Papers by Christoph Hauert
Spontaneous symmetry breaking of cooperation between species
Data from: Social evolution in structured populations
<b>Abstract</b><br/>Understanding the evolution of social behaviours such as al... more <b>Abstract</b><br/>Understanding the evolution of social behaviours such as altruism and spite is a long-standing problem that has generated thousands of articles and heated debates. Previous theoretical studies showed that whether altruism and spite evolve may be contingent on seemingly artificial model features, such as which rule is chosen to update the population (e.g., Birth-Death or Death-Birth), and whether the benefits and costs of sociality affect fecundity or survival. Here we unify these features in a single comprehensive framework. We derive a general condition for social behaviour to be favoured over non-social behaviour, which is applicable in a large class of models for structured populations of fixed size. We recover previous results as special cases, and we are able to evaluate the relative effects of benefits and costs of social interactions on fecundity and survival. Our results highlight the crucial importance of identifying the relative scale at which competition occurs.
Essential versus Non-Essential Strategies 1.1 Game 1: Evolution of Deception Extensive form of th... more Essential versus Non-Essential Strategies 1.1 Game 1: Evolution of Deception Extensive form of this game, presented in Figure S1, results in six strategies. These strategies can be identified from the figure by determining all possible combination of actions that a player can take. For example, the prey can either choose to mimic or to not mimic. Hence, there are two strategies for prey. The predator can either pursue or not pursue. However, the predator can distinguish between prey that have cue of defendedness and those that don't. Hence, pursuing or not pursuing can be conditioned on whether the cue of defendedness was detected or not. Thus, this results in four strategies for the predator. Two of the predator strategies were excluded from the analysis in the main text: 1) pursue only when there is cue of defendedness, and 2) never pursue. To justify the exclusion of these strategies, we numerically analyze the full system under a mutation-selection regime, and show that the frequency of these non-essential strategies rarely rises above the mutation rate, and when it does it is due to weak selection.
arXiv (Cornell University), Dec 22, 2013
Population structures can be crucial determinants of evolutionary processes. For the Moran proces... more Population structures can be crucial determinants of evolutionary processes. For the Moran process on graphs certain structures suppress selective pressure, while others amplify it (Lieberman et al. 2005 Nature 433 312-316). Evolutionary amplifiers suppress random drift and enhance selection. Recently, some results for the most powerful known evolutionary amplifier, the superstar, have been invalidated by a counter example (Díaz et al. 2013 Proc. R. Soc. A 469 20130193). Here we correct the original proof and derive improved upper and lower bounds, which indicate that the fixation probability remains close to 1 − 1/(r 4 H) for population size N → ∞ and structural parameter H 1. This correction resolves the differences between the two aforementioned papers. We also confirm that in the limit N, H → ∞ superstars remain capable of eliminating random drift and hence of providing arbitrarily strong selective advantages to any beneficial mutation. In addition, we investigate the robustness of amplification in superstars and find that it appears to be a fragile phenomenon with respect to changes in the selection or mutation processes.
Supplementary material from "A sheep in wolf’s clothing: levels of deceit and detection in the evolution of cue-mimicry
In an evolutionary context, trusted signals or cues provide individuals with the opportunity to m... more In an evolutionary context, trusted signals or cues provide individuals with the opportunity to manipulate them to their advantage by deceiving others. The deceived can then respond to the deception by either ignoring the signals or cues or evolving means of deception–detection. If the latter happens, it can result in an arms race between deception and detection. Here, we formally analyse these possibilities in the context of cue-mimicry in prey–predator interactions. We demonstrate that two extrinsic parameters control whether and for how long an arms race continues: the benefits of deception, and the cost of ignoring signals and cues and having an indiscriminate response. As long as the cost of new forms of deception is less than its benefits and the cost of new forms of detection is less than the cost of an indiscriminate response, an arms race results in the perpetual evolution of better forms of detection and deception. When novel forms of deception or detection become too costly to evolve, the population settles on a polymorphic equilibrium involving multiple strategies of deception and honesty, and multiple strategies of detection and trust.
