Evolutionary game dynamics of two players with two strategies has been studied in great detail. These games have been used to model many biologically relevant scenarios, ranging from social dilemmas in mammals to microbial diversity. Some of these games may, in fact, take place between a number of individuals and not just between two. Here we address one-shot games with multiple players. As long as we have only two strategies, many results from twoplayer games can be generalized to multiple players. For games with multiple players and more than two strategies, we show that statements derived for pairwise interactions no longer hold. For twoplayer games with any number of strategies there can be at most one isolated internal equilibrium. For any number of players d with any number of strategies n, there can be at most ðd − 1Þ n − 1 isolated internal equilibria. Multiplayer games show a great dynamical complexity that cannot be captured based on pairwise interactions. Our results hold for any game and can easily be applied to specific cases, such as public goods games or multiplayer stag hunts.evolutionary dynamics | multiplayer games | multiple strategies | replicator dynamics | finite populations G ame theory was developed in economics to describe social interactions, but it took the genius of John Maynard Smith and George Price to transfer this idea to biology and develop evolutionary game theory (1-3). Numerous books and articles have been written since. Typically, they begin with an introduction about evolutionary game theory and go on to describe the Prisoner's Dilemma, which is one of the most intriguing games because rational individual decisions lead to a deviation from the social optimum. In an evolutionary setting, the average welfare of the population decreases, because defection is selected over cooperation. How can a strategy spread that decreases the fitness of an actor but increases the fitness of its interaction partner? Various ways to solve such social dilemmas have been proposed (4, 5). In the multiplayer version of the Prisoner's Dilemma, the public goods game, a number of players take part by contributing to a common pot. Interest is added to it and then the amount is split equally among all, regardless of whether they have contributed or not. Because only a fraction of one's own investment goes back to each player, there is no incentive to deposit anything. Instead, it is tempting only to take the profits of the investments of others. This scenario has been analyzed in a variety of contexts (6, 7). The evolutionary dynamics of more general multiplayer games has received considerably less attention, and we can guess why from the way William Donald Hamilton put it: "The theory of many-person games may seem to stand to that of two-person games in the relation of sea-sickness to a headache" (8). Only recently, this topic has attracted renewed interest (9-14).As shown by Broom et al. (9), the most general form of multiplayer games, a straightforward generalization of the payoff matrix concept, le...

BackgroundHost-parasite coevolution is generally believed to follow Red Queen dynamics consisting of ongoing oscillations in the frequencies of interacting host and parasite alleles. This belief is founded on previous theoretical work, which assumes infinite or constant population size. To what extent are such sustained oscillations realistic?ResultsHere, we use a related mathematical modeling approach to demonstrate that ongoing Red Queen dynamics is unlikely. In fact, they collapse rapidly when two critical pieces of realism are acknowledged: (i) population size fluctuations, caused by the antagonism of the interaction in concordance with the Lotka-Volterra relationship; and (ii) stochasticity, acting in any finite population. Together, these two factors cause fast allele fixation. Fixation is not restricted to common alleles, as expected from drift, but also seen for originally rare alleles under a wide parameter space, potentially facilitating spread of novel variants.ConclusionOur results call for a paradigm shift in our understanding of host-parasite coevolution, strongly suggesting that these are driven by recurrent selective sweeps rather than continuous allele oscillations.

SignificanceRapid evolution is central to the current antibiotic crisis. Sustainable treatments must thus take account of the bacteria’s potential for adaptation. We identified cellular hysteresis as a principle to constrain bacterial evolution. Cellular hysteresis is a persistent change in bacterial physiology, reminiscent of cellular memory, which is induced by one antibiotic and enhances susceptibility toward another antibiotic. Cellular hysteresis increases bacterial extinction in fast sequential treatments and reduces selection of resistance by favoring responses specific to the induced physiological effects. Fast changes between antibiotics are key, because they create the continuously high selection conditions that are difficult to counter by bacteria. Our study highlights how an understanding of evolutionary processes can help to outsmart human pathogens.

We consider evolutionary game dynamics in a finite population of size N. When mutations are rare, the population is monomorphic most of the time. Occasionally a mutation arises. It can either reach fixation or go extinct. The evolutionary dynamics of the process under small mutation rates can be approximated by an embedded Markov chain on the pure states. Here we analyze how small the mutation rate should be to make the embedded Markov chain a good approximation by calculating the difference between the real stationary distribution and the approximated one. While for a coexistence game, where the best reply to any strategy is the opposite strategy, it is necessary that the mutation rate μ is less than N (-1/2)exp[-N] to ensure that the approximation is good, for all other games, it is sufficient if the mutation rate is smaller than (N ln N)(-1). Our results also hold for a wide class of imitation processes under arbitrary selection intensity.