Online Altruism Lab

Results and Discussion

The results for the multi-agent model and the cellular automaton are very similar, although in one case the population is structured in groups and in the other by neighborhood.

The simulations show that altruistic traits can prevail in both the Prisoner's Dilemma (T > R > P > S) and the Game of Chicken (T > R > S > P) when P < threshold ≤ R. In other words: The threshold must be higher than the payoff for D when it interacts with another D (→ P or punishment), and less than or equal to the payoff for a C against another C (→ R or reward).

In the deer hunting game (R > T ≥ P > S), altruism prevails at T or P < threshold < R as well. This means that the threshold must be greater than the minimum payoff of D and less than or equal to the payoff for C when competing against another C.

For a better overview, here is the general matrix from the introduction again:

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)

Prisoner's dilemma

Population development at different selection thresholds:

A prisoner's dilemma in a well-mixed, unstructured population has only one Nash equilibrium, namely when all players are D players.

However, our simulation of a group-structured population with a global selection threshold shows that a population of C players develops stably if the selection threshold is greater P (value for D playing against another D) but less or equal R (the value a C player receives when interacting with a D player).

Game of Chicken

Population development at different selection thresholds: For the game of chicken a mixed strategy of randomly C and D is the only EES (evolutionary stable strategy) in unstructured populations.

Like in prisoner's dilemma, the dynamics of a structured population also dramatically supports the C players in the game of chicken. Here, stable C populations develop even at threshold values from P (D against other D) to R (C against C). Stable D populations, however, are not possible in the chicken game.

Stag Hunt

Population development at different selection thresholds: Stable C populations also develop at certain thresholds in Stag Hunt games. Specifically, this occurs when the threshold is greater than T=P (D player against C or D) but less than or equal to R (C against C).

Figure 1: Threshold ranges for the outcomes of both the Matrix Games simulation and the Cellular Automaton. S, P, R and T are the specified payoff options, and the color-coded areas indicate how the populations develop in the given games. Stable altruistic populations (blue ranges) are possible with all the three games at threshold values between P and R. In Game of Chicken, a stable population of egoists is not possible, but it is possible with the other two strategies (yellow range).

Example "evo prison" in the multi-agent model

The example setup “evo prison” shows that under the preset prisoner's dilemma, a stable C population can develop even from a pure D population. Under the same conditions, a game of chicken also leads to this result.

This proves that structuring the population can not only preserve altruistic traits, but also that these traits can spread even in a population that was originally egotistical.

References and further reading

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