Evolution of cooperation in two-level Prisoner's Dilemma

Michael Doebeli; Yaroslav Ispolatov

arxiv: 2606.03071 · v1 · pith:KBT75OK4new · submitted 2026-06-02 · 🧬 q-bio.PE · nlin.AO

Evolution of cooperation in two-level Prisoner's Dilemma

Yaroslav Ispolatov , Michael Doebeli This is my paper

Pith reviewed 2026-06-28 07:52 UTC · model grok-4.3

classification 🧬 q-bio.PE nlin.AO

keywords evolution of cooperationprisoner's dilemmamultilevel selectionspatial structuregroup fissiongroup extinctioncontinuous prisoner's dilemma

0 comments

The pith

Spatial between-group dynamics maintain cooperation in the Prisoner's Dilemma despite within-group selection against it.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a model of continuous Prisoner's Dilemma in spatially structured populations divided into groups. Individuals play the game within their groups, affecting their birth rates, while groups play against neighboring groups, affecting their probabilities of fissioning or going extinct. Although defectors always take over within any group, the overall population maintains a positive level of cooperation through these group-level events. The spatial nature is crucial, as removing space or making selection global eliminates cooperation entirely. Local between-group selection leads to higher cooperation levels than global selection.

Core claim

We show that a certain level of cooperation is maintained due to specific between-group dynamics even though the within-group evolution by itself always results in a complete loss of cooperation. The spatial nature of games and resulting fissioning and extinction events is essential for the evolution of cooperation: without it cooperation is never maintained. Analyzing various scenarios of between-group fission and extinction events, we find that higher levels of cooperation evolve when the selection affecting fission and extinction events is local rather than global.

What carries the argument

Two-level Prisoner's Dilemma with individual payoffs modulating birth rates and group payoffs modulating fission and extinction probabilities in a spatial setting.

If this is right

Cooperation persists at the population level due to between-group selection.
The effect vanishes without spatial structure or when group selection is global.
Local selection on groups produces higher cooperation than global selection.
Within-group dynamics alone always eliminate cooperation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This mechanism may explain cooperation in biological systems with group turnover, such as microbial populations or social groups that split or dissolve based on performance.
Similar dynamics could be explored in models where groups compete for resources rather than through direct games.
The model highlights the importance of modularity and spatial embedding in sustaining cooperation against individual-level selection pressures.

Load-bearing premise

Group-level game payoffs directly control the rates of group fission and extinction, and this control is strong enough to offset the complete loss of cooperation within groups.

What would settle it

Running the model with group payoffs set to have no effect on fission or extinction probabilities, or making all interactions global instead of spatial nearest-neighbor, should result in zero cooperation maintained.

Figures

Figures reproduced from arXiv: 2606.03071 by Michael Doebeli, Yaroslav Ispolatov.

**Figure 2.** Figure 2: The figure illustrates that cooperation evolves under the BW [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: The steady state system-averaged distribution of within-group [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: The level of cooperation in groups on a 2-dimensional square [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗

**Figure 5.** Figure 5: The average level of cooperation for the WB update: A) [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

**Figure 6.** Figure 6: The dependence of the nearest neighbour correlation coefficient on [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗

read the original abstract

We consider continuous Prisoner's Dilemma played in spatial setting by group-structured populations. The population dynamics consists of individual-level birth and death and group-level fission and extinction events. Each individual plays games with all other individuals within their group, while groups play games against their nearest neighbours. Payoffs from individual-level games affect birth rates of individuals, and payoffs from group-level games affect group extinction and fission probabilities. We show that a certain level of cooperation is maintained due to specific between-group dynamics even though the within-group evolution by itself always results in a complete loss of cooperation. The spatial nature of games and resulting fissioning and extinction events is essential for the evolution of cooperation: without it cooperation is never maintained. Analyzing various scenarios of between-group fission and extinction events, we find that higher levels of cooperation evolve when the selection affecting fission and extinction events is local rather than global.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Spatial between-group fission/extinction sustains some cooperation in this two-level continuous PD where within-group dynamics alone drive it to zero.

read the letter

The central result is that local group-level selection via fission and extinction probabilities, modulated by nearest-neighbor group payoffs, keeps cooperation from disappearing even though every within-group game is a standard continuous PD that always favors defectors. Removing the spatial structure and making group events global eliminates the effect, which the authors use to argue that the between-group spatial dynamics are doing the work.

