Cooperation usually becomes harder to sustain as groups become larger because incentives to shirk increase with the number of potential contributors to collective action.
But is this always the case?
Here we study a binary-action cooperative dilemma
where a public good is provided as long as not more than a given number of players shirk from a costly cooperative task.
We find that at the stable polymorphic equilibrium, which exists when the cost of cooperation is low enough, the probability of cooperating increases with group size and reaches a limit of one when the group size tends to infinity.
Nevertheless, increasing the group size may increase or decrease the probability that the public good is provided at such an equilibrium, depending on the cost value.
We also prove that the expected payoff to individuals at the stable polymorphic equilibrium (i.e., their fitness) decreases with group size.
For low enough costs of cooperation, both the probability of provision of the public good and the expected payoff converge to positive values in the limit of large group sizes.
However, we also find that the basin of attraction of the stable polymorphic equilibrium is a decreasing function of group size and shrinks to zero in the limit of very large groups.
Overall, we demonstrate non-trivial comparative statics with respect to group size in an otherwise simple collective action problem.
@article{Pena2024,author={PeƱa, Jorge and Heifetz, Aviad and Nƶldeke, Georg},journal={Theoretical Population Biology,},title={The shirker's dilemma and the prospect of cooperation in large groups},year={2024},volume={155},pages={10--23},issn={0040-5809},abbr={Theor Popul Biol},preprint={https://doi.org/10.1101/2023.04.06.535867},bibtex_show={true},code={https://github.com/jorgeapenas/ShirkersDilemma},doi={10.1016/j.tpb.2023.11.001},html={https://doi.org/10.1016/j.tpb.2023.11.001},pdf={Pena2024.pdf},refid={Pena2024}}
2023
bioRxiv
The evolution of environmentally mediated social interactions and posthumous spite under isolation by distance
Many social interactions happen indirectly via modifications of the environment, e.g. through the secretion of functional compounds or the depletion of renewable resources. Here, we derive the selection gradient on a quantitative trait affecting dynamical environmental variables that feed back on reproduction and survival in a finite patch-structured population subject to isolation by distance. Our analysis shows that the selection gradient depends on how a focal individual influences the fitness of all future individuals in the population through modifications of the environmental variables they experience, weighted by the neutral relatedness between recipients and the focal. The evolutionary relevant trait-driven environmental modifications are formalized as the extended phenotypic effects of an individual, quantifying how a trait change in an individual in the present affects the environmental variables in all patches at all future times. When the trait affects reproduction and survival through a payoff function, the selection gradient can be expressed in terms of extended phenotypic effects weighted by scaled-relatedness. We show how to compute extended phenotypic effects, relatedness, and scaled-relatedness using Fourier analysis, which allow us to investigate a broad class of environmentally mediated social interactions in a tractable way. We use our approach to study the evolution of a trait controlling the costly production of some lasting commons (e.g. a common-pool resource or a toxic compound) that can diffuse in space and persist in time. We show that indiscriminate posthumous spite can evolve readily in this scenario, though more generally whether selection favours environmentally mediated altruism or spite is determined by the spatial correlation between an individual's lineage and the commons originating from its patch. The sign of this correlation depends on interactions between dispersal patterns and the commons' renewal dynamics. More broadly, we suggest that selection can favour a wide range of social behaviours when these have carry-over effects in space and time.
@article{Mullon2023,author={Mullon, Charles and PeƱa, Jorge and Lehmann, Laurent},journal={bioRxiv,},title={The evolution of environmentally mediated social interactions and posthumous spite under isolation by distance},year={2023},abbr={bioRxiv},preprint={https://doi.org/10.1101/2023.09.22.558951},bibtex_show={true},refid={Mullon2023}}
Dyn Games Appl
Cooperative dilemmas with binary actions and multiple players
The prisoner's dilemma, the snowdrift game, and the stag hunt are two-player symmetric games that are often considered as prototypical examples of cooperative dilemmas across disciplines. However, surprisingly little consensus exists about the precise mathematical meaning of the words ācooperationā and ācooperative dilemmaā for these and other binary-action symmetric games, in particular when considering interactions among more than two players. Here, we propose definitions of these terms and explore their evolutionary consequences on the equilibrium structure of cooperative dilemmas in relation to social optimality. We show that our definition of cooperative dilemma encompasses a large class of collective action games often discussed in the literature, including congestion games, games with participation synergies, and public goods games. One of our main results is that regardless of the number of players, all cooperative dilemmasāincluding multi-player generalizations of the prisoner's dilemma, the snowdrift game, and the stag huntāfeature inefficient equilibria where cooperation is underprovided, but cannot have equilibria in which cooperation is overprovided. We also find simple conditions for full cooperation to be socially optimal in a cooperative dilemma. Our framework and results unify, simplify, and extend previous work on the structure and properties of cooperative dilemmas with binary actions and two or more players.
@article{Pena2023,author={PeƱa, Jorge and Nƶldeke, Georg},journal={Dynamic Games and Applications,},title={Cooperative dilemmas with binary actions and multiple players},year={2023},volume={13},number={4},pages={1156-1193},abbr={Dyn Games Appl},bibtex_show={true},code={https://github.com/jorgeapenas/CooperativeDilemmas},doi={10.1007/s13235-023-00524-9},html={https://doi.org/10.1007/s13235-023-00524-9},pdf={Pena2023.pdf},refid={Pena2023},preprint={https://osf.io/8y2z5}}
2022
Nat Commun
The evolution of mechanisms to produce phenotypic heterogeneity in microorganisms
In bacteria and other microorganisms, the cells within a population often show extreme phenotypic variation. Different species use different mechanisms to determine how distinct phenotypes are allocated between individuals, including coordinated, random, and genetic determination. However, it is not clear if this diversity in mechanisms is adaptive--arising because different mechanisms are favoured in different environments--or is merely the result of non-adaptive artifacts of evolution. We use theoretical models to analyse the relative advantages of the two dominant mechanisms to divide labour between reproductives and helpers in microorganisms. We show that coordinated specialisation is more likely to evolve over random specialisation in well-mixed groups when: (i) social groups are small; (ii) helping is more āessentialā; and (iii) there is a low metabolic cost to coordination. We find analogous results when we allow for spatial structure with a more detailed model of cellular filaments. More generally, this work shows how diversity in the mechanisms to produce phenotypic heterogeneity could have arisen as adaptations to different environments.
