Cooperation In Finite Populations Research Articles

Cost Optimisation of Individual-Based Institutional Reward Incentives for Promoting Cooperation in Finite Populations

In this paper, we study the problem of cost optimisation of individual-based institutional incentives (reward, punishment, and hybrid) for guaranteeing a certain minimal level of cooperative behaviour in a well-mixed, finite population. In this scheme, the individuals in the population interact via cooperation dilemmas (Donation Game or Public Goods Game) in which institutional reward is carried out only if cooperation is not abundant enough (i.e., the number of cooperators is below a threshold 1≤t≤N-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le t\\le N-1$$\\end{document}, where N is the population size); and similarly, institutional punishment is carried out only when defection is too abundant. We study analytically the cases t=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t=1$$\\end{document} for the reward incentive under the small mutation limit assumption and two different initial states, showing that the cost function is always non-decreasing. We derive the neutral drift and strong selection limits when the intensity of selection tends to zero and infinity, respectively. We numerically investigate the problem for other values of t and for population dynamics with arbitrary mutation rates.

Mixed strategy approach destabilizes cooperation in finite populations with clustering coefficient.

Evolutionary game theory, encompassing discrete, continuous, and mixed strategies, is pivotal for understanding cooperation dynamics. Discrete strategies involve deterministic actions with a fixed probability of one, whereas continuous strategies employ intermediate probabilities to convey the extent of cooperation and emphasize expected payoffs. Mixed strategies, though akin to continuous ones, calculate immediate payoffs based on the action chosen at a given moment within intermediate probabilities. Although previous research has highlighted the distinct impacts of these strategic approaches on fostering cooperation, the reasons behind the differing levels of cooperation among these approaches have remained somewhat unclear. This study explores how these strategic approaches influence cooperation in the context of the prisoner's dilemma game, particularly in networked populations with varying clustering coefficients. Our research goes beyond existing studies by revealing that the differences in cooperation levels between these strategic approaches are not confined to finite populations; they also depend on the clustering coefficients of these populations. In populations with nonzero clustering coefficients, we observed varying degrees of stable cooperation for each strategic approach across multiple simulations, with mixed strategies showing the most variability, followed by continuous and discrete strategies. However, this variability in cooperation evolution decreased in populations with a clustering coefficient of zero, narrowing the differences in cooperation levels among the strategies. These findings suggest that in more realistic settings, the robustness of cooperation systems may be compromised, as the evolution of cooperation through mixed and continuous strategies introduces a degree of unpredictability.

Cost optimisation of hybrid institutional incentives for promoting cooperation in finite populations

In this paper, we rigorously study the problem of cost optimisation of hybrid (mixed) institutional incentives, which are a plan of actions involving the use of reward and punishment by an external decision-maker, for maximising the level (or guaranteeing at least a certain level) of cooperative behaviour in a well-mixed, finite population of self-regarding individuals who interact via cooperation dilemmas (Donation Game or Public Goods Game). We show that a mixed incentive scheme can offer a more cost-efficient approach for providing incentives while ensuring the same level or standard of cooperation in the long-run. We establish the asymptotic behaviour (namely neutral drift, strong selection, and infinite-population limits). We prove the existence of a phase transition, obtaining the critical threshold of the strength of selection at which the monotonicity of the cost function changes and providing an algorithm for finding the optimal value of the individual incentive cost. Our analytical results are illustrated with numerical investigations. Overall, our analysis provides novel theoretical insights into the design of cost-efficient institutional incentive mechanisms for promoting the evolution of cooperation in stochastic systems.

Cost efficiency of institutional incentives for promoting cooperation in finite populations.

Institutions can provide incentives to enhance cooperation in a population where this behaviour is infrequent. This process is costly, and it is thus important to optimize the overall spending. This problem can be mathematically formulated as a multi-objective optimization problem where one wishes to minimize the cost of providing incentives while ensuring a minimum level of cooperation, sustained over time. Prior works that consider this question usually omit the stochastic effects that drive population dynamics. In this paper, we provide a rigorous analysis of this optimization problem, in a finite population and stochastic setting, studying both pairwise and multi-player cooperation dilemmas. We prove the regularity of the cost functions for providing incentives over time, characterize their asymptotic limits (infinite population size, weak selection and large selection) and show exactly when reward or punishment is more cost efficient. We show that these cost functions exhibit a phase transition phenomenon when the intensity of selection varies. By determining the critical threshold of this phase transition, we provide exact calculations for the optimal cost of the incentive, for any given intensity of selection. Numerical simulations are also provided to demonstrate analytical observations. Overall, our analysis provides for the first time a selection-dependent calculation of the optimal cost of institutional incentives (for both reward and punishment) that guarantees a minimum level of cooperation over time. It is of crucial importance for real-world applications of institutional incentives since the intensity of selection is often found to be non-extreme and specific for a given population.

Emergence of cooperation in finite populations under biased selection

We study the effects of biased selection on evolutionary games in finite populations. Biased selection is induced by including a number of invariant agents, who do not evolve, into an otherwise competing and evolving population. The invariant agents react differently when they encounter different types of variant agents, helping the cooperators to attain a higher payoff than the defectors by a difference Δ. The probability of a single cooperator invading and taking over a population of all defectors is evaluated based on the Moran process. When M invariant agents are introduced into the evolving population of size N, their presence helps promote the replacement of defectors by cooperators, in comparison with the case of unbiased selection with M=0. While the prisoner’s dilemma cannot sustain a cooperative population without the invariant agents, it is possible for cooperation to emerge with biased selection. We show that for a given N (M), there is a threshold of M (N) above (below) which an initial population of all defectors evolve into one of all cooperators. The dependence of the threshold on the game’s payoffs, intensity of selection, and the values of M and Δ is studied. Beyond the prisoner’s dilemma, biased selection also promotes cooperation in a bigger part of payoff parameter space corresponding to other types of games.