Journal of Statistical Physics, Jun 13, 2014
Population structure affects both the outcome and the speed of evolutionary dynamics. Here we con... more Population structure affects both the outcome and the speed of evolutionary dynamics. Here we consider a finite population that is divided into subpopulations called demes. The dynamics within the demes are stochastic and frequency-dependent. Individuals can adopt one of two strategic types, A or B. The fitness of each individual is determined by interactions with other individuals in the same deme. With small probability, proportional to fitness, individuals migrate to other demes. The outcome of these dynamics has been studied earlier by analyzing the fixation probability of a single mutant in an otherwise homogeneous population. These results give only a partial picture of the dynamics, because the time when fixation occurs can be exceedingly large. In this paper, we study the impact of deme structures on the speed of evolution. We derive analytical approximations of fixation times in the limit of rare migration and rare mutation. In this limit, the conditional fixation time of a single A mutant in a B population is the same as that of a single B in an A population. For the prisoner's dilemma game, simulation results fit very well with our analytical predictions and demonstrate that fixation takes place in a moderate amount of time as compared to the expected waiting time until a mutant successfully invades and fixates. The simulations also confirm that the conditional fixation time of a single cooperator is indeed the same as that of a single defector.
Evolutionary Dynamics
NATO science for peace and security series, Apr 1, 2009
Evolutionary dynamics in finite populations reflects a balance between Darwinian selection and ra... more Evolutionary dynamics in finite populations reflects a balance between Darwinian selection and random drift. For a long time population structures were assumed to leave this balance unaffected provided that the mutants and residents have fixed fitness values. This result indeed holds for a certain (large) class of population structures or graphs. However, other structures can tilt the balance to the
PLOS Computational Biology, Mar 4, 2022
Most microbes live in spatially structured communities (e.g., biofilms) in which they interact wi... more Most microbes live in spatially structured communities (e.g., biofilms) in which they interact with their neighbors through the local exchange of diffusible molecules. To understand the functioning of these communities, it is essential to uncover how these local interactions shape community-level properties, such as the community composition, spatial arrangement, and growth rate. Here, we present a mathematical framework to derive communitylevel properties from the molecular mechanisms underlying the cell-cell interactions for systems consisting of two cell types. Our framework consists of two parts: a biophysical model to derive the local interaction rules (i.e. interaction range and strength) from the molecular parameters underlying the cell-cell interactions and a graph based model to derive the equilibrium properties of the community (i.e. composition, spatial arrangement, and growth rate) from these local interaction rules. Our framework shows that key molecular parameters underlying the cell-cell interactions (e.g., the uptake and leakage rates of molecules) determine community-level properties. We apply our model to mutualistic cross-feeding communities and show that spatial structure can be detrimental for these communities. Moreover, our model can qualitatively recapitulate the properties of an experimental microbial community. Our framework can be extended to a variety of systems of two interacting cell types, within and beyond the microbial world, and contributes to our understanding of how community-level properties emerge from microscopic interactions between cells.
Self-organized criticality in a nutshell
Physical review, Sep 1, 1999
Journal of Evolutionary Biology, Sep 1, 2006
Theoretical Population Biology, Feb 1, 2017
Repeated games have a long tradition in the behavioral sciences and evolutionary biology. Recentl... more Repeated games have a long tradition in the behavioral sciences and evolutionary biology. Recently, strategies were discovered that permit an unprecedented level of control over repeated interactions by enabling a player to unilaterally enforce linear constraints on payoffs. Here, we extend this theory of "zerodeterminant" (or, more generally, "autocratic") strategies to alternating games, which are often biologically more relevant than traditional synchronous games. Alternating games naturally result in asymmetries between players because the first move matters or because players might not move with equal probabilities. In a strictly-alternating game with two players, X and Y , we give conditions for the existence of autocratic strategies for player X when (i) X moves first and (ii) Y moves first. Furthermore, we show that autocratic strategies exist even for (iii) games with randomly-alternating moves. Particularly important categories of autocratic strategies are extortionate and generous strategies, which enforce unfavorable and favorable outcomes for the opponent, respectively. We illustrate these strategies using the continuous Donation Game, in which a player pays a cost to provide a benefit to the opponent according to a continuous cooperative investment level. Asymmetries due to alternating moves could easily arise from dominance hierarchies, and we show that they can endow subordinate players with more autocratic strategies than dominant players.