@article{Cooper2022,author={Cooper, Guy A. and Liu, Ming and PeƱa, Jorge and West, Stuart A.},journal={Nature Communications,},title={The evolution of mechanisms to produce phenotypic heterogeneity in microorganisms},year={2022},issn={2041-1723},number={1},pages={195},volume={13},abbr={Nat Commun},bibtex_show={true},code={https://github.com/mingpapilio/Codes_DOL_Mechanisms},doi={10.1038/s41467-021-27902-4},html={https://www.nature.com/articles/s41467-021-27902-4},pdf={Cooper2022.pdf},refid={Cooper2022},preprint={https://www.biorxiv.org/node/2060541.abstract}}
Eusociality, where largely unreproductive offspring help their mothers reproduce, is a major form of social organization. An increasingly documented feature of eusociality is that mothers induce their offspring to help by means of hormones, pheromones or behavioural displays, with evidence often indicating that offspring help voluntarily. The co-occurrence of maternal influence and offspring voluntary help may be explained by what we call the converted helping hypothesis, whereby maternally manipulated helping subsequently becomes voluntary. Such hypothesis requires that parent-offspring conflict is eventually dissolved---for instance, if the benefit of helping increases sufficiently over evolutionary time. We show that help provided by maternally manipulated offspring can enable the mother to sufficiently increase her fertility to transform parent-offspring conflict into parent-offspring agreement. This conflict-dissolution mechanism requires that helpers alleviate maternal life-history trade-offs, and results in reproductive division of labour, high queen fertility and honest queen signalling suppressing worker reproductionthus exceptionally recovering diverse features of eusociality. As such trade-off alleviation seemingly holds widely across eusocial taxa, this mechanism offers a potentially general explanation for the origin of eusociality, the prevalence of maternal influence, and the offspringās willingness to help. Overall, our results explain how a major evolutionary transition can happen from ancestral conflict.
@article{GonzalezForero2021,author={GonzƔlez-Forero, Mauricio and PeƱa, Jorge},journal={Proceedings of the Royal Society B: Biological Sciences,},title={Eusociality through conflict dissolution},year={2021},month=apr,number={1949},volume={288},abbr={Proc R Soc B},bibtex_show={true},code={https://github.com/jorgeapenas/conflictdissolution},doi={10.1098/rspb.2021.0386},html={https://royalsocietypublishing.org/doi/10.1098/rspb.2021.0386},pdf={GonzalezForero2021.pdf},preprint={https://www.biorxiv.org/content/10.1101/2020.09.29.316877v4},pressrelease={https://www.iast.fr/press-release-researchers-develop-new-model-evolution-eusociality},pressrelease2={https://news.st-andrews.ac.uk/archive/insect-youths-give-in-to-parasitic-mum/},publisher={The Royal Society}}
2020
J Math Econ
Group size and collective action in a binary contribution game
We consider how group size affects the private provision of a public good with non-refundable binary contributions. A fixed amount of the good is provided if and only if the number of contributors reaches an exogenous threshold. The threshold, the group size, and the identical, non-refundable cost of contributing to the public good are common knowledge. Our focus is on the case in which the threshold is larger than one, so that teamwork is required to produce the public good. We show that both expected payoffs and the probability that the public good is obtained in the best symmetric equilibrium are decreasing in group size. We also characterize the limit outcome when group size converges to infinity and provide precise conditions under which the expected number of contributors is decreasing or increasing in group size for sufficiently large groups.
@article{Noeldeke2020,author={Nƶldeke, Georg and PeƱa, Jorge},journal={Journal of Mathematical Economics,},title={Group size and collective action in a binary contribution game},year={2020},issn={0304-4068},pages={42--51},volume={88},abbr={J Math Econ},bibtex_show={true},code={https://github.com/jorgeapenas/olsonconjecture},doi={10.1016/j.jmateco.2020.02.003},html={https://doi.org/10.1016/j.jmateco.2020.02.003},keywords={Private provision of public goods, Group size effects, Collective action, Threshold games, Participation games, Teamwork dilemma},pdf={Noeldeke2020.pdf},preprint={https://www.biorxiv.org/content/10.1101/483149v2},url={https://www.sciencedirect.com/science/article/pii/S0304406820300288}}
Am Nat
The evolution of egg trading in simultaneous hermaphrodites
Egg trading---whereby simultaneous hermaphrodites exchange each other's eggs for fertilization---constitutes one of the few rigorously documented and most widely cited examples of direct reciprocity among unrelated individuals. Yet how egg trading may initially invade a population of nontrading simultaneous hermaphrodites is still unresolved. Here, we address this question with an analytical model that considers mate encounter rates and costs of egg production in a population that may include traders (who provide eggs for fertilization only if their partners also have eggs to reciprocate), providers (who provide eggs regardless of whether their partners have eggs to reciprocate), and withholders (cheaters who mate only in the male role and just use their eggs to elicit egg release from traders). Our results indicate that a combination of intermediate mate encounter rates, sufficiently high costs of egg production, and a sufficiently high probability that traders detect withholders (in which case eggs are not provided) is conducive to the evolution of egg trading. Under these conditions, traders can invade---and resist invasion from---providers and withholders alike. The prediction that egg trading evolves only under these specific conditions is consistent with the rare occurrence of this mating system among simultaneous hermaphrodites.