The evolution of cooperation in finite populations with synergistic payoffs

In a series of papers, Forber and Smead (J Philos 111(3):151–166, 2014, Biol Philos 30(3):405–421, 2015) and Smead and Forber (Evolution 67(3):698–707, 2013) make a valuable contribution to the study of cooperation in finite populations by analyzing an understudied model: the prisoner’s delight. It always pays to cooperate in the one-shot prisoner’s delight, so this model presents a best-case scenario for the evolution of cooperation. Yet, what Forber and Smead find is highly counterintuitive. In finite populations playing the prisoner’s delight, increasing the benefit of cooperation causes selection to favor defection. Here, I extend their model by considering the effects of non-linear payoffs. In particular, I show that interesting subtleties arise when payoffs are synergistic. Indeed, analysis reveals that increasing the benefit of cooperation does not always favor the spread of defection if payoffs are synergistic. I conclude by drawing some general considerations about robustness analysis in evolutionary models.

Evolution of Cooperation in Public Goods Games with Stochastic Opting-Out

We study the evolution of cooperation in group interactions where players are randomly drawn from well-mixed populations of finite size to participate in a public goods game. However, due to the possibility of unforeseen circumstances, each player has a fixed probability of being unable to participate in the game, unlike previous models which assume voluntary participation. We first study how prescribed stochastic opting-out affects cooperation in finite populations, and then generalize for the limiting case of large populations. Because we use a pairwise comparison updating rule, our results apply to both genetic and behavioral evolution mechanisms. Moreover, in the model, cooperation is favored by natural selection over both neutral drift and defection if the return on investment exceeds a threshold value depending on the population size, the game size, and a player’s probability of opting-out. Our analysis further shows that, due to the stochastic nature of the opting-out in finite populations, the threshold of return on investment needed for natural selection to favor cooperation is actually greater than the one corresponding to compulsory games with the equal expected game size. We also use adaptive dynamics to study the co-evolution of cooperation and opting-out behavior. Indeed, given rare mutations minutely different from the resident population, an analysis based on adaptive dynamics suggests that over time the population will tend towards complete defection and non-participation, and subsequently cooperators abstaining from the public goods game will stand a chance to emerge by neutral drift, thereby paving the way for the rise of participating cooperators. Nevertheless, increasing the probability of non-participation decreases the rate at which the population tends towards defection when participating. Our work sheds light on understanding how stochastic opting-out emerges in the first place and on its role in the evolution of cooperation.

Errors can increase cooperation in finite populations

I use an evolutionary game to investigate how the level of noise influences cooperation and efficiency in a dynamic setting. Players choose strategies to play indefinitely repeated prisoner's dilemmas; the strategies are represented by finite automata, and complexity costs are imposed. Players update their strategies based on the successfulness of the strategies. Using both theoretical analysis and computational experiments, I show that the presence of noise dramatically changes the system dynamics. The effect of noise interacts with the benefit of cooperation: noise can increase cooperation, but only when its level is low and the benefit of cooperation is high. In the noise-free environment, I observe constant oscillations between cooperation and defection. In contrast, the presence of noise makes Win-Stay Lose-Shift (WSLS) a successful strategy when the benefit of cooperation is sufficiently high, making cooperation relatively stable and leading to an efficient outcome.

Fixation of strategies driven by switching probabilities in evolutionary games

We study the evolutionary dynamics of strategies in finite populations which are homogeneous and well mixed by means of the pairwise comparison process, the core of which is the proposed switching probability. Previous studies about this subject are usually based on the known payoff comparison of the related players, which is an ideal assumption. In real social systems, acquiring the accurate payoffs of partners at each round of interaction may be not easy. So we bypass the need of explicit knowledge of payoffs, and encode the payoffs into the willingness of any individual shift from her current strategy to the competing one, and the switching probabilities are wholly independent of payoffs. Along this way, the strategy updating can be performed when game models are fixed and payoffs are unclear, expected to extend ideal assumptions to be more realistic one. We explore the impact of the switching probability on the fixation probability and derive a simple formula which determines the fixation probability. Moreover we find that cooperation dominates defection if the probability of cooperation replacing defection is always larger than the probability of defection replacing cooperation in finite populations. Last, we investigate the influences of model parameters on the fixation of strategies in the framework of three concrete game models: prisoner's dilemma, snowdrift game and stag-hunt game, which effectively portray the characteristics of cooperative dilemmas in real social systems.

Cooperation in finite populations: Being alone helps

We consider the evolution of cooperation in finite populations and we model a scenario where two individuals can interact only if both intend to do so with their counterpart. This feature allows a possibility for individuals to remain alone for a given round and not interact with anybody. Such an individual receives a baseline payoff rather than one based upon a matrix game. We provide sufficient conditions on the payoff matrix that will guarantee fixation probabilities to be monotone relative to the baseline payoff. We then apply the findings to the Prisoner’s Dilemma and Hawk-Dove games. In both cases, the possibility that an individual might remain alone increases the chances that cooperation or non-aggression fixes within the population. Moreover, weak selection models overlