Evolutionary dynamics of public goods games with diverse contributions in finite populations

Stochastic evolutionary dynamics of trust games with asymmetric parameters

Trusting in others and reciprocating that trust with trustworthy actions are crucial to successful and prosperous societies. The trust game has been widely used to quantitatively study trust and trustworthiness, involving a sequential exchange between an investor and a trustee. Deterministic evolutionary game theory predicts no trust and no trustworthiness, whereas the behavioral experiments with the one-shot anonymous trust game show that people substantially trust and respond trustworthily. To explain these discrepancies, previous works often turn to additional mechanisms, which are borrowed from other games such as the prisoner's dilemma. Although these mechanisms lead to the evolution of trust and trustworthiness to an extent, the optimal or the most common strategy often involves no trustworthiness. In this paper, we study the impact of asymmetric demographic parameters (e.g., different population sizes) on game dynamics of the trust game. We show that, in a weak-mutation limit, stochastic evolutionary dynamics with the asymmetric parameters can lead to the evolution of high trust and high trustworthiness without any additional mechanisms in well-mixed finite populations. Even full trust and near full trustworthiness can be the most common strategies. These results are qualitatively different from those of the previous works. Our results thereby demonstrate rich evolutionary dynamics of the asymmetric trust game.

Symmetries in the time-averaged dynamics of networks: Reducing unnecessary complexity through minimal network models

Complex networks are the subject of fundamental interest from the scientific community at large. Several metrics have been introduced to characterize the structure of these networks, such as the degree distribution, degree correlation, path length, clustering coefficient, centrality measures, etc. Another important feature is the presence of network symmetries. In particular, the effect of these symmetries has been studied in the context of network synchronization, where they have been used to predict the emergence and stability of cluster synchronous states. Here, we provide theoretical, numerical, and experimental evidence that network symmetries play a role in a substantially broader class of dynamical models on networks, including epidemics, game theory, communication, and coupled excitable systems; namely, we see that in all these models, nodes that are related by a symmetry relation show the same time-averaged dynamical properties. This discovery leads us to propose reduction techniques for exact, yet minimal, simulation of complex networks dynamics, which we show are effective in order to optimize the use of computational resources, such as computation time and memory.

Cyclic public goods games: Compensated coexistence among mutual cheaters stabilized by optimized penalty taxation

We study the problem of stabilized coexistence in a three-species public goods game in which each species simultaneously contributes to one public good while freeloading off another public good ("cheating"). The proportional population growth is governed by an appropriately modified replicator equation, depending on the returns from the public goods and the cost. We show that the replicator dynamic has at most one interior unstable fixed point and that the population becomes dominated by a single species. We then show that by applying an externally imposed penalty, or "tax" on success can stabilize the interior fixed point, allowing for the symbiotic coexistence of all species. We show that the interior fixed point is the point of globally minimal total population growth in both the taxed and untaxed cases. We then formulate an optimal taxation problem and show that it admits a quasilinearization, resulting in novel necessary conditions for the optimal control. In particular, the optimal control problem governing the tax rate must solve a certain second-order ordinary differential equation.

Extrapolating weak selection in evolutionary games

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

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

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.

How small are small mutation rates?

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.