@article{Pena2020,author={PeƱa, Jorge and Nƶldeke, Georg and Puebla, Oscar},journal={The American Naturalist,},title={The evolution of egg trading in simultaneous hermaphrodites},year={2020},issn={0003-0147},number={3},pages={524--533},volume={195},abbr={Am Nat},bibtex_show={true},blog={https://marketdesigner.blogspot.com/search?q=egg+trading},code={https://github.com/jorgeapenas/eggtrading},comment={doi: 10.1086/707016},doi={10.1086/707016},html={https://doi.org/10.1086/707016},pdf={Pena2020.pdf},pressrelease={http://13xy.mj.am/nl2/13xy/m6kwt.html?hl=fr},pressrelease2={https://www.unibas.ch/en/News-Events/News/Uni-Research/Egg-trading-between-hermaphroditic-fish--Why-would-you-give-when-you-can-just-take-.html},publisher={The University of Chicago Press},summary={https://www.amnat.org/an/newpapers/Mar-Pena.html},preprint={https://www.biorxiv.org/content/10.1101/460386v4},url={https://doi.org/10.1086/707016}}
By consuming and producing environmental resources, organisms inevitably change their habitats. The consequences of such environmental modifications can be detrimental or beneficial not only to the focal organism but also to other organisms sharing the same environment. Social evolution theory has been very influential in studying how social interactions mediated by public "goods" or "bads" evolve by emphasizing the role of spatial structure. The environmental dimensions driving these interactions, however, are typically abstracted away. We propose here a new, environment-mediated taxonomy of social behaviors where organisms are categorized by their production or consumption of environmental factors that can help or harm others in the environment. We discuss microbial examples of our classification and highlight the importance of environmental intermediates more generally.
How the size of social groups affects the evolution of cooperative behaviors is a classic question in evolutionary biology. Here we investigate group size effects in the evolutionary dynamics of games in which individuals choose whether to cooperate or defect and payoffs do not depend directly on the size of the group. We find that increasing the group size decreases the proportion of cooperators at both stable and unstable rest points of the replicator dynamics. This implies that larger group sizes can have negative effects (by reducing the amount of cooperation at stable polymorphisms) and positive effects (by enlarging the basin of attraction of more cooperative outcomes) on the evolution of cooperation. These two effects can be simultaneously present in games whose evolutionary dynamics feature both stable and unstable rest points, such as public goods games with participation thresholds. Our theory recovers and generalizes previous results and is applicable to a broad variety of social interactions that have been studied in the literature.
@article{Pena2018,author={PeƱa, Jorge and Nƶldeke, Georg},journal={Journal of Theoretical Biology,},title={Group size effects in social evolution},year={2018},issn={0022-5193},pages={211--220},volume={457},abbr={J Theor Biol},bibtex_show={true},doi={10.1016/j.jtbi.2018.08.004},html={https://www.sciencedirect.com/science/article/pii/S0022519318303746},keywords={Evolution of cooperation, Evolutionary game theory, Replicator dynamics, Public goods games},pdf={Pena2018.pdf},preprint={https://www.biorxiv.org/content/10.1101/319350v2},url={https://www.sciencedirect.com/science/article/pii/S0022519318303746}}
2017
PLOS Comput Biol
Fragmentation modes and the evolution of life cycles
Mode of reproduction is a defining trait of all organisms, including colonial bacteria and multicellular organisms. To produce offspring, aggregates must fragment by splitting into two or more groups. The particular way that a given group fragments defines the life cycle of the organism. For instance, insect colonies can reproduce by splitting or by producing individuals that found new colonies. Similarly, some colonial bacteria propagate by fission or by releasing single cells, while others split in highly sophisticated ways; in multicellular organisms reproduction typically proceeds via a single-cell bottleneck phase. The space of possibilities for fragmentation is so vast that an exhaustive analysis seems daunting. Focusing on fragmentation modes of a simple kind we parametrise all possible modes of group fragmentation and identify those modes leading to the fastest population growth rate. Two kinds of life cycle dominate: one involving division into two equal size groups, and the other involving production of a unicellular propagule. The prevalence of these life cycles in nature is consistent with our null model and suggests that benefits accruing from population growth rate alone may have shaped the evolution of fragmentation mode.
@article{Pichugin2017,author={Pichugin, Yuriy and PeƱa, Jorge and Rainey, Paul B. and Traulsen, Arne},journal={PLOS Computational Biology,},title={Fragmentation modes and the evolution of life cycles},year={2017},month=nov,number={11},pages={e1005860},volume={13},abbr={PLOS Comput Biol},bibtex_show={true},doi={10.1371/journal.pcbi.1005860},html={https://doi.org/10.1371/journal.pcbi.1005860},pdf={Pichugin2017.pdf},preprint={https://www.biorxiv.org/content/10.1101/120097v3},publisher={Public Library of Science},url={https://doi.org/10.1371/journal.pcbi.1005860}}
Cooperation in collective action dilemmas usually breaks down in the absence of additional incentive mechanisms. This tragedy can be escaped if cooperators have the possibility to invest in reward funds that are shared exclusively among cooperators (prosocial rewarding). Yet, the presence of defectors who do not contribute to the public good but do reward themselves (antisocial rewarding) deters cooperation in the absence of additional countermeasures. A recent simulation study suggests that spatial structure is sufficient to prevent antisocial rewarding from deterring cooperation. Here we reinvestigate this issue assuming mixed strategies and weak selection on a game-theoretic model of social interactions, which we also validate using individual-based simulations. We show that increasing reward funds facilitates the maintenance of prosocial rewarding but prevents its invasion, and that spatial structure can sometimes select against the evolution of prosocial rewarding. Our results suggest that, even in spatially structured populations, additional mechanisms are required to prevent antisocial rewarding from deterring cooperation in public goods dilemmas.
