Evolutionary game with stochastic payoffs in a finite island model

arXiv:2311.00269 · q-bio.PE · Submitted 2023-11-01 · Read on arXiv

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Introduction to the show: ident: Genomics Radio. Generated commentary on the latest computational biology and genomics papers.

Ines: Hi, I'm Ines.

Marcus: And I'm Marcus, here with Yuki.

Ines: We're looking at the paper: "Evolutionary game with stochastic payoffs in a finite island model".

Marcus: Here's the gist of the paper. The gist In this paper, we consider a two-player two-strategy game with random payoffs in a population subdivided into d demes, each containing N individuals at the beginning of any given generation and experiencing local extinction and recolonization with some fixed probability m after reproduction and selection among offspring.

Ines: Model Setup The model considers a haploid population structured into d isolated demes, where each deme starts with N individuals in every generation, producing a large number of offspring which is assumed to be the same for each individual in the population The payoffs are given by the entries of the game matrix (S1 S2 S1 a1,1 a1,2 S2 a2,1 a2,2) = (ak,l) In this paper, we suppose that the payoffs randomly fluctuate over successive generations in such a way that ak,l < 1 and E

ak,l: = µk,l N d + o 1/d The viability of an individual I is given by ωI = 1 + aI A deme goes to extinction with a fixed probability m ∈ (0, 1) and is repopulated by offspring coming from the migrant pool when this occurs Diffusion Approximation The discrete-time evolution of the population converges to a continuous-time diffusion as d → ∞, verified by Ethier and Nagylaki (1980) The infinitesimal mean and variance of this diffusion are expressed in terms of the population-scaled means and variances of the payoffs besides identity-by-descent measures between offspring in the same deme in a neutral population The first conditional moment of the change in the frequency of S1 is Ez

X(1) − X(0): = 1/NdM(z) + o 1/d The second result concerns the variance of the change in the frequency of S1, which is V arz

X(1) − X(0): = 1/NdQ(z) + o 1/d Fixation Probability Analysis The fixation probability of S1 introduced as a single mutant, F1, is approximately given by F1 ≈ u1 / Nd = R (Nd)-1 / 0 Ψ(y)dy R 1/0 Ψ(y)dy ≈ Ψ(0) Nd R 1/0 Ψ(y)dy The condition for selection to favour the evolution of S1 more than the evolution of S2 is Psi(1) < 1 Stochastic Payoff Variability Effects The results show that differences in variability can make this fixation probability for cooperation generally increase as the payoffs to that strategy exhibit less variability or the payoffs to the other strategy more variability The analysis in Case 1 shows that an increase in σ2 will increase the fixation probability F1 and decrease the fixation probability F2 In general, assuming that all population-scaled means and variances of the payoffs are of the same small order, we have shown that decreasing the population-scaled variance of any payoff to S1 or increasing the population-scaled variance of any payoff to S2 increases the fixation probability F1 and decreases the fixation probability F2 Convergence in Large Populations As ν → ∞, the limiting diffusion process is exactly the one obtained in a large well-mixed population (Li and Lessard

34: ) The results extend previous studies that have explored the stochastic Prisoner’s dilemma in a large but finite well-mixed population which corresponds to the case where ν → ∞ (see Li and Lessard

34: for a Wright-Fisher model and by Kroumi et al.

27: for a Moran model) The findings indicate that selection fully disfavours the evolution of cooperation in the sense that F1 0 Threshold Values There exist three thresholds, namely σ2∗(ν), σ2∗∗(ν) and σ2∗∗∗(ν), such that the conditions F1 > (Nd)-1, F1 > F2 and F2 σ2∗(ν), σ2 > σ2∗−>(N d)−1, and σ2 > sigma 3(ν), respectively The order of these threshold values is not clear in Case 3, but for the additive Prisoner’s dilemma case, we have sigma 3(ν) < sigma 2∗(ν) < sigma 2∗∗(ν) <sigma 2∗∗∗(ν), which implies that the conditions for F1 0 and any ν > 0 Final Summary The paper demonstrates how stochastic fluctuations in the payoffs impact evolutionary dynamics by showing that more variability in the payoffs for defection than in the payoffs for cooperation is required for selection to favour the evolution of cooperation in any sense The analysis provides three scenarios based on which payoff variances are considered negligible, and concludes that increasing σ2 will increase F1 and decrease F2 The results extend previous studies that have explored the stochastic Prisoner’s dilemma in a large but finite well-mixed population which corresponds to the case where ν → ∞ (see Li and Lessard

