Bipartite graphs as models of population structures in evolutionary multiplayer games

Strategy evolution on higher-order networks

Cooperation is key to prosperity in human societies. Population structure is well understood as a catalyst for cooperation, where research has focused on pairwise interactions. But cooperative behaviors are not simply dyadic, and they often involve coordinated behavior in larger groups. Here we develop a framework to study the evolution of behavioral strategies in higher-order population structures, which include pairwise and multi-way interactions. We provide an analytical treatment of when cooperation will be favored by higher-order interactions, accounting for arbitrary spatial heterogeneity and nonlinear rewards for cooperation in larger groups. Our results indicate that higher-order interactions can act to promote the evolution of cooperation across a broad range of networks, in public goods games. Higher-order interactions consistently provide an advantage for cooperation when interaction hyper-networks feature multiple conjoined communities. Our analysis provides a systematic account of how higher-order interactions modulate the evolution of prosocial traits.

Stability analysis of multiplayer games on adaptive simplicial complexes

We analyze the influence of multiplayer interactions and network adaptation on the stability of equilibrium points in evolutionary games. We consider the Snowdrift game on simplicial complexes. In particular, we consider as a starting point the extension from only two-player interactions to coexistence of two- and three-player interactions. The state of the system and the topology of the interactions are both adaptive through best-response strategies of nodes and rewiring strategies of edges, respectively. We derive a closed set of low-dimensional differential equations using pairwise moment closure, which yields an approximation of the lower moments of the system. We numerically confirm the validity of these moment equations. Moreover, we demonstrate that the stability of the fixed points remains unchanged for the considered adaption process. This stability result indicates that rational best-response strategies in games are very difficult to destabilize, even if higher-order multiplayer interactions are taken into account.

Introducing a graph topology for robust cooperation

Identifying the conditions that support cooperation in spatial evolutionary game theory has been the focus of a large body of work. In this paper, the classical Prisoner's Dilemma is adopted as an interaction model; agents are placed on graphs and their interactions are constrained by a graph topology. A simple strategy update mechanism is used where agents copy the best performing strategy of their neighbourhood (including themselves). In this paper, we begin with a fully cooperative population and explore the robustness of the population to the introduction of defectors. We introduce a graph structure that has the property that the initial fully cooperative population is robust to any one perturbation (a change of any cooperator to a defector). We present a proof of this property and specify the necessary constraints on the graph. Furthermore, given the standard game payoffs, we calculate the smallest graph which possesses this property. We present an approach for increasing the size of the graph and we show empirically that this extended graph is robust to an increasing percentage of perturbations. We define a new class of graphs for the purpose of future work.

Evolutionary dynamics of higher-order interactions in social networks

We live and cooperate in networks. However, links in networks only allow for pairwise interactions, thus making the framework suitable for dyadic games, but not for games that are played in larger groups. Here, we study the evolutionary dynamics of a public goods game in social systems with higher-order interactions. First, we show that the game on uniform hypergraphs corresponds to the replicator dynamics in the well-mixed limit, providing a formal theoretical foundation to study cooperation in networked groups. Second, we unveil how the presence of hubs and the coexistence of interactions in groups of different sizes affects the evolution of cooperation. Finally, we apply the proposed framework to extract the actual dependence of the synergy factor on the size of a group from real-world collaboration data in science and technology. Our work provides a way to implement informed actions to boost cooperation in social groups.

Computational Behavioral Models for Public Goods Games on Social Networks

Cooperation is a fundamental aspect of well-organized societies and public good games are a useful metaphor for modeling cooperative behavior in the presence of strong incentives to free ride. Usually, social agents interact to play a public good game through network structures. Here, we use social network structures and computational agent rules inspired by recent experimental work in order to develop models of agent behavior playing public goods games. The results of our numerical simulations based on a couple of simple models show that agents behave in a manner qualitatively similar to what has been observed experimentally. Computational models such as those presented here are very useful to interpret observed behavior and to enhance computationally the limited variation that is possible in the experimental domain. By assuming a priori reasonable individual behaviors, the easiness of running simulations could also facilitate exploration prior to any experimental work in order to vary and estimate a number of key parameters that would be very difficult, if not impossible, to change during the actual experiment.

Cooperation in the spatial prisoner's dilemma game with probabilistic abstention

Research has shown that the addition of abstention as an option transforms social dilemmas to rock-paper-scissor type games, where defectors dominate cooperators, cooperators dominate abstainers (loners), and abstainers (loners), in turn, dominate defectors. In this way, abstention can sustain cooperation even under adverse conditions, although defection also persists due to cyclic dominance. However, to abstain or to act as a loner has, to date, always been considered as an independent, third strategy to complement traditional cooperation and defection. Here we consider probabilistic abstention, where each player is assigned a probability to abstain in a particular instance of the game. In the two limiting cases, the studied game reverts to the prisoner's dilemma game without loners or to the optional prisoner's dilemma game. For intermediate probabilities, we have a new hybrid game, which turns out to be most favorable for the successful evolution of cooperation. We hope this novel hybrid game provides a more realistic view of the dilemma of optional/voluntary participation.

Antisocial rewarding in structured populations

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.

Evolution of gossip-based indirect reciprocity on a bipartite network

Cooperation can be supported by indirect reciprocity via reputation. Thanks to gossip, reputations are built and circulated and humans can identify defectors and ostracise them. However, the evolutionary stability of gossip is allegedly undermined by the fact that it is more error-prone that direct observation, whereas ostracism could be ineffective if the partner selection mechanism is not robust. The aim of this work is to investigate the conditions under which the combination of gossip and ostracism might support cooperation in groups of different sizes. We are also interested in exploring the extent to which errors in transmission might undermine the reliability of gossip as a mechanism for identifying defectors. Our results show that a large quantity of gossip is necessary to support cooperation, and that group structure can mitigate the effects of errors in transmission.

Evolutionary Games of Multiplayer Cooperation on Graphs

There has been much interest in studying evolutionary games in structured populations, often modeled as graphs. However, most analytical results so far have only been obtained for two-player or linear games, while the study of more complex multiplayer games has been usually tackled by computer simulations. Here we investigate evolutionary multiplayer games on graphs updated with a Moran death-Birth process. For cycles, we obtain an exact analytical condition for cooperation to be favored by natural selection, given in terms of the payoffs of the game and a set of structure coefficients. For regular graphs of degree three and larger, we estimate this condition using a combination of pair approximation and diffusion approximation. For a large class of cooperation games, our approximations suggest that graph-structured populations are stronger promoters of cooperation than populations lacking spatial structure. Computer simulations validate our analytical approximations for random regular graphs and cycles, but show systematic differences for graphs with many loops such as lattices. In particular, our simulation results show that these kinds of graphs can even lead to more stringent conditions for the evolution of cooperation than well-mixed populations. Overall, we provide evidence suggesting that the complexity arising from many-player interactions and spatial structure can be captured by pair approximation in the case of random graphs, but that it need to be handled with care for graphs with high clustering.

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.

Intergroup information exchange drives cooperation in the public goods game

In this paper we explore the onset of cooperative traits in the public goods game. This well-known game involves N-agent interactions and thus reproduces a large number of social scenarios in which cooperation appears to be essential. Many studies have recently addressed how the structure of the interaction patterns influences the emergence of cooperation. Here we study how information about the payoffs collected by each individual in the different groups it participates in influences the decisions made by its group partners. Our results point out that cross-information plays a fundamental and positive role in the evolution of cooperation for different versions of the public goods game and different interaction structures.

Evolutionary dynamics of group interactions on structured populations: a review

Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.