@article{Santos2017,author={Dos Santos, Miguel and PeƱa, Jorge},journal={Scientific Reports,},title={Antisocial rewarding in structured populations},year={2017},issn={2045-2322},number={1},pages={6212},volume={7},abbr={Sci Rep},bibtex_show={true},doi={10.1038/s41598-017-06063-9},html={https://doi.org/10.1038/s41598-017-06063-9},pdf={dosSantos2017.pdf},preprint={https://www.biorxiv.org/content/10.1101/092932v4.full},refid={dos Santos2017},url={https://doi.org/10.1038/s41598-017-06063-9}}
2016
PLOS Comput Biol
Evolutionary games of multiplayer cooperation on graphs
Cooperation can be defined as the act of providing fitness benefits to other individuals, often at a personal cost. When interactions occur mainly with neighbors, assortment of strategies can favor cooperation but local competition can undermine it. Previous research has shown that a single coefficient can capture this trade-off when cooperative interactions take place between two players. More complicated, but also more realistic, models of cooperative interactions involving multiple players instead require several such coefficients, making it difficult to assess the effects of population structure. Here, we obtain analytical approximations for the coefficients of multiplayer games in graph-structured populations. Computer simulations show that, for particular instances of multiplayer games, these approximate coefficients predict the condition for cooperation to be promoted in random graphs well, but fail to do so in graphs with more structure, such as lattices. Our work extends and generalizes established results on the evolution of cooperation on graphs, but also highlights the importance of explicitly taking into account higher-order statistical associations in order to assess the evolutionary dynamics of cooperation in spatially structured populations.
@article{Pena2016,author={PeƱa, Jorge and Wu, Bin and Arranz, Jordi and Traulsen, Arne},journal={PLOS Computational Biology,},title={Evolutionary games of multiplayer cooperation on graphs},year={2016},month=aug,number={8},pages={e1005059},volume={12},abbr={PLOS Comput Biol},bibtex_show={true},doi={10.1371/journal.pcbi.1005059},html={https://doi.org/10.1371/journal.pcbi.1005059},pdf={Pena2016PLOSCB.pdf},preprint={https://www.biorxiv.org/content/10.1101/038505v2},publisher={Public Library of Science},url={https://doi.org/10.1371/journal.pcbi.1005059}}
Games Econ Behav
The symmetric equilibria of symmetric voter participation games with complete information
We characterize the symmetric Nash equilibria of the symmetric voter participation game with complete information from Palfrey and Rosenthal (1983). To do so, we use methods based on polynomials in Bernstein form to determine how the probability that a voter is pivotal depends on the participation probability and the number of players in the game.
@article{Noeldeke2016,author={Nƶldeke, Georg and PeƱa, Jorge},journal={Games and Economic Behavior,},title={The symmetric equilibria of symmetric voter participation games with complete information},year={2016},issn={0899-8256},pages={71--81},volume={99},abbr={Games Econ Behav},bibtex_show={true},doi={10.1016/j.geb.2016.06.016},html={https://www.sciencedirect.com/science/article/pii/S0899825616300641},keywords={Costly voting, Participation games, Mixed strategy equilibrium, Polynomials in Bernstein form},pdf={Noeldeke2016.pdf},preprint={https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2635697},url={https://www.sciencedirect.com/science/article/pii/S0899825616300641}}
J R Soc Interface
Ordering structured populations in multiplayer cooperation games
Spatial structure greatly affects the evolution of cooperation. While in two-player games the condition for cooperation to evolve depends on a single structure coefficient, in multiplayer games the condition might depend on several structure coefficients, making it difficult to compare different population structures. We propose a solution to this issue by introducing two simple ways of ordering population structures: the containment order and the volume order. If population structure $\mathcal{S}_1$ is greater than population structure $\mathcal{S}_2$ in the containment or the volume order, then $\mathcal{S}_1$ can be considered a stronger promoter of cooperation. We provide conditions for establishing the containment order, give general results on the volume order, and illustrate our theory by comparing different models of spatial games and associated update rules. Our results hold for a large class of population structures and can be easily applied to specific cases once the structure coefficients have been calculated or estimated.
@article{Pena2016a,author={PeƱa, Jorge and Wu, Bin and Traulsen, Arne},journal={Journal of The Royal Society Interface,},title={Ordering structured populations in multiplayer cooperation games},year={2016},month=jan,number={114},pages={20150881},volume={13},abbr={J R Soc Interface},bibtex_show={true},comment={doi: 10.1098/rsif.2015.0881},doi={10.1098/rsif.2015.0881},html={https://doi.org/10.1098/rsif.2015.0881},pdf={Pena2016JRSI.pdf},preprint={https://www.biorxiv.org/content/10.1101/021550v3},publisher={Royal Society},url={https://doi.org/10.1098/rsif.2015.0881}}
J Theor Biol
Variability in group size and the evolution of collective action
Models of the evolution of collective action typically assume that interactions occur in groups of identical size. In contrast, social interactions between animals occur in groups of widely dispersed size. This paper models collective action problems as two-strategy multiplayer games and studies the effect of variability in group size on the evolution of cooperative behavior under the replicator dynamics. The analysis identifies elementary conditions on the payoff structure of the game implying that the evolution of cooperative behavior is promoted or inhibited when the group size experienced by a focal player is more or less variable. Similar but more stringent conditions are applicable when the confounding effect of size-biased sampling, which causes the group-size distribution experienced by a focal player to differ from the statistical distribution of group sizes, is taken into account.
