Introduction
As with Public Goods, the Matrix Games model simulates the behavior of a population of altruists and egoists, whereby the population is divided into groups with limited migration. However, deposits and withdrawals are not made via a common pot, but via individual interactions and payouts specified by a payoff matrix.This matrix can be used to define well-known payout scenarios, such as a Prisoner's Dilemma, a Game of Chicken, or a Stag Hunt.
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.
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 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.
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.
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.
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.
Payoff Matrix
C
D
C
R
(reward)S
(sucker)
D
T
(temptation)P
(punishment)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.
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.
C
D
C
3
0
D
4
1
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.
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.
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).
C
D
C
D
References and further reading
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