Average abundancy of cooperation in multi-player games with random payoffs

arXiv:2202.11930 · q-bio.PE · Submitted 2022-02-24 · 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: We're looking at the paper: "Average abundancy of cooperation in multi-player games with random payoffs".

Marcus: Here's the gist of the paper. The gist The study derives conditions for weak selection to favor the abundance of cooperation in multi-player games with random payoffs by analyzing a Moran model in mutation-selection equilibrium Model Formulation The model considers interactions between players in groups of size d ≥ 2 with payoffs that fluctuate randomly over time, updated by a birth-death event according to a Moran model The payoffs are random variables whose first and second moments are given by specific scaling parameters related to the order of selection intensity δ The reproductive fitnesses of cooperators and defectors depend on the group composition through functions fC(x) and fD(x) The population state frequency X is an aperiodic irreducible Markov chain on a finite state space S = 0, 1/N,..., (N − 1)/N, 1 > Weak Selection Conditions Weak selection favors the abundance of C if its average abundance under weak enough selection exceeds what it would be under neutrality, defined by Eδ

X: > E0

X: = 1/2 The first-order approximation for the expected frequency of C in the selection-mutation equilibrium is given by Eδ

X: ≈ 1/2 + δ(1 − u) / u E0

X(1 − X)m(X): (where m(x) is defined based on payoff moments Weak selection favors the abundance of C if E0

X(1 − X)m(X): > 0 Effects of Payoff Variability and Covariance The analysis shows that increasing the variance of any payoff for defection or decreasing the variance of any payoff for cooperation increases the average abundance of cooperation Increasing the scaled mutation rate or the group size can enhance or lessen the condition for weak selection to favor the abundance of C Furthermore, in large populations, decreasing any covariance σCC,kl between two payoffs to C, or increasing any covariance σDD,kl between two payoffs to D increases the average abundance of C Increasing any covariance σCD,kl between a payoff to C and a payoff to D for k + l > d − 1 increases the average abundance of C, while decreasing it for k + l < d − 1 increases it, with no effect when k + l = d − 1 Game Specific Results The results are applied to three social dilemmas: Public Goods Game: For the public goods game, weak selection favors the abundance of C if σbc - σ2c > 2µc However, for large group size d and any scaled mutation rate θ > 0 weak selection will never favor the abundance of C Snowdrift Game: In the snowdrift game and in the case of interactions in large groups, weak selection favors the abundance of C as long as σbc > µc for any mutation probability Conclusion on Group Size and Mutation Rate Increasing the group size d decreases the weights of µb, σbc, and σ2b on the average abundance of C while the weights of µc and σ2c remain the same For large group sizes, weak selection can only favor the abundance of D for any scaled mutation rate θ > 0 In the limit of a low scaled mutation rate, weak selection favors C if Xd-1k=0 µC,k > Xd-1k=0 µD,k 0 and the condition σbc > µc for weak selection to favor C in that scenario

Title and authors: Ines: So we started by looking at "Average abundancy of cooperation in multi-player games with random payoffs" and who wrote it. Marcus It's Kroumi and Lessard, and they are diving into how randomness in payoffs affects cooperation in these multi-player settings.

Yuki: For me, the authors are connecting this to the broader evolutionary history of social structures; I wonder what kind of population dynamics this mathematical model can really support when we look at real species. Ines Right, and they’re using a Moran model to simulate how players switch between cooperation and defection in groups where those payoffs aren't fixed but fluctuate randomly over time >

Marcus: So the paper is essentially building a statistical framework to predict whether cooperation wins or loses when the environment itself is unpredictable, even if you know the average outcomes. Ines That’s a good way to put it. It moves beyond just assuming fixed payoffs and tests how variability plays a role in selection >

Yuki: I think what they’re doing is providing tools to test those hypotheses in ways that might be hard to do with just deterministic models, especially when you consider things like unknown risk of predation or resource variability >

Marcus: Right. And this framework helps us see how those underlying environmental risks translate into observable changes in the average abundance of cooperation in a large population >

Ines: So, moving on from the setup, let's talk about what they actually found and what that summary means for us as computational biologists or data scientists >

The paper's summary: Marcus: Okay, so the actual finding is that weak selection favors cooperation if the average abundance under weak selection is greater than what it would be under neutrality, which is defined as EδX > E0X equals one half >

Ines: That’s the starting point. They give us a first-order approximation for how the expected frequency of cooperation looks in that selection-mutation equilibrium, which involves terms like X times one minus X multiplied by m(X) defined based on payoff moments >

Yuki: I’m trying to connect this back to biology here. This isn't just abstract math; it’s about how these fluctuating payoffs influence the stability of a strategy within a group dynamic, which is crucial for understanding social structures >

Marcus: Exactly. The key condition they derive for weak selection favoring cooperation boils down to checking if E0

X times one minus X m(X): is greater than zero >

Ines: And they then look at how payoff variability and covariance play into this. They show that increasing the variance of any payoff for defection or decreasing the variance of any payoff for cooperation increases the average abundance of cooperation >