@article{Pena2016b,author={PeƱa, Jorge and Nƶldeke, Georg},journal={Journal of Theoretical Biology,},title={Variability in group size and the evolution of collective action},year={2016},issn={0022-5193},pages={72--82},volume={389},abbr={J Theor Biol},bibtex_show={true},doi={10.1016/j.jtbi.2015.10.023},html={https://www.sciencedirect.com/science/article/pii/S0022519315005226},keywords={Evolutionary game theory, Public goods games, Replicator dynamics, Stochastic orders},pdf={Pena2016JTB.pdf},preprint={https://www.biorxiv.org/content/10.1101/021485v2},url={https://www.sciencedirect.com/science/article/pii/S0022519315005226}}
2015
J Theor Biol
Evolutionary dynamics of collective action in spatially structured populations
Many models proposed to study the evolution of collective action rely on a formalism that represents social interactions as n-player games between individuals adopting discrete actions such as cooperate and defect. Despite the importance of spatial structure in biological collective action, the analysis of n-player games games in spatially structured populations has so far proved elusive. We address this problem by considering mixed strategies and by integrating discrete-action n-player games into the direct fitness approach of social evolution theory. This allows to conveniently identify convergence stable strategies and to capture the effect of population structure by a single structure coefficient, namely, the pairwise (scaled) relatedness among interacting individuals. As an application, we use our mathematical framework to investigate collective action problems associated with the provision of three different kinds of collective goods, paradigmatic of a vast array of helping traits in nature: "public goods" (both providers and shirkers can use the good, e.g., alarm calls), "club goods" (only providers can use the good, e.g., participation in collective hunting), and "charity goods" (only shirkers can use the good, e.g., altruistic sacrifice). We show that relatedness promotes the evolution of collective action in different ways depending on the kind of collective good and its economies of scale. Our findings highlight the importance of explicitly accounting for relatedness, the kind of collective good, and the economies of scale in theoretical and empirical studies of the evolution of collective action.
@article{Pena2015,author={PeƱa, Jorge and Nƶldeke, Georg and Lehmann, Laurent},journal={Journal of Theoretical Biology,},title={Evolutionary dynamics of collective action in spatially structured populations},year={2015},issn={0022-5193},pages={122--136},volume={382},abbr={J Theor Biol},bibtex_show={true},doi={10.1016/j.jtbi.2015.06.039},html={https://www.sciencedirect.com/science/article/pii/S0022519315003185},keywords={-player games, Games between relatives, Relatedness, Inclusive fitness},pdf={Pena2015.pdf},preprint={https://www.biorxiv.org/content/10.1101/012229v3.full},url={https://www.sciencedirect.com/science/article/pii/S0022519315003185}}
2014
J Theor Biol
Gains from switching and evolutionary stability in multi-player matrix games
In this paper we unify, simplify, and extend previous work on the evolutionary dynamics of symmetric N-player matrix games with two pure strategies. In such games, gains from switching strategies depend, in general, on how many other individuals in the group play a given strategy. As a consequence, the gain function determining the gradient of selection can be a polynomial of degree Nā1. In order to deal with the intricacy of the resulting evolutionary dynamics, we make use of the theory of polynomials in Bernstein form. This theory implies a tight link between the sign pattern of the gains from switching on the one hand and the number and stability of the rest points of the replicator dynamics on the other hand. While this relationship is a general one, it is most informative if gains from switching have at most two sign changes, as is the case for most multi-player matrix games considered in the literature. We demonstrate that previous results for public goods games are easily recovered and extended using this observation. Further examples illustrate how focusing on the sign pattern of the gains from switching obviates the need for a more involved analysis.
@article{Pena2014,author={PeƱa, Jorge and Lehmann, Laurent and Nƶldeke, Georg},journal={Journal of Theoretical Biology,},title={Gains from switching and evolutionary stability in multi-player matrix games},year={2014},issn={0022-5193},pages={23--33},volume={346},abbr={J Theor Biol},bibtex_show={true},doi={10.1016/j.jtbi.2013.12.016},html={https://www.sciencedirect.com/science/article/pii/S0022519313005675},keywords={Evolutionary game theory, Replicator dynamics, Polynomials in Bernstein form, Public goods games},pdf={Pena2014.pdf},preprint={https://arxiv.org/abs/1310.0498},url={https://www.sciencedirect.com/science/article/pii/S0022519313005675}}
2012
PLOS ONE
Bipartite graphs as models of population structures in evolutionary multiplayer games
By combining evolutionary game theory and graph theory, "games on graphs" study the evolutionary dynamics of frequency-dependent selection in population structures modeled as geographical or social networks. Networks are usually represented by means of unipartite graphs, and social interactions by two-person games such as the famous prisonerās dilemma. Unipartite graphs have also been used for modeling interactions going beyond pairwise interactions. In this paper, we argue that bipartite graphs are a better alternative to unipartite graphs for describing population structures in evolutionary multiplayer games. To illustrate this point, we make use of bipartite graphs to investigate, by means of computer simulations, the evolution of cooperation under the conventional and the distributed N-person prisonerās dilemma. We show that several implicit assumptions arising from the standard approach based on unipartite graphs (such as the definition of replacement neighborhoods, the intertwining of individual and group diversity, and the large overlap of interaction neighborhoods) can have a large impact on the resulting evolutionary dynamics. Our work provides a clear example of the importance of construction procedures in games on graphs, of the suitability of bigraphs and hypergraphs for computational modeling, and of the importance of concepts from social network analysis such as centrality, centralization and bipartite clustering for the understanding of dynamical processes occurring on networked population structures.
@article{Pena2012,author={PeƱa, Jorge and Rochat, Yannick},journal={PLOS ONE,},title={Bipartite graphs as models of population structures in evolutionary multiplayer games},year={2012},month=sep,number={9},pages={e44514},volume={7},abbr={PLOS ONE},bibtex_show={true},doi={10.1371/journal.pone.0044514},html={https://doi.org/10.1371/journal.pone.0044514},pdf={Pena2012PLOSONE.pdf},publisher={Public Library of Science},url={https://doi.org/10.1371/journal.pone.0044514}}
Public goods games are models of social dilemmas where cooperators pay a cost for the production of a public good while defectors free ride on the contributions of cooperators. In the traditional framework of evolutionary game theory, the payoffs of cooperators and defectors result from interactions in groups formed by binomial sampling from an infinite population. Despite empirical evidence showing that group-size distributions in nature are highly heterogeneous, most models of social evolution assume that the group size is constant. In this article, I remove this assumption and explore the effects of having random group sizes on the evolutionary dynamics of public goods games. By a straightforward application of Jensen's inequality, I show that the outcome of general nonlinear public goods games depends not only on the average group size but also on the variance of the group-size distribution. This general result is illustrated with two nonlinear public goods games (the public goods game with discounting or synergy and the N-person volunteer's dilemma) and three different group-size distributions (Poisson, geometric, and Waring). The results suggest that failing to acknowledge the natural variation of group sizes can lead to an underestimation of the actual level of cooperation exhibited in evolving populations.
@article{Pena2012a,author={PeƱa, Jorge},journal={Evolution,},title={Group-size diversity in public goods games},year={2012},issn={0014-3820},month=mar,number={3},pages={623--636},volume={66},abbr={Evolution},bibtex_show={true},comment={https://doi.org/10.1111/j.1558-5646.2011.01504.x},doi={10.1111/j.1558-5646.2011.01504.x},html={https://doi.org/10.1111/j.1558-5646.2011.01504.x},keywords={Cooperation, game theory, Jensenās inequality, prisonerās dilemma, public goods game, variance},pdf={Pena2012Evolution.pdf},publisher={John Wiley & Sons, Ltd},url={https://doi.org/10.1111/j.1558-5646.2011.01504.x}}
2011
Physica A
The influence of tie strength on evolutionary games on networks: An empirical investigation
Extending previous work on unweighted networks, we present here a systematic numerical investigation of standard evolutionary games on weighted networks. In the absence of any reliable model for generating weighted social networks, we attribute weights to links in a few ways supported by empirical data ranging from totally uncorrelated to weighted bipartite networks. The results of the extensive simulation work on standard complex network models show that, except in a case that does not seem to be common in social networks, taking the tie strength into account does not change in a radical manner the long-run steady-state behavior of the studied games. Besides model networks, we also included a real-life case drawn from a coauthorship network. In this case also, taking the weights into account only changes the results slightly with respect to the raw unweighted graph, although to draw more reliable conclusions on real social networks many more cases should be studied as these weighted networks become available.
@article{Buesser2011,author={Buesser, Pierre and PeƱa, Jorge and Pestelacci, Enea and Tomassini, Marco},journal={Physica A: Statistical Mechanics and its Applications,},title={The influence of tie strength on evolutionary games on networks: An empirical investigation},year={2011},issn={0378-4371},number={23},pages={4502--4513},volume={390},abbr={Physica A},bibtex_show={true},doi={10.1016/j.physa.2011.07.021},html={https://www.sciencedirect.com/science/article/pii/S0378437111005681},keywords={Evolutionary games, Complex networks, Weighted networks},pdf={Buesser2011.pdf},url={https://www.sciencedirect.com/science/article/pii/S0378437111005681}}
J Theor Biol
Participation costs can suppress the evolution of upstream reciprocity
Indirect reciprocity, one of the many mechanisms proposed to explain the evolution of cooperation, is the idea that altruistic actions can be rewarded by third parties. Upstream or generalized reciprocity is one type of indirect reciprocity in which individuals help someone if they have been helped by somebody else in the past. Although empirically found to be at work in humans, the evolution of upstream reciprocity is difficult to explain from a theoretical point of view. A recent model of upstream reciprocity, first proposed by Nowak and Roch (2007) and further analyzed by Iwagami and Masuda (2010), shows that while upstream reciprocity alone does not lead to the evolution of cooperation, it can act in tandem with mechanisms such as network reciprocity and increase the total level of cooperativity in the population. We argue, however, that Nowak and Roch's model systematically leads to non-uniform interaction rates, where more cooperative individuals take part in more games than less cooperative ones. As a result, the critical benefit-to-cost ratios derived under this model in previous studies are not invariant with respect to the addition of participation costs. We show that accounting for these costs can hinder and even suppress the evolution of upstream reciprocity, both for populations with non-random encounters and graph-structured populations.
We study the effects of conformity, the tendency of humans to imitate locally common behaviors, in the evolution of cooperation when individuals occupy the vertices of a graph and engage in the one-shot prisonerās dilemma or the snowdrift game with their neighbors. Two different graphs are studied: rings (one-dimensional lattices with cyclic boundary conditions) and scale-free networks of the BarabĆ”si-Albert type. The proposed evolutionary-graph model is studied both by means of Monte Carlo simulations and an extended pair-approximation technique. We find improved levels of cooperation when evolution is carried on rings and individuals imitate according to both the traditional payoff bias and a conformist bias. More importantly, we show that scale-free networks are no longer powerful amplifiers of cooperation when fair amounts of conformity are introduced in the imitation rules of the players. Such weakening of the cooperation-promoting abilities of scale-free networks is the result of a less biased flow of information in scale-free topologies, making hubs more susceptible of being influenced by less-connected neighbors.