34: for a Wright-Fisher model and by Kroumi et al.

27: for a Moran model) The general conclusion still holds.

Paper summary: Ines: So, wrapping up the "Evolutionary game with stochastic payoffs in a finite island model," the authors Dhaker Kroumi and Sabin Lessard have shown how payoff variability impacts evolution in this structured setting. The core finding is that more variability in the defection payoffs than in cooperation payoffs is required for selection to favor cooperation.

Marcus: They established that even with random payoffs, structure matters because it creates these differences in payoff variance interactions between different types of interacting offspring within the demes. It’s not just about the average payoff anymore.

Yuki: What this means for species evolution is that spatial or social structure doesn't just dilute selection; it can amplify specific noise patterns to favor one strategy over another under certain conditions involving population size and extinction rates.

Ines: The paper provides a framework where we can use diffusion approximations to study these dynamics, linking the discrete time steps to a continuous process as the number of demes increases. It gives us concrete mathematical tools for analyzing this kind of noise in evolutionary models.

Marcus: And while the results are specific to this island model setup, they extend previous work on stochastic Prisoner’s dilemmas in large but finite populations, like the Wright-Fisher and Moran models, when you take the population size parameter to infinity.

Yuki: The paper helps us understand how demographic processes like local extinction and recolonization can create niche conditions that favor diversity or specific strategies within a species over long evolutionary timescales.

Conclusion: Ines: So, this paper, "Evolutionary game with stochastic payoffs in a finite island model," it basically looks at how random payoff changes affect evolution when you have these isolated groups of individuals.

Marcus: Yeah, what I'm seeing is they take a standard two-strategy game, right? and instead of fixed payoffs, they let them fluctuate randomly over generations.

Yuki: It’s about testing if that randomness, that noise in the payoffs, can actually push the population toward cooperation or defection differently than if everything was stable.

Ines: The authors are using this setup to see how much variability in one strategy's payoff compared to the other strategy's payoff matters for fixation probability.

Marcus: They find that if you make the payoffs for one strategy more variable than the other, that can actually increase its chance of fixing in the whole population.

Yuki: That suggests structure and noise work together to create selection pressures that aren't just about who has a higher average payoff, but about how much risk is involved.

Ines: It’s like saying the stability of an island doesn't just buffer you; it can amplify the effects of noise on your strategy choice.

Marcus: And they do this by looking at those population-scaled means and variances, which is a specific way to normalize the data so you can compare things across different group sizes.

Yuki: So what does this mean for species evolution in real life? It suggests that environmental or social fluctuations could play a bigger role in maintaining diversity than we usually think.

Ines: Exactly, it shifts our focus from just the average fitness to understanding the distribution of fitness outcomes under noise.

King Fahd University of Petroleum and Minerals · University of Montreal

q-bio.PE

Submitted: 2023-11-01

Updated: 2023-11-01

Journal ref: Journal of Dynamics and Games, Vol. 11, No. 4, pp. 295-334 (2024)

DOI: 10.3934/jdg.2024006

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 69/100

The gist: The gist In this paper, we consider a two-player two-strategy game with random payoffs in a population subdivided into d demes, each containing N individuals at the beginning of any given generation