The setup adds explicit group games on top of individual ones inside groups, with payoffs feeding separately into individual birth rates and group extinction/fission rates. This is a straightforward extension of existing spatial PD work rather than a new framework, but the concrete demonstration that locality at the group level raises cooperation compared with global group selection is a clean, usable finding.

The main limitation visible from the description is that the exact mapping from group payoffs to fission and extinction probabilities is not spelled out in the abstract, so robustness to those functional forms and to group size or interaction range would need checking in the full simulations. No internal contradiction appears in the level separation or the spatial-vs-global contrast.

This is for people already working on multi-level selection or spatial evolutionary games who want a simulation example of how local group events can counteract within-group defection. It is not a first-principles derivation but adds a testable case to the literature.

I would send it for peer review. The claim is narrow enough to be evaluated directly against the simulations, and the local-versus-global comparison gives referees something concrete to assess.

Referee Report

0 major / 1 minor

Summary. The manuscript models continuous Prisoner's Dilemma in a spatial, group-structured population where individuals play within-group games (payoffs affect birth rates) and groups play nearest-neighbor games (payoffs affect fission and extinction probabilities). The central claim is that between-group spatial dynamics maintain a positive level of cooperation even though within-group evolution always drives cooperation to zero; the effect requires spatial structure, as global (non-spatial) interactions eliminate maintenance of cooperation. Higher cooperation evolves under local rather than global selection on fission/extinction events.

Significance. If the simulation results are robust, the work provides a concrete demonstration of multilevel selection sustaining cooperation via spatial group fission and extinction in the PD. The explicit separation of within-group defection from between-group maintenance and the contrast between local and global between-group selection are useful contributions to evolutionary game theory on cooperation.

minor comments (1)

The abstract provides no equations, payoff functions, parameter values, group sizes, or simulation details, making it impossible to assess whether the reported maintenance of cooperation is independent of specific parameter choices or follows directly from the stated rules.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting its potential contribution to multilevel selection in evolutionary game theory. The report lists no specific major comments, so we provide no point-by-point responses below. We remain available to address any additional questions or clarifications the referee may have.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper describes an agent-based simulation model with explicitly defined rules for individual PD payoffs affecting birth rates and group payoffs affecting fission/extinction probabilities in a spatial lattice. The central result—that within-group PD drives cooperation to zero while between-group spatial dynamics maintain positive levels—is reported as an emergent outcome of running the simulations under those rules, with explicit controls (global vs local interactions) confirming the spatial component is necessary. No equations, parameters, or predictions reduce to their inputs by construction; the separation of within- and between-group effects is a direct consequence of the stated payoff and event rules rather than a fitted or self-referential claim. No self-citations or uniqueness theorems are invoked as load-bearing steps.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Review based solely on abstract; full model equations, parameter choices, and simulation protocols unavailable. Free parameters and axioms cannot be enumerated precisely.

free parameters (2)

payoff-to-fission/extinction mapping parameters
Abstract states payoffs affect fission and extinction probabilities but provides no values or functional forms; these must be chosen or fitted to produce the reported cooperation levels.
spatial interaction range and group size parameters
Nearest-neighbor group games and group sizes are invoked but not quantified.

axioms (2)

domain assumption Within-group continuous PD always drives cooperation to zero under individual birth-death dynamics.
Stated as background fact that the model relies on to highlight the between-group rescue effect.
domain assumption Group-level payoffs modulate extinction and fission probabilities.
Core modeling choice that enables the between-group selection mechanism.

pith-pipeline@v0.9.1-grok · 5677 in / 1426 out tokens · 18449 ms · 2026-06-28T07:52:11.638488+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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and Nielsen, A

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[1] [1]

and van Veelen, M

Akdeniz, A. and van Veelen, M. (2020). The cancellation effect at the group level. Evolution , 74(7):1246--1254

2020

[2] [2]

and Hamilton, W

Axelrod, R. and Hamilton, W. D. (1981). The evolution of cooperation. science , 211(4489):1390--1396

1981

[3] [3]

Champagnat, N., Ferri \`e re, R., and M \'e l \'e ard, S. (2006). Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models. Theoretical population biology , 69(3):297--321