Marcus: That’s a direct mechanism: if you make it riskier for defectors by making their payoffs more spread out, cooperation gets a boost >

Yuki: But then they also look at those covariances between payoffs. It suggests that how the rewards or costs of different strategies interact with each other in the group matters significantly >

Ines: They found specific rules based on group size d and whether k plus l is greater than or less than d minus one to determine which covariance structures help cooperation >

The paper's improvements: Ines: Now, the authors suggest several ways this analysis can be used to improve our understanding of these games or perhaps how we design systems where cooperation is more likely to emerge >

Marcus: They point out that we can dynamically adjust cooperation strategies based on payoff uncertainty by exploiting specific covariance structures, for instance, increasing the covariance between a payoff to C and any payoff to D when k plus l is greater than d minus one >

Yuki: So if we know those structural rules, an AI could theoretically use them to adapt its strategy based on how uncertain the payoffs are in real-time situations >

Ines: That’s a big idea. It means the AI wouldn't just pick a fixed strategy; it would exploit the specific way uncertainty is distributed across different payoff interactions >

Marcus: They also suggest optimizing resource allocation in multi-player games by tailoring your strategy to the specific game structure, like in public goods games where you can increase cooperation if sigma bc minus sigma 2c is greater than two mu c > <ref:2202.11930#pg1>

Yuki: That's useful because it means we don't have to assume a universal rule for how cooperation works; we need to know the specific payoff landscape of the game we are modeling >

Ines: And in games like stag hunt and snowdrift, they suggest that increasing the covariance between payoffs for C and D when k plus l is less than d minus one also increases the average abundance of cooperation >

Conclusion: Marcus: So to wrap up, this paper shows that cooperation can be favored by carefully structuring payoff uncertainty in multi-player games, depending on the group size and the specific covariance terms >

Ines: In essence, if you increase the variance of defector payoffs or decrease the variance of cooperators' payoffs, cooperation gets stronger >

Yuki: For me, this is really about giving us more predictive power. It suggests that when we look at population genetic data, we need to account for these payoff fluctuations rather than just focusing on fixed average fitness metrics >

Marcus: Right. And the implications are that for things like the public goods game, there’s a specific threshold based on sigma bc minus sigma 2c being greater than two mu c that can favor cooperation regardless of mutation rate or group size > <ref:2202.11930#pg1>

Ines: And for very large group sizes, they show that even with high scaled mutation rates, if you maintain the condition sigma bc greater than mu c, weak selection will still favor cooperation >

Yuki: That gives us a concrete check to see what kind of evolutionary pressures are necessary for cooperation to persist in those larger systems >

Marcus: This paper on the "Average abundancy of cooperation in multi-player games with random payoffs" suggests that uncertainty isn't just noise; it can be a factor we can potentially exploit to drive evolutionary outcomes toward cooperation >

King Fahd University of Petroleum and Minerals · University of Montreal

q-bio.PE

Submitted: 2022-02-24

Updated: 2022-02-24

Journal ref: Journal of Mathematical Biology 85, 27 (2022)

DOI: 10.1007/s00285-022-01789-1

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 66/100

The gist: The gist The study derives conditions for weak selection to favor the abundance of cooperation in multi-player games with random payoffs by analyzing a Moran model in mutation-selection equilibrium

Key concepts

Weak Selection Conditions: This refers to the mathematical criteria that determine whether a slight advantage for cooperation exists when payoffs are random. The core condition is that the average abundance of cooperators under weak selection must be greater than what would occur under neutral conditions, which is defined as an expected frequency of 1/2.
Payoff Covariance Structures: The paper examines how the correlation (covariance) between different payoffs affects cooperation. Specifically, increasing the covariance between a payoff to C and a payoff to D in certain scenarios can favor cooperation, while decreasing it in others. This suggests that the relationship between different player interactions is crucial for evolutionary outcomes.
Moran Model: The Moran model is a mathematical framework used to simulate how a population changes over time through birth and death events. In this context, it tracks the frequency of cooperators versus defectors in a group of size $d$, allowing researchers to analyze the dynamics under fluctuating, random payoffs.
Group Size ($d$): The group size represents the number of players interacting simultaneously in the game. The analysis shows that increasing this size generally reduces the impact of certain payoff variances on cooperation, and for very large groups, specific conditions involving mutation rates become critical for maintaining cooperation.