Key concepts

Diffusion Approximation: This technique simplifies complex, discrete-time evolutionary processes into a continuous-time mathematical model (a diffusion process). It allows researchers to analyze the long-term behavior of the population dynamics by focusing on the infinitesimal changes in strategy frequencies rather than tracking every single generation.
Stochastic Payoff Variability: This refers to how much payoffs fluctuate randomly over generations. The paper investigates how different levels of variability in payoffs for cooperation versus defection affect whether cooperation or defection becomes fixed in the population.
Fixation Probability (F1, F2): This measures the likelihood that one strategy (e.g., cooperation, S1) will completely take over the entire population from a starting frequency. The analysis shows that changes in payoff variability directly influence these probabilities, determining which strategy is more likely to evolve.
Identity-by-Descent Measures: These are metrics used to quantify how related offspring are within the same deme. They are important because they help determine the level of identity among offspring, which influences how payoffs vary between individuals in different demes.

Terminology

Cooperation — 1× in this paper · explained in 2 episodes

Summary

The gist In this paper, we consider a two-player two-strategy game with random payoffs in a population subdivided into d demes, each containing N individuals at the beginning of any given generation and experiencing local extinction and recolonization with some fixed probability m after reproduction and selection among offspring.

Model Setup

The model considers a haploid population structured into d isolated demes, where each deme starts with N individuals in every generation, producing a large number of offspring which is assumed to be the same for each individual in the population The payoffs are given by the entries of the game matrix (S1 S2 S1 a1,1 a1,2 S2 a2,1 a2,2) = (ak,l) In this paper, we suppose that the payoffs randomly fluctuate over successive generations in such a way that ak,l < 1 and E [ak,l] = µk,l N d + o 1/d The viability of an individual I is given by ωI = 1 + aI A deme goes to extinction with a fixed probability m ∈ (0, 1) and is repopulated by offspring coming from the migrant pool when this occurs

Diffusion Approximation

The discrete-time evolution of the population converges to a continuous-time diffusion as d → ∞, verified by Ethier and Nagylaki (1980) The infinitesimal mean and variance of this diffusion are expressed in terms of the population-scaled means and variances of the payoffs besides identity-by-descent measures between offspring in the same deme in a neutral population The first conditional moment of the change in the frequency of S1 is Ez [X(1) − X(0)] = 1/NdM(z) + o 1/d The second result concerns the variance of the change in the frequency of S1, which is V arz [X(1) − X(0)] = 1/NdQ(z) + o 1/d

Fixation Probability Analysis

The fixation probability of S1 introduced as a single mutant, F1, is approximately given by F1 ≈ u1 / Nd = R (Nd)-1 / 0 Ψ(y)dy R 1/0 Ψ(y)dy ≈ Ψ(0) Nd R 1/0 Ψ(y)dy The condition for selection to favour the evolution of S1 more than the evolution of S2 is Psi(1) < 1

Stochastic Payoff Variability Effects

The results show that differences in variability can make this fixation probability for cooperation generally increase as the payoffs to that strategy exhibit less variability or the payoffs to the other strategy more variability The analysis in Case 1 shows that an increase in σ2 will increase the fixation probability F1 and decrease the fixation probability F2

Conclusion on Selection

In general, assuming that all population-scaled means and variances of the payoffs are of the same small order, we have shown that decreasing the population-scaled variance of any payoff to S1 or increasing the population-scaled variance of any payoff to S2 increases the fixation probability F1 and decreases the fixation probability F2

Convergence in Large Populations

As ν → ∞, the limiting diffusion process is exactly the one obtained in a large well-mixed population (Li and Lessard [34]) The results extend previous studies that have explored the stochastic Prisoner’s dilemma in a large but finite well-mixed population which corresponds to the case where ν → ∞ (see Li and Lessard [34] for a Wright-Fisher model and by Kroumi et al. [27] for a Moran model) The findings indicate that selection fully disfavours the evolution of cooperation in the sense that F1 0

