} \dot{\left( \frac{x_i}{x_j}\right) } = \left( \frac{x_i}{x_j}\right) [(A{\mathbf{x}})_i - (A{\mathbf{x}})_j]. \end{aligned}$$

where the two average fitness terms, being the same for \(x_i\) and \(x_j\), cancel out. Then, using \({\mathbf{x}} \mapsto {\mathbf{y}}\) we have,

$$\begin{aligned} {\dot{y}}_i = \dot{\left( \frac{x_i}{x_n}\right) } = \left( \frac{x_i}{x_n}\right) [(A{\mathbf{x}})_i - (A{\mathbf{x}})_n]. \end{aligned}$$

We are going from a n-dimensional simplex to an n-dimensional open space with \(y_n = 1\), and it is well known that A is invariant under the addition of a constant on each column in the matrix (that’s how A can always be equiv-transformed into a matrix whose diagonal elements are all 0). So, we imagine A in the replicator equation as having already been transformed to a matrix with only 0 for the nth row. Thus, we have \((A{\mathbf{x}})_n = 0\) and therefore,

$$\begin{aligned} {\dot{y}}_i = \left( \frac{x_i}{x_n} \right) [(A{\mathbf{x}})_i] = y_i \left( \sum _i a_{ij} y_j \right) x_n, \;\;\; i = 1,\ldots ,n \end{aligned}$$

which can be written more explicitly as (for \(x_n = {\text{constant}}\)),

$$\begin{aligned} {\dot{y}}_i = y_i \left( a_{in} + \sum _{j=1}^{n-1} a_{ij}y_j \right) , \;\;\; i = 1,\ldots ,n-1 \end{aligned}$$

And this is the LV equation in the theorem since \(a_{nn} = a_{nj} = 0, \; \forall j = 1,\ldots ,n-1.\)

LV equation\(\Rightarrow\)replicator equation (Hofbauer 1981)

We begins with the first mapping

$$\begin{aligned} {\dot{x}}_i = \frac{{\dot{y}}_i \sum y_j - y_i \sum {\dot{y}}_j}{(\sum y_j)^2.}. \end{aligned}$$

Going backwards with the above proof, we have the following:

$$\begin{aligned} {\dot{y}}_i = y_i [(A\mathbf{y })_i] = y_i \left( \sum _i a_{ij} y_j \right) , \;\;\; i = 1,\ldots ,n \end{aligned}$$

and

$$\begin{aligned} \sum {\dot{y}}_j = \sum _j y_j \sum _k a_{jk} y_k = \sum _{j,k} a_{jk}y_j y_k \end{aligned}$$

where \(i, j, k = 1,\ldots ,n\), and the \(a_{ni}\)’s are all 0.

Substituting these two expressions back to the first, and noticing that \(x_i = y_i / \sum y\), we have

$$\begin{aligned} {\dot{x}}_i = x_i \left( \sum _{j=1}^n a_{ij}x_j \right) - \frac{x_i}{(\sum y)} \left( \sum _{j,k = 1}^n x_j a_{jk} x_k \right) . \end{aligned}$$

where all the summations are from 1 to n, and \(\sum y\) is a number.

By absorbing \(\sum y\) into the coefficients, we have recovered the replicator equation as,

$$\begin{aligned} {\dot{x}}_i = x_i \left( \sum _{j=1}^n a_{ij}x_j - \sum _{j,k = 1}^n x_j a_{jk} x_k \right) = x_i[(A{\mathbf{x}})_i - {\mathbf{x}} \cdot A{\mathbf{x}})]. \end{aligned}$$

Because the solutions of a replicator equation live on a n-dimensional simplex, which only has \(n-1\) degrees of freedom, \(y_n = 1\) is defined arbitrarily to lock out one degree of freedom in the LV equation’s solution space.

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Liu, C. A Tale of Two Equations Ludi Vitae or Motus Vita?. Fudan J. Hum. Soc. Sci. (2020). https://doi.org/10.1007/s40647-020-00284-5

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Keywords

  • Philosophy of biology
  • Evolutionary game
  • Equivalence
  • Reduction