Figure 1.
Replicator dynamics of the simplified collective-risk dilemma without timing.
For three strategies, the state space takes the form of a triangle, the simplex . The corners of this triangle correspond to homogeneous populations, where all individuals use the same strategy, whereas points in the interior correspond to mixed populations. (a) If the risk probability
, then all interior initial populations eventually converge to a population of defectors. (b) For
a bistable situation emerges: If the initial frequency of fair sharers is sufficiently high, then the subjects learn to coordinate on the beneficial fair share equilibrium. (c) The fraction of initial populations that converge towards the fair share equilibrium increases with
, reaching
in the limit of full risk. For this graph, we have simulated the replicator dynamics for 20,000 randomly chosen initial populations.
Figure 2.
Replicator dynamics of the collective-risk dilemma with timing.
a) Monte-Carlo simulations for 20,000 randomly chosen initial populations confirm that individuals are most likely to adopt the defector's strategy if , whereas subjects tend to use cooperative strategies for higher risk values. (b) An analysis of the timing of contributions reveals that for high risk values, individuals tend to make their contributions in the second rather than in the first round. (c) Average payoffs in the game with timing are above the payoffs in the game without timing (the grey shaded area represents the set of all possible average payoffs).
Figure 3.
Simulations of the evolutionary dynamics for the collective-risk dilemma with multiple players and multiple rounds.
Each graph depicts the average payoff for various , measured in fractions of the initial endowment (the grey shaded area represents the set of all possible average payoffs). (a) A collective-risk game with
rounds and varying group size, (b) a collective-risk game with
players and varying round number (averages over
generations, number of games per generation
, mutation rate
, and the standard deviation for mutations in the thresholds
is set to
).
Figure 4.
Timing of contributions in the collective-risk dilemma.
Simulations of the evolutionary dynamics of a collective-risk game with , and round number
of
and
depicting the average contribution per round. We consider two treatments: (a) Possible contributions 0 or
. (b) Possible contributions 0,
, or
. In both treatments we observe delayed contributions, irrespective of the total game length (averages over
generations,
,
,
,
,
,
).
Figure 5.
Comparison of the expected timing of contributions according to the simulations with the observed timing in the experiments of Milinski et al.[6].
The bold dashed lines show the linear trend, indicating that in the experiments and in our simulations contributions tend to be delayed towards later stages of the game. Parameters were chosen to fit the rules of the experiment, i.e. group size , number of rounds
, initial endowment
, and individuals are allowed to contribute either
,
or
monetary units per round. The other parameters are set to the values in the previous figures.