Online Altruism Lab

Introduction

Cellular automata (CA) are computational models in which individuals occupy discrete sites on a grid (or lattice) and interact only with their local neighbors according to simple, rule-based dynamics. Each site can represent a cell, organism, or strategy (e.g., cooperator or defector). Despite their simplicity, cellular automata can generate complex, emergent patterns of behavior.

CA provide a powerful way to study an other kind of structured populations. While "Matrix Games" examines populations that are separated into groups, here it is the direct neighborhood of the agents. Each cell (agent) has eight neighbors and interacts only with these neighbors. The rewards are according to the used payoff matrix, just like in the Matrix Games model.

Nowak & May (1992) used such simple two-dimensional spacial arrays (e.g., the prisoner's dilemma on a chessboard) to demonstrate computationally how altruists can survive in clusters even when surrounded by egoists.

Payoff Matrix

This section applies to both Matrix Games and Cellular Automata, and is therefore included in both introductions.

Matrix or normal-form games are a mathematical framework for analyzing strategic interactions between individuals. In evolutionary biology, they are used to model how different behavioral strategies perform when competing in a population. A matrix game is defined by a payoff matrix, where the rows represent the strategies of the focal individual and the columns represent the strategies of its opponent. Each entry gives the payoff (e.g., fitness, reproductive success) to the focal individual for that interaction.

Table 1: Payoff matrix for two-strategy game with strategies C (= cooperator, altruist) and D (defector, egoist). It shows that a C-player interacting with another C-player earns payoff R, while against a D-player it earns S, and so on. R, S, T and P are the values that the focal players listed in the left column receive when interacting with an opponent listed in the row above.
C D
C R
(reward)		 S
(sucker)
D T
(temptation)		 P
(punishment)

Matrix games form the foundation of evolutionary game theory, which studies how strategies spread under natural selection. Concepts like evolutionarily stable strategies (ESS) arise from this framework and help explain the persistence of cooperation, competition, and other social behaviors in biology and sociology.

Prisoner's Dilemma

The Prisoner's Dilemma is a classic example in game theory that shows why two rational individuals might not cooperate, even if it’s in their best interest.

Prisoner's Dilemmas have the payoff values T > R > P > S.

The idea: Two prisoners are arrested for a crime. They’re interrogated separately and can either cooperate with each other by staying silent or defect by betraying the other.

If both stay silent, both get light sentences. If one betrays and the other stays silent, the betrayer goes free, and the silent one gets a heavy sentence. If both betray, both get moderate sentences.

The paradox is that betraying (defecting) is the rational choice for each individual, but if both choose to defect, they end up worse off than if they had cooperated.

Table 2: The example payoff matrix in the simulation model for a prisoner's dilemma with strategies C and D.
C D
C 3 0
D 4 1
Incidentally, a Public Goods Game with two participants and high costs for the Cs (specifically: if benefit/2 < costs < benefit) is a prisoner's dilemma, as you can easily see below using the Public Goods to Matrix calculator in table 5.

Game of Chicken

Chicken is a further classic model from game theory used to analyze conflict and cooperation in situations where two parties are competing for a resource or dominance. It is also known as snowdrift game or the hawk-dove game, which is particularly well known in evolutionary biology.

The game of chicken is a situation where two sides are on a collision course, and the one who "swerves" first is seen as the "chicken" (dovish behavior), while the one who stays the course wins, but with high risk if both stay.

The payoff values for a Chicken Game are T > R > S > P.

An Evolutionarily Stable Strategy (ESS) is a strategy that, if adopted by a mixed (= not structured) population, cannot be invaded by a new strategy. In the chicken or hawk-dove game, the ESS is often a mixed strategy where both D- (aka hawks) and C-players (aka doves) coexist, as the optimal choice depends on the frequency of the other strategy.

Table 3: Example payoff matrix for a game of chicken in the simulation model.
C D
C 3 1
D 4 0

Stag Hunt

The stag hunt model dates back to Jean-Jacques Rousseau and describes the conflict between safety and cooperation.

The payoff values for the Stag Hunt are R > T ≥ P > S.

We can imagine it like this: The C player hunts a stag together with another C player, but comes away empty-handed with a D player. D players only hunt hares, regardless of who they are with.

Table 4: Example payoff matrix for a stag hunt.
C D
C 4 0
D 2 2

Public Goods to Matrix calculator

In the special case where the group consists of only two players, i.e., for pairs, the benefits and costs of a public goods game can be converted into a payoff matrix (but not the other way around).

This is a calculator for that purpose. Enter the costs for C and benefits by C players and click “Calculate.” As in the models presented here, the defectors D have no costs and provide no benefit to the group.

Benefit: ... Costs: ...

Table 5: Public goods to payoff matrix calculator. Valid only for groups of two!
C D
C
D

References and further reading

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Cremer, J., Melbinger, A., Wienand, K., Henriquez, T., Jung, H., Frey, E., 2019. Cooperation in Microbial Populations: Theory and Experimental Model Systems. Journal of Molecular Biology 431, 4599–4644. https://doi.org/10.1016/j.jmb.2019.09.023

Darwin, C., 1888. The descent of man: and selection in relation to sex. John Murray, Albemarle Street.

Dugatkin, L.A., 2017. The evolution of altruism. Vestn. VOGiS 21, 487–491. https://doi.org/10.18699/VJ17.267

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Hardin, G., The Tragedy of the Commons. Science162,1243-1248(1968). DOI:10.1126/science.162.3859.1243

Killingback, T., Bieri, J., Flatt, T., 2006. Evolution in group-structured populations can resolve the tragedy of the commons. Proc. R. Soc. B. 273, 1477–1481. https://doi.org/10.1098/rspb.2006.3476

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Maynard Smith, J., Price, G. The Logic of Animal Conflict. Nature 246, 15–18 (1973). https://doi.org/10.1038/246015a0

Nowak, M.A., 2012. Evolving cooperation. J Theor Biol 299, 1–8. https://doi.org/10.1016/j.jtbi.2012.01.014

Nowak, M. A. & May, R. M., 1992. Evolutionary games and spatial chaos. Nature 359, 826–829. (doi:10.1038/ 359826a0)

Nowak, M. A., Tarnita, C. E., Antal T., 2010. Evolutionary dynamics in structured populations. Phil. Trans. R. Soc. B. 365, 19–30 (doi:10.1098/rstb.2009.0215)

Okasha, S., 2020. Biological Altruism, in: Zalta, E.N. (Ed.), The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.

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Steiner, K.F., 2024. Altruism pays off in group-structured populations through probable reciprocity. bioRxiv 2024.01.20.575560. doi: https://doi.org/10.1101/2024.01.20.575560

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