Proceedings of the National Academy of Sciences of the United States of America, Mar 14, 2016
The recent discovery of zero-determinant strategies for the iterated Prisoner's Dilemma sparked a... more The recent discovery of zero-determinant strategies for the iterated Prisoner's Dilemma sparked a surge of interest in the surprising fact that a player can exert unilateral control over iterated interactions. These remarkable strategies, however, are known to exist only in games in which players choose between two alternative actions such as "cooperate" and "defect." Here we introduce a broader class of autocratic strategies by extending zero-determinant strategies to iterated games with more general action spaces. We use the continuous Donation Game as an example, which represents an instance of the Prisoner's Dilemma that intuitively extends to a continuous range of cooperation levels. Surprisingly, despite the fact that the opponent has infinitely many donation levels from which to choose, a player can devise an autocratic strategy to enforce a linear relationship between his or her payoff and that of the opponent even when restricting his or her actions to merely two discrete levels of cooperation. In particular, a player can use such a strategy to extort an unfair share of the payoffs from the opponent. Therefore, although the action space of the continuous Donation Game dwarfs that of the classical Prisoner's Dilemma, players can still devise relatively simple autocratic and, in particular, extortionate strategies.
Journal of Theoretical Biology, Jun 1, 2006
Social dilemmas and the evolutionary conundrum of cooperation are traditionally studied through v... more Social dilemmas and the evolutionary conundrum of cooperation are traditionally studied through various kinds of game theoretical models such as the prisoner's dilemma, public goods games, snowdrift games or by-product mutualism. All of them exemplify situations which are characterized by different degrees of conflicting interests between the individuals and the community. In groups of interacting individuals, cooperators produce a common good benefitting the entire group at some cost to themselves, whereas defectors attempt to exploit the resource by avoiding the costly contributions. Based on synergistic or discounted accumulation of cooperative benefits a unifying theoretical framework was recently introduced that encompasses all games that have traditionally been studied separately (Hauert, Michor, Nowak, Doebeli, 2005. Synergy and discounting of cooperation in social dilemmas. J. Theor. Biol., in press.). Within this framework we investigate the effects of spatial structure with limited local interactions on the evolutionary fate of cooperators and defectors. The quantitative effects of space turn out to be quite sensitive to the underlying microscopic update mechanisms but, more general, we demonstrate that in prisoner's dilemma type interactions spatial structure benefits cooperation-although the parameter range is quite limited-whereas in snowdrift type interactions spatial structure may be beneficial too, but often turns out to be detrimental to cooperation.
Proceedings of The Royal Society B: Biological Sciences, Apr 7, 2001
The notion of fundamental clusters is introduced, serving as a rule of thumb to characterize the ... more The notion of fundamental clusters is introduced, serving as a rule of thumb to characterize the statistical properties of the complex behaviour of cellular automata such as spatial 2 Â 2 games. They represent the smallest cluster size determining the fate of the entire system. Checking simple growth criteria allows us to decide whether the cluster-individuals, e.g. some mutant family, are capable of surviving and invading a resident population. In biology, spatial 2 Â 2 games have a broad spectrum of applications ranging from the evolution of cooperation and intraspecies competition to disease spread. This methodological study allows simple classi¢cations and long-term predictions in various biological and social models to be made. For minimal neighbourhood types, we show that the intuitive candidate, a 3 Â 3 cluster, turns out to be fundamental with certain weak limitations for the Moore neighbourhood but not for the Von Neumann neighbourhood. However, in the latter case, 2 Â 2 clusters generally serve as reliable indicators to whether a strategy survives. Stochasticity is added to investigate the e¡ects of varying fractions of one strategy present at initialization time and to discuss the rich dynamic properties in greater detail. Finally, we derive Liapunov exponents for the system and show that chaos reigns in a small region where the two strategies coexist in dynamical equilibrium.