Threshold Values

There exist three thresholds, namely σ2∗(ν), σ2∗∗(ν) and σ2∗∗∗(ν), such that the conditions F1 > (Nd)-1, F1 > F2 and F2 σ2∗(ν), σ2 > σ2∗−>(N d)−1, and σ2 > sigma 3(ν), respectively The order of these threshold values is not clear in Case 3, but for the additive Prisoner’s dilemma case, we have sigma 3(ν) < sigma 2∗(ν) < sigma 2∗∗(ν) <sigma 2∗∗∗(ν), which implies that the conditions for F1 0 and any ν > 0

Final Summary

The paper demonstrates how stochastic fluctuations in the payoffs impact evolutionary dynamics by showing that more variability in the payoffs for defection than in the payoffs for cooperation is required for selection to favour the evolution of cooperation in any sense The analysis provides three scenarios based on which payoff variances are considered negligible, and concludes that increasing σ2 will increase F1 and decrease F2 The results extend previous studies that have explored the stochastic Prisoner’s dilemma in a large but finite well-mixed population which corresponds to the case where ν → ∞ (see Li and Lessard [34] for a Wright-Fisher model and by Kroumi et al. [27] for a Moran model) The general conclusion still holds.

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Improvements for AI systems

Increased Robustness to Stochastic Payoffs in Structured Environments: The improved AI system can model evolutionary dynamics in populations subdivided into demes subject to local extinction and recolonization with probability m, incorporating random payoffs whose means and variances are proportional to the inverse of population size N d, leading to a diffusion approximation for strategy frequencies as d → ∞. This allows the system to predict fixation probabilities based on the relationship between payoff variability and identity-by-descent measures between offspring in demes.
Quantifiable Selection Favoring Cooperation: The system can determine that "the probability for a strategy to fix in the population starting from an initial frequency (N d) − 1 generally increases as the payoffs to that strategy exhibit less variability or the payoffs to the other strategy more variability," providing a precise metric for assessing selection pressure in stochastic settings.
Adaptive Threshold Prediction: The system can calculate specific threshold values, such as σ2∗(ν), σ2∗∗(ν), and σ2∗∗∗(ν), which determine whether "F1 > (N d)−1, F1 > F2 and F2 < (N d)−1" holds for the evolution of strategies S1 (cooperation) and S2 (defection) under different scenarios of payoff variances.
Sensitivity Analysis on Extinction Rates: The system can analyze how the effect of the deme-scaled extinction rate ν increases, showing that increasing the deme-scaled extinction rate will lessen the conditions for selection to favour the evolution of S1 and to favour the evolution of S1 more than the evolution of S2.
Characterization of Payoff Variance Impact: The system can differentiate between scenarios based on payoff variance, concluding that increasing σ22 increases F1 and decreases F2 when all other variances are negligible, enabling targeted strategies for maximizing cooperation fixation probability.

Abstract

In this paper, we consider a two-player two-strategy game with random payoffs in a population subdivided into d demes, each containing N individuals at the beginning of any given generation and experiencing local extinction and recolonization with some fixed probability m after reproduction and selection among offspring. Within each deme, offspring engage in random pairwise interactions, and the payoffs are assumed to have means and variances proportional to the inverse of the population size. By verifying the conditions given in Ethier and Nagylaki (1980) to approximate Markov chains with two time scales, we establish that the discrete-time evolutionary dynamics with Nd generations as unit of time converges to a continuous-time diffusion as d to infinity. The infinitesimal mean and variance of this diffusion are expressed in terms of the population-scaled means and variances of the payoffs besides identity-by-descent measures between offspring in the same deme in a neutral population. We show that the probability for a strategy to fix in the population starting from an initial frequency (Nd)-1 generally increases as the payoffs to that strategy exhibit less variability or the payoffs to the other strategy more variability. As a result, differences in variability can make this fixation probability for cooperation larger than the corresponding one for defection. As the deme-scaled extinction rate ν=mN decreases for N large enough and m small enough, creating a higher level of identity among offspring within demes, the differences between the population-scaled variances of the payoffs for interacting offspring of different types increases this effect to a greater extent than the differences for interacting offspring of the same type.