@article{Pena2009,author={PeƱa, Jorge and Volken, Henri and Pestelacci, Enea and Tomassini, Marco},journal={Physical Review E,},title={Conformity hinders the evolution of cooperation on scale-free networks},year={2009},month=jul,number={1},pages={016110},volume={80},abbr={Phys Rev E},bibtex_show={true},doi={10.1103/PhysRevE.80.016110},html={https://link.aps.org/doi/10.1103/PhysRevE.80.016110},pdf={Pena2009.pdf},publisher={American Physical Society},refid={10.1103/PhysRevE.80.016110},url={https://link.aps.org/doi/10.1103/PhysRevE.80.016110}}
CEC 2009
Conformity and network effects in the Prisoner's Dilemma
We study the evolution of cooperation using the prisoner's dilemma as a metaphor of the tensions between cooperators and non-cooperators, and evolutionary game theory as the mathematical framework for modeling the cultural evolutionary dynamics of imitation in a population of unrelated individuals. We investigate the interplay between network reciprocity (a mechanism that promotes cooperation in the prisoner's dilemma by restricting interactions to adjacent sites in spatial structures or neighbors in social networks) and conformity (the tendency of imitating common behaviors). We confirm previous results on the improved levels of cooperation when both network reciprocity and conformity are present in the model and evolution is carried on top of degree-homogeneous graphs, such as rings and grids. However, we also find that scale-free networks are no longer powerful amplifiers of cooperation when fair amounts of conformity are introduced in the imitation rules of the players. Such weakening of the cooperation-promoting abilities of scale-free networks is the result of a less biased flow of information in such topologies, making hubs more susceptible of being influenced by lessconnected neighbors.
@inproceedings{Pena2009a,author={PeƱa, J. and Pestelacci, E. and Tomassini, M. and Volken, H.},booktitle={2009 IEEE Congress on Evolutionary Computation,},title={Conformity and network effects in the Prisoner's Dilemma},year={2009},pages={506--513},abbr={CEC 2009},bibtex_show={true},doi={10.1109/CEC.2009.4982988},html={https://ieeexplore.ieee.org/document/4982988},issn={1941-0026},journal={2009 IEEE Congress on Evolutionary Computation},pdf={Pena2009CEC.pdf}}
Particle swarm optimization (PSO) is a swarm intelligence technique originally inspired by models of flocking and of social influence that assumed homogeneous individuals. During its evolution to become a practical optimization tool, some heterogeneous variants have been proposed. However, heterogeneity in PSO algorithms has never been explicitly studied and some of its potential effects have therefore been overlooked. In this paper, we identify some of the most relevant types of heterogeneity that can be ascribed to particle swarms. A number of particle swarms are classified according to the type of heterogeneity they exhibit, which allows us to identify some gaps in current knowledge about heterogeneity in PSO algorithms. Motivated by these observations, we carry out an experimental study of two heterogeneous particle swarms each of which is composed of two kinds of particles. Directions for future developments on heterogeneous particle swarms are outlined.
@inproceedings{Oca2009,author={Montes De Oca, M. A. and PeƱa, J. and StĆ¼tzle, T. and Pinciroli, C. and Dorigo, M.},booktitle={2009 IEEE Congress on Evolutionary Computation},title={Heterogeneous particle swarm optimizers},year={2009},pages={698--705},abbr={CEC 2009},bibtex_show={true},doi={10.1109/CEC.2009.4983013},html={https://ieeexplore.ieee.org/document/4983013},issn={1941-0026},journal={2009 IEEE Congress on Evolutionary Computation},pdf={MontesdeOca2009.pdf}}
2008
ALife XI
Conformist transmission and the evolution of cooperation
PeƱa, J.
In Artificial Life XI: Proceedings of the Eleventh International on the Simulation and Synthesis of Living Systems,
2008
We study the effects of conformist transmission on the evolutionary dynamics of the Prisonerās Dilemma, the Snowdrift and the Stag Hunt games in both well-mixed and spatially structured populations. The addition of conformism introduces a transformation of the payoff matrix that favours the stability of pure equilibria and reduces the basin of attraction of risk dominant equilibria. When both conformism and local interactions are present, the system can exhibit higher levels of cooperation than those obtained in the absence of either of the two mechanisms.
@inproceedings{Pena2008,author={PeƱa, J.},booktitle={Artificial Life XI: Proceedings of the Eleventh International on the Simulation and Synthesis of Living Systems,},title={Conformist transmission and the evolution of cooperation},year={2008},address={Cambridge, MA},pages={458-465},publisher={MIT Press},abbr={ALife XI},bibtex_show={true},pdf={Pena2008AlifeXI.pdf}}
ICES 2008
Evolutionary graph models with dynamic topologies on the ubichip
The ubichip is a reconfigurable digital circuit with special reconfiguration mechanisms, such as dynamic routing and self-replication, for supporting the implementation of bio-inspired hardware systems. The dynamic routing mechanism allows to create and destroy interconnections between remote units in a distributed fashion, thus proving useful for implementing cellular systems featuring dynamic topologies. Evolutionary graph theory investigates genetic and cultural evolution processes using the mathematical formalism of both evolutionary game and graph theory. Populations are embedded in graphs representing interaction and imitation links. Payoffs are assigned and successful individuals are imitated with high probability. This paper describes the hardware implementation of a general evolutionary graph model in which the imitation network changes over time by exploiting the dynamic routing capabilities of the ubichip. As a particular example, we analyze the case of a coordination game played by agents arranged in a cycle in which imitation links are randomly created so as to simulate dynamic small-world networks.
@inproceedings{Pena2008a,author={PeƱa, J. C. and PeƱa, J. and Upegui, A.},booktitle={Evolvable Systems: From Biology to Hardware. ICES 2008. Lecture Notes in Computer Science, vol 5216,},title={Evolutionary graph models with dynamic topologies on the ubichip},year={2008},address={Berlin, Heidelberg},publisher={Springer},abbr={ICES 2008},bibtex_show={true},html={https://doi.org/10.1007/978-3-540-85857-7_6},pdf={Pena2008ICES},url={https://doi.org/10.1007/978-3-540-85857-7_6}}
ANTS 2008
Simple dynamic particle swarms without velocity
PeƱa, J.
In Ant Colony Optimization and Swarm Intelligence. ANTS 2008. Lecture Notes in Computer Science, vol 5217,
2008
The standard particle swarm optimiser uses update rules including both multiplicative randomness and velocity. In this paper, we look into a general particle swarm model that removes these two features, and study it mathematically. We derive the recursions and fixed points for the first four moments of the sampling distribution, and analyse the transient behaviour of the mean and the variance. Then we define actual instances of the algorithm by coupling the general update rule with specific recombination operators, and empirically test their optimisation efficiency.
