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The authors have declared that no competing interests exist.

Conceived and designed the experiments: BW JG CH AT. Performed the experiments: BW JG CH AT. Analyzed the data: BW JG CH AT. Contributed reagents/materials/analysis tools: BW JG CH AT. Wrote the paper: BW JG CH AT.

In evolutionary games, reproductive success is determined by payoffs. Weak selection means that even large differences in game outcomes translate into small fitness differences. Many results have been derived using weak selection approximations, in which perturbation analysis facilitates the derivation of analytical results. Here, we ask whether results derived under weak selection are also qualitatively valid for intermediate and strong selection. By “qualitatively valid” we mean that the ranking of strategies induced by an evolutionary process does not change when the intensity of selection increases. For two-strategy games, we show that the ranking obtained under weak selection cannot be carried over to higher selection intensity if the number of players exceeds two. For games with three (or more) strategies, previous examples for multiplayer games have shown that the ranking of strategies can change with the intensity of selection. In particular, rank changes imply that the most abundant strategy at one intensity of selection can become the least abundant for another. We show that this applies already to pairwise interactions for a broad class of evolutionary processes. Even when both weak and strong selection limits lead to consistent predictions, rank changes can occur for intermediate intensities of selection. To analyze how common such games are, we show numerically that for randomly drawn two-player games with three or more strategies, rank changes frequently occur and their likelihood increases rapidly with the number of strategies

In evolutionary game dynamics in finite populations, selection intensity plays a key role in determining the impact of the game on reproductive success. Weak selection is often employed to obtain analytical results in evolutionary game theory. We investigate the validity of weak selection predictions for stronger intensities of selection. We prove that in general qualitative results obtained under weak selection fail to extend even to moderate selection strengths for games with either more than two strategies or more than two players. In particular, we find that even in pairwise interactions qualitative changes with changing selection intensity arise almost certainly in the case of a large number of strategies.

In evolutionary theory, weak selection means that differences in reproductive success are small. If fitness differences are close enough to zero, perturbation analysis allows to derive analytical results in models of population dynamics. This approach has a long standing history in population genetics, where selection is typically frequency independent

In infinitely large populations, the intensity of selection merely results in a rescaling of time, but does not affect the outcome of the evolutionary dynamics

Let us illustrate this idea with an example. Consider the public goods game discussed in

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

In the example above, focusing only on the weak selection leads to results that do not even qualitatively hold for higher intensities of selection. The change in the order of strategies shows that, in this case, the predictive power of weak selection to higher intensities of selection is limited. However, many results on the selection of strategies are based on weak selection

We study imitation dynamics in finite populations using pure strategies. While we could work with the Moran process discussed above, we choose for convenience a slightly different process, which is based on the pairwise comparison of two individuals. In this case, only payoff differences matter. Thus the effective parameter number is smaller (in a

The abundance ranking of strategies is invariant under changes of the selection intensity in

We depict the average abundance of strategy

Why do similar functions lead to radically different results when selection is not weak? The intuition behind is as follows: As shown in the SI, the stationary distribution depends only on the product

In the SI, we show that even the monotonicity in the payoff difference cannot ensure the invariance of ranking for any two-strategy game and any imitation function (see Section 3 in SI). Yet this monotonicity applies for all

For games with more than two strategies, i.e.,

An example in which the ranking of strategies changes with the intensity of selection was already provided in the introduction. To go one step further, we provide a theorem for a more challenging constraint in which the limits of both weak and strong selection are identical, yet rank changes occur at intermediate selection strengths.

Theorem 1 states that weak selection results cannot be extrapolated to non-weak selection for

In order to determine how frequent such rank changes occur or how generic these games are, we analyze changes in the ranking of strategies in random games

The numerical approach shows that the construction provided in Theorem 1 is relevant for a substantial fraction of random games and does not merely represent a non-generic, special case. It also shows that a larger number of rank changes may occur as illustrated in

Theorem 1 states that

Games with more than

In the first row, we plot the estimated probability

Similarly, the expected number of rank changes for random

For two-strategy multiplayer games in well-mixed populations under small mutation rates

In evolutionary games in finite populations the assumption that mutation rates are sufficiently rare to consider pairwise invasions between strategies is popular

Here, we have shown that already for

An intuitive reason for changes in the abundance ranking of strategies for

We have focused solely on well-mixed populations and our analytical considerations cannot easily be generalized to structured populations. However, several papers on the evolution of cooperation have shown that the ranking of the average abundance of strategies can change in structured populations even in

Our results have been obtained for imitation processes, i.e. processes in which one individual probabilistically compares its performance to another one and tends to adopt strategies of better performing members of the population. The results derived for three or more strategies assume rare mutations such that the transition matrix of the embedded Markov chain

We assume a finite well-mixed population of size

Variation in the population is generated by mutations. That is, the imitation step described above happens with probability

In a large class of evolutionary processes (where all transitions between states are possible), the transition matrix

For the computational results, we determine the strategy abundances for a random game as a function of the selection strength,

Our source code in Python is publicly available on figshare (

Supplementary Information: Extrapolating weak selection in evolutionary games.

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