Effects of sampling interaction partners and competitors in evolutionary games

Christoph Hauert and Jacek Miȩkisz

Phys. Rev. E 98, 052301 – Published 5 November 2018

Abstract

The sampling of interaction partners depends on often implicit modeling assumptions, yet has marked effects on the dynamics in evolutionary games. One particularly important aspect is whether or not competitors also interact. Population structures naturally affect sampling such that in a microscopic interpretation of the replicator dynamics in well-mixed populations competing individuals do not interact but do interact in structured populations. In social dilemmas interactions with competitors invariably inhibit cooperation, while limited local interactions in structured populations support cooperation by reducing exploitation through cluster formation. These antagonistic effects of population structures on cooperation affect interpretations and the conclusions depend on the details of the comparison. For example, in the snowdrift game, spatial structure may inhibit cooperation when compared to the replicator dynamics. However, modifying the replicator dynamics to include interactions between competitors lowers the equilibrium frequency of cooperators, which changes the conclusions, and space is invariably beneficial, just as in the prisoner's dilemma. These conclusions are confirmed by comparisons with random-matching models, which mimic population structures but randomly reshuffle individuals to inhibit spatial correlations. Finally, the differences in the dynamics with and without interactions among competing individuals underlie the differences between death-birth and birth-death updating in the spatial Moran process: death-birth updating supports cooperation because competitors tend not to interact whereas they tend to do for birth-death updating and hence cooperators provide direct support to competitors to their own detriment.

Received 30 May 2018

DOI:https://doi.org/10.1103/PhysRevE.98.052301

Physics Subject Headings (PhySH)

Animal behavior, social interactions & emergent phenomena Evolutionary & population dynamics Stochastic processes

Physics of Living SystemsStatistical Physics & ThermodynamicsInterdisciplinary Physics

Authors & Affiliations

Christoph Hauert¹ and Jacek Miȩkisz²

¹Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2
²Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

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Vol. 98, Iss. 5 — November 2018

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Images

Figure 1
Sampling of interaction partners in (a) and (b) well-mixed and (c) spatially structured populations. (a) The two competitors (red and blue) both interact with a random sample of $k = 4$ other members of the population (light red and light blue, respectively). (b) The two competitors interact with each other plus $k - 1 = 3$ random members of the population. (c) In spatially structured populations competitors are always neighbors and interact with each other as well as with their other neighbors, some of which may be shared depending on the underlying geometry.
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Figure 2
Equilibrium fraction of cooperators in well-mixed populations interacting in the snowdrift game, $x^{*}$ (dashed line) and $x_{k}^{*}$ (dash-dotted line) for $k = 4$ , as a function of the cost-to-benefit ratio $r$ compared to individual based simulations ( $▪$ ) for $N = 10^{4}$ and selection strength $ω = 0.01$ . (a) Random sampling of interaction partners matches $x^{*}$ even close to 0 or 1, indicating that for $N = 10^{4}$ stochastic fluctuations, which may result in the extinction or fixation of cooperators, are negligible. (b) The modified sampling scheme to include the competitor among the interaction partners matches $x_{k}^{*}$ and is consistently lower than $x^{*}$ . Above $r_{c} = (k - 1) / (k + 1) = 3 / 5$ cooperators can no longer persist.
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Figure 3
Equilibrium fraction of cooperators in structured populations interacting in the snowdrift game as a function of the cost-to-benefit ratio $r$ from individual based simulations ( $▪$ ) with selection strength $ω = 0.01$ . As a reference, the equilibrium of the replicator dynamics is shown for the standard sampling scheme $x^{*}$ (dashed line) as well as for the modified sampling scheme $x_{k}^{*}$ (dash-dotted line) with $k = 4$ . (a) Lattice population ( $N = 100 \times 100$ ) interacting with the four nearest neighbors ( $k = 4$ ). Compared to $x^{*}$ , spatial structure turns out to be detrimental to cooperation for larger $r$ ; however, compared to $x_{k}^{*}$ , spatial structure remains beneficial. (b) A random regular graph with four neighbors ( $N = 10^{4}$ and $k = 4$ ) results in equilibrium frequencies of cooperators between the lattice and $x^{*}$ due to the decrease in ordered structure and hence local clustering.
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Figure 4
Equilibrium fraction of cooperators in the spatial Moran process for the snowdrift game as a function of the cost-to-benefit ratio $r$ from individual-based simulations ( $•$ , standard deviation as error bars) for (a) death-birth updating and (b) birth-death updating with selection strength $ω = 0.01$ on an $N = 100 \times 100$ lattice with four nearest neighbors ( $k = 4$ ). The simulation results are averaged over 100 runs each, relaxed over 5000 generations, and then taking the mean and standard deviations over the next 5000 generations. The equilibria for the Moran process in well-mixed populations in the limit $N \to \infty$ are the same as for the replicator dynamics for standard sampling (gray dashed line). Accounting for the fact that competitors also interact for birth-death updating recovers modified sampling (gray dash-dotted line). For the spatial Moran process the equilibria are derived from pair approximation in the limit of weak selection $ω ≪ 1$ for birth-death updating (black dashed line) and death-birth updating (black dash-dotted line).
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Figure 5
Stochastic stability in the stag-hunt game for random matching in finite populations ( $N = 100$ ) for (a) standard sampling and (b) modified sampling (focal and model individuals always interact) with $k = 4$ interaction partners and $ε = 0.001$ as a function of the payoff for a hare $a$ [see Eq. (15)] ( $•$ , mean of the stationary distribution with standard deviation as error bars). (a) For standard sampling, the risk dominant strategy is stochastically stable: For $a < 1 / 2$ cooperation is risk dominant, whereas for $a > 1 / 2$ defection is risk dominant. (b) In contrast, for the modified sampling, cooperation is stochastically stable only for $a < 1 / 4$ , whereas defection remains the only stable state for $a > 1 / 2$ . However, for $1 / 4 < a < 1 / 2$ both homogeneous states are stochastically stable and attract the same probability mass, which results in a mean of $1 / 2$ .
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Figure 6
Stochastic stability in the snowdrift game for random matching in finite populations ( $N = 100$ ) for (a) standard sampling and (b) modified sampling (focal and model individuals always interact) with $k = 4$ interaction partners and $ε = 0.001$ as a function of the cost-to-benefit-ratio $r$ [see Eq. (13)] ( $•$ , mean of the stationary distribution with standard deviation as error bars). (a) For standard sampling the interior equilibrium $x^{*}$ is always stochastically stable. Note that $x^{*}$ coincides with the homogeneous equilibria for $r = 0$ and $r = 1$ . (b) In contrast, for the modified sampling, defection is stochastically stable for $r > 3 / 5$ , whereas the interior equilibrium remains stable for $r < 3 / 5$ .
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