@inproceedings{Pena2008b,author={PeƱa, J.},booktitle={Ant Colony Optimization and Swarm Intelligence. ANTS 2008. Lecture Notes in Computer Science, vol 5217,},title={Simple dynamic particle swarms without velocity},year={2008},address={Berlin, Heidelberg},publisher={Springer},abbr={ANTS 2008},bibtex_show={true},html={https://doi.org/10.1007/978-3-540-87527-7_13},url={https://doi.org/10.1007/978-3-540-87527-7_13}}
GECCO 2008
Theoretical and empirical study of particle swarms with additive stochasticity and different recombination operators
PeƱa, J.
In Proceedings of the 10th Annual Conference on Genetic and Evolutionary Computation,
2008
Standard particle swarms exhibit both multiplicative and additive stochasticity in their update equations. Recently, a simpler particle swarm with just additive stochasticity has been proposed and studied using a new theoretical approach. In this paper we extend the main results of that study to a large number of existing particle swarm optimisers by defining a general update rule from which actual algorithms can be instantiated via the choice of specific recombination operators. In particular, we derive the stability conditions and the dynamic equations for the first two moments of the sampling distribution during stagnation, and show how they depend on the used recombination operator. Finally, the optimisation efficiency of several particle swarms with additive stochasticity is compared in a suite of 16 benchmark functions.
@inproceedings{Pena2008c,author={PeƱa, J.},booktitle={Proceedings of the 10th Annual Conference on Genetic and Evolutionary Computation,},title={Theoretical and empirical study of particle swarms with additive stochasticity and different recombination operators},year={2008},address={New York, NY},pages={95-102},publisher={ACM},abbr={GECCO 2008},bibtex_show={true},html={https://dl.acm.org/doi/10.1145/1389095.1389109},pdf={Pena2008GECCO.pdf},url={https://dl.acm.org/doi/10.1145/1389095.1389109}}
2007
AHS 2007
A population-oriented architecture for particle swarms
Self-adaptive autonomous hardware systems require on-chip heuristics to generate the circuit that constitutes the desired solution. In this paper, we present a population-oriented hardware architecture for particle swarm optimization with discrete recombination (PSO-DR), a hardware-friendly particle swarm that has shown to perform better than the standard PSO for certain parameter values and test functions. We present simulation and synthesis results showing the feasibility, performance, and advantages of the proposed architecture.
@inproceedings{Pena2007,author={PeƱa, J. and Upegui, A.},booktitle={Second NASA/ESA Conference on Adaptive Hardware and Systems,},title={A population-oriented architecture for particle swarms},year={2007},pages={563--570},abbr={AHS 2007},bibtex_show={true},doi={10.1109/AHS.2007.14},html={https://ieeexplore.ieee.org/document/4291968},journal={Second NASA/ESA Conference on Adaptive Hardware and Systems (AHS 2007)},pdf={Pena2007.pdf}}
2006
AHS 2006
Particle swarm optimization with discrete recombination: an online optimizer for evolvable hardware
Self-reconfigurable adaptive systems have the possibility of adapting their own hardware configuration. This feature provides enhanced performance and flexibility, reflected in computational cost reductions. Self-reconfigurable adaptation requires powerful optimization algorithms in order to search in a space of possible hardware configurations. If such algorithms are to be implemented on chip, they must also be as simple as possible, so the best performance can be achieved with the less cost in terms of logic resources, convergence speed, and power consumption. This paper presents hybrid bio-inspired optimization technique that introduces the concept of discrete recombination in a particle swarm optimizer, obtaining a simple and powerful algorithm, well suited for embedded applications. The proposed algorithm is validated using standard benchmark functions and used for training a neural network-based adaptive equalizer for communications systems.
@inproceedings{Pena2006,author={PeƱa, J. and Upegui, A. and Sanchez, E.},booktitle={First NASA/ESA Conference on Adaptive Hardware and Systems,},title={Particle swarm optimization with discrete recombination: an online optimizer for evolvable hardware},year={2006},pages={163--170},abbr={AHS 2006},bibtex_show={true},doi={10.1109/AHS.2006.56},html={https://ieeexplore.ieee.org/document/1638155},journal={First NASA/ESA Conference on Adaptive Hardware and Systems (AHS'06)},pdf={Pena2006AHS.pdf},url={https://ieeexplore.ieee.org/document/1638155}}
ReConfig 2006
Digital hardware architectures of Kohonen's self organizing feature maps with exponential neighboring function
Kohonen maps are self-organizing neural networks that categorize input data, capturing its topology and probability distribution. Efficient hardware implementations of such maps require the definition of a certain number of simplifications to the original algorithm. In particular, multiplications have to be avoided by means of choices in the distance metric, the neighborhood function and the set of learning parameter values. In this paper, one-dimensional and bi-dimensional Kohonen maps with exponential neighboring function and Cityblock and Chessboard norms are defined, and their hardware architecture is presented. VHDL simulations and synthesis on an FPGA of the proposed architectures demonstrate both satisfactory functionality and feasibility.
@inproceedings{Pena2006a,author={PeƱa, J. and Vanegas, M. and Valencia, A.},booktitle={2006 IEEE International Conference on Reconfigurable Computing and FPGA's,},title={Digital hardware architectures of Kohonen's self organizing feature maps with exponential neighboring function},year={2006},pages={1--8},abbr={ReConfig 2006},bibtex_show={true},doi={10.1109/RECONF.2006.307761},html={https://ieeexplore.ieee.org/document/4099981},issn={2325-6532},journal={2006 IEEE International Conference on Reconfigurable Computing and FPGA's (ReConFig 2006)},pdf={Pena2006ReConfig.pdf},url={https://ieeexplore.ieee.org/document/4099981}}