Proceedings of The Royal Society B: Biological Sciences, May 22, 2003
The puzzle of the emergence of cooperation between unrelated individuals is shared across diverse... more The puzzle of the emergence of cooperation between unrelated individuals is shared across diverse fields of behavioural sciences and economics. In this article we combine the public goods game originating in economics with evolutionary approaches traditionally used in biology. Instead of pairwise encounters, we consider the more complex case of groups of three interacting individuals. We show that territoriality is capable of promoting cooperative behaviour, as in the case of the Prisoner's Dilemma. Moreover, by adding punishment opportunities, the readiness to cooperate is greatly enhanced and asocial strategies can be largely suppressed. Finally, as soon as players carry a reputation for being willing or unwilling to punish, highly cooperative and fair outcomes are achieved. This group-beneficial result is obtained, intriguingly, by making individuals more likely to exploit their co-players if they can get away with it. Thus, less-cooperative individuals make more-cooperative societies.
Proceedings of the National Academy of Sciences of the United States of America, Sep 11, 2001
Nature, May 1, 2006
A fundamental aspect of all biological systems is cooperation. Cooperative interactions are requi... more A fundamental aspect of all biological systems is cooperation. Cooperative interactions are required for many levels of biological organization ranging from single cells to groups of animals 1-4. Human society is based to a large extent on mechanisms that promote cooperation 5-7. It is well known that in unstructured populations, natural selection favors defectors over cooperators. There is much current interest, however, for studying evolutionary games in structured populations and on graphs 8-17. These efforts recognize the fact that who-meets-whom is not random, but determined by spatial relationships or social networks 18-24. Here we describe a surprisingly simple rule, which is a good approximation for all graphs that we have analyzed, including cycles, spatial lattices, random regular graphs, random graphs and scale-free networks 25,26 : natural selection favors cooperation, if the benefit of the altruistic act, b, divided by the cost, c, exceeds the average number of neighbors, k. Therefore, cooperation can evolve as a consequence of 'social viscosity' even in the absence of reputation effects or strategic complexity. A cooperator is someone who pays a cost, c, for another individual to receive a benefit, b. A defector pays no cost and does not distribute any benefits. In evolutionary biology, cost and benefit are measured in terms of fitness. Reproduction can be genetic or cultural. In the latter case, the strategy of someone who does well is imitated by others. In an unstructured population, where all individuals interact equally likely with each other, defectors have a higher average payoff than unconditional cooperators. Therefore, natural selection increases the relative abundance of defectors and drives cooperators to extinction. These evolutionary dynamics hold for the deterministic setting of the replicator equation 27,28 and for stochastic game dynamics of finite populations 29. In our model, the players of an evolutionary game occupy the vertices of a graph. The edges denote links between individuals in terms of game dynamical interaction and biological reproduction. We assume that the graph is fixed for the duration of the evolutionary dynamics. Consider a population of N individuals consisting of cooperators and defectors. A cooperator helps all individuals to whom it is connected. If a cooperator is connected to k other individuals and i of those are cooperators, then its payoff is bi-ck. A defector does not provide any help, and therefore has no costs, but it can receive the benefit from neighboring cooperators. If a defector is connected to j cooperators, then its payoff is bj. The fitness of an individual is given by a constant term, denoting the baseline fitness, plus the payoff that arises from the game. Strong selection means that the payoff is large compared to the baseline fitness; weak selection means the payoff is small compared to the baseline fitness.
This work is inspired by a 2013 paper from Arne Traulsen's lab at the Max Plank Institute for Evo... more This work is inspired by a 2013 paper from Arne Traulsen's lab at the Max Plank Institute for Evolutionary Biology (Wu et al. in PLoS Comput Biol 9:e1003381, 2013). They studied evolutionary games when the mutation rate is so small that each mutation goes to fixation before the next one occurs. It has been shown that for 2 × 2 games the ranking of the strategies does not change as strength of selection is increased (Wu et al. in Phys Rev 82:046106, 2010). The point of the 2013 paper is that when there are three or more strategies the ordering can change as selection is increased. Wu et al. (2013) did numerical computations for a fixed population size N. Here, we will instead let the strength of selection β = c/N where c is fixed and let N → ∞ to obtain formulas for the invadability probabilities φ i j that determine the rankings. These formulas, which are integrals on [0, 1], are intractable calculus problems, but can be easily evaluated numerically. Here, we use them to derive simple formulas for the ranking order when c is small or c is large.
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Papers by Christoph Hauert