Terminology

Cooperation — 8× in this paper · explained in 2 episodes
Abundance — 5× in this paper · explained in 1 episode

Summary

The gist The study derives conditions for weak selection to favor the abundance of cooperation in multi-player games with random payoffs by analyzing a Moran model in mutation-selection equilibrium

Model Formulation

The model considers interactions between players in groups of size d ≥ 2 with payoffs that fluctuate randomly over time, updated by a birth-death event according to a Moran model The payoffs are random variables whose first and second moments are given by specific scaling parameters related to the order of selection intensity δ The reproductive fitnesses of cooperators and defectors depend on the group composition through functions fC(x) and fD(x) The population state frequency X is an aperiodic irreducible Markov chain on a finite state space S = 0, 1/N,..., (N − 1)/N, 1 >

Weak Selection Conditions

Weak selection favors the abundance of C if its average abundance under weak enough selection exceeds what it would be under neutrality, defined by Eδ[X] > E0[X] = 1/2 The first-order approximation for the expected frequency of C in the selection-mutation equilibrium is given by Eδ[X] ≈ 1/2 + δ(1 − u) / u E0[X(1 − X)m(X)] (where m(x) is defined based on payoff moments Weak selection favors the abundance of C if E0[X(1 − X)m(X)] > 0

Effects of Payoff Variability and Covariance

The analysis shows that increasing the variance of any payoff for defection or decreasing the variance of any payoff for cooperation increases the average abundance of cooperation Increasing the scaled mutation rate or the group size can enhance or lessen the condition for weak selection to favor the abundance of C Furthermore, in large populations, decreasing any covariance σCC,kl between two payoffs to C, or increasing any covariance σDD,kl between two payoffs to D increases the average abundance of C Increasing any covariance σCD,kl between a payoff to C and a payoff to D for k + l > d − 1 increases the average abundance of C, while decreasing it for k + l < d − 1 increases it, with no effect when k + l = d − 1

Game Specific Results

The results are applied to three social dilemmas:

** Public Goods Game:**

For the public goods game, weak selection favors the abundance of C if σbc - σ2c > 2µc However, for large group size d and any scaled mutation rate θ > 0 weak selection will never favor the abundance of C

** Snowdrift Game:**

In the snowdrift game and in the case of interactions in large groups, weak selection favors the abundance of C as long as σbc > µc for any mutation probability

Conclusion on Group Size and Mutation Rate

Increasing the group size d decreases the weights of µb, σbc, and σ2b on the average abundance of C while the weights of µc and σ2c remain the same For large group sizes, weak selection can only favor the abundance of D for any scaled mutation rate θ > 0 In the limit of a low scaled mutation rate, weak selection favors C if Xd-1k=0 µC,k > Xd-1k=0 µD,k 0 and the condition σbc > µc for weak selection to favor C in that scenario

Improvements for AI systems

The AI system can dynamically adjust cooperation strategies based on payoff uncertainty to favor cooperation under weak selection by exploiting specific covariance structures; for instance, it can utilize Result 1 which states that "increasing any covariance σCD,kl between a payoff to C and a payoff to D for k + l > d − 1, or decreasing it for k + l < d − 1 increases the average abundance of C."
The system can optimize resource allocation in multi-player games by adjusting its strategy based on the specific game structure; for example, in the public goods game, it can increase cooperation if "σbc − σ2c > 2µc" (Result 2), a condition that is independent of mutation rate or group size.
The AI can adapt its decision-making in stag hunt and snowdrift games by prioritizing strategies based on payoff variances and covariances; specifically, it can increase cooperation if decreasing the variance of the cost c, σ2c, or the variance of the benefit b, σ2b, or increasing their covariance (Result 3).
For very large group sizes (d → ∞), a high scaled mutation rate can be used to maintain cooperation by exploiting approximations showing that "weak selection favors the abundance of C as long as σbc > µc" (Result 6), regardless of the scaled mutation rate.
The AI can predict the optimal strategy in large group snowdrift games under low mutation rates by using the condition derived in section 7.2: "weak selection favors the abundance of C as long as µb − Hdµc − σ2c2−2Hd − H2d/dσ2b + 2Hd−H2d/dσbc > 0" (Result 66).
The system can determine the threshold for cooperation in the public goods game by calculating "σ2/µc > ψ1/2 d−1 k02/ψ2k0+1 2d+1" (Result 27), allowing it to assess whether uncertainty in costs or benefits is sufficient to favor cooperation.

Abstract

We consider interactions between players in groups of size d at least2 with payoffs that not only depend on the strategies used in the group but also fluctuate at random over time. An individual can adopt either cooperation or defection as strategy and the population is updated from one-time step to the next by a birth-death event according to a Moran model. Assuming recurrent symmetric mutation and payoffs with expected values, variances, and covariances of the same small order, we derive a first-order approximation of the average abundance of cooperation in the selection-mutation equilibrium. We show that increasing the variance of any payoff for defection or decreasing the variance of any payoff for cooperation increases the average abundance of cooperation. As for the effect of the covariance between any payoff for cooperation and any payoff for defection, we show that it depends on the number of cooperators in the group associated with these payoffs. We study in particular the public goods game, the stag hunt game, and the snowdrift game, all social dilemmas based on random benefit b and cost c for cooperation. We show that a decrease in the scaled variance of b or c, or an increase in their scaled covariance, makes it easier for weak selection to favor the abundance of cooperation in the stag hunt game and the snowdrift game. The same conclusion holds for the public goods game except that the covariance of b has no effect on the average abundance of C. On the other hand, increasing the scaled mutation rate or the group size can enhance or lessen the condition for weak selection to favor the abundance of C.