2006

[4] [4]

Cooney, D. B. (2019). The replicator dynamics for multilevel selection in evolutionary games. Journal of mathematical biology , 79(1):101--154

2019

[5] [5]

Cooney, D. B. and Mori, Y. (2022). Long-time behavior of a pde replicator equation for multilevel selection in group-structured populations. Journal of Mathematical Biology , 85(2):12

2022

[6] [6]

and Hauert, C

Doebeli, M. and Hauert, C. (2005). Models of cooperation based on the prisoner's dilemma and the snowdrift game. Ecology letters , 8(7):748--766

2005

[7] [7]

A., Cavanaugh, J., Mustoe, A

French, J. A., Cavanaugh, J., Mustoe, A. C., Carp, S. B., and Womack, S. L. (2018). Social monogamy in nonhuman primates: phylogeny, phenotype, and physiology. The Journal of Sex Research , 55(4-5):410--434

2018

[8] [8]

and Doebeli, M

Hauert, C. and Doebeli, M. (2021). Spatial social dilemmas promote diversity. Proceedings of the National Academy of Sciences , 118(42):e2105252118

2021

[9] [9]

and Pletscher, D

Hebblewhite, M. and Pletscher, D. H. (2002). Effects of elk group size on predation by wolves. Canadian Journal of Zoology , 80(5):800--809

2002

[10] [10]

Jackson, J. (1977). Competition on marine hard substrata: the adaptive significance of solitary and colonial strategies. The American Naturalist , 111(980):743--767

1977

[11] [11]

Lang, S. D. and Farine, D. R. (2017). A multidimensional framework for studying social predation strategies. Nature ecology & evolution , 1(9):1230--1239

2017

[12] [12]

Luo, S. (2014). A unifying framework reveals key properties of multilevel selection. Journal of Theoretical Biology , 341:41--52

2014

[13] [13]

Ohtsuki, H., Hauert, C., Lieberman, E., and Nowak, M. A. (2006). A simple rule for the evolution of cooperation on graphs and social networks. Nature , 441(7092):502--505

2006

[14] [14]

and Nowak, M

Sigmund, K. and Nowak, M. A. (1999). Evolutionary game theory. Current Biology , 9(14):R503--R505

1999

[15] [15]

Simon, B. (2010). A dynamical model of two-level selection. Evolutionary Ecology Research , 12(5):555--588

2010

[16] [16]

A., and Doebeli, M

Simon, B., Fletcher, J. A., and Doebeli, M. (2013). Towards a general theory of group selection. Evolution , 67(6):1561--1572

2013

[17] [17]

Simon, B., Ispolatov, Y., and Doebeli, M. (2024a). Evolutionary branching in multi-level selection models. Journal of Mathematical Biology , 89(5):52

[18] [18]

Simon, B., Ispolatov, Y., and Doebeli, M. (2024b). Fission as a source of variation for group selection. Evolution , 78(9):1583--1593

[19] [19]

and Nielsen, A

Simon, B. and Nielsen, A. (2012). Numerical solutions and animations of group selection dynamics. Evolutionary Ecology Research , 14(6):757--768

2012

[20] [20]

and Pilosov, M

Simon, B. and Pilosov, M. (2016). Group-level events are catalysts in the evolution of cooperation. Journal of Theoretical Biology , 410:125--136

2016

[21] [21]

D., Day, T., and Wild, G

Taylor, P. D., Day, T., and Wild, G. (2007). Evolution of cooperation in a finite homogeneous graph. Nature , 447(7143):469--472

2007

[22] [22]

and Glynatsi, N

Traulsen, A. and Glynatsi, N. E. (2023). The future of theoretical evolutionary game theory. Philosophical Transactions of the Royal Society B: Biological Sciences , 378(1876)

2023

[23] [23]

and Nowak, M

Traulsen, A. and Nowak, M. A. (2006). Evolution of cooperation by multilevel selection. Proceedings of the National Academy of Sciences , 103(29):10952--10955

2006

[24] [24]

Zukewich, J., Kurella, V., Doebeli, M., and Hauert, C. (2013). Consolidating birth-death and death-birth processes in structured populations. PLoS One , 8(1):e54639

2013