Figure 1.
Average strategy abundance in compulsory public goods games with punishment [11], where ,
,
,
and
.
The game has three strategies: cooperators contribute to the common pool, defectors exploit cooperators, and altruistic punishers contribute to the common pool and punish defectors. The evolutionary dynamics are based on the Moran process in a population of size , where an individual is chosen for reproduction with a probability proportional to its fitness
, which is an increasing function of its payoff
. Weak selection implies that large payoff differences result in small fitness differences: A exponential payoff-to-fitness mapping,
[20], [50] and B linear payoff-to-fitness mapping,
[11]. The dashed lines represents weak selection approximations. Vertical lines indicate the two selection intensities where the ranking of strategies changes. In both cases, most favored strategy changes at moderate intensities of selection. Thus, predictions based on weak selection results do not carry over to higher intensities of selection.
Figure 2.
The rank invariance property is sensitive to the imitation function for two-strategy multiplayer games.
We depict the average abundance of strategy in a 2-strategy 3-player game in a population of size
as a function of selection intensity
for two imitation functions,
and
, where
is the error function (see inset). The game is given by the table in the figure. Invariance of ranking holds if and only if the curves never cross the
threshold. This threshold is crossed for imitation function
but not for
, despite their similarity, see main text for details.
Figure 3.
Number of changes in the abundance ranking of strategies in games.
A Illustration of a particular game where selection curves intersect times, giving rise to
different rankings (from right to left population sizes
– thick lines,
,
,
–thinner lines). B Statistics over the number of rank changes in games with randomly drawn payoff entries. At least one rank change is obtained in about one quarter of random games. The frequency of
rank changes decreases approximately exponentially with
. As an imitation function, we used the Fermi function
. Parameters: Uniform distribution with payoff values in (0,1), Gaussian distribution with mean 0 and variance 1, frequencies obtained by averaging over
independent samples.
Figure 4.
Occurrence of rank changes in random games.
In the first row, we plot the estimated probability of getting at least
rank changes as a function of the number of strategies
for uniformly distributed payoffs (Panel A) and Gaussian distributed payoffs (Panel B). In the second row, we plot the estimated probability of getting at least
changes in the most abundant strategy as a function of the number of strategies
for uniformly distributed payoffs (Panel C) and Gaussian distributed payoffs (Panel D). Finally, on the third row we show the expected total number of rank changes for uniformly and Gaussian distributed payoffs (Panels E and F). Here, we used a Fermi imitation function
in a population of size
. Simulations: For each
, (
),
random
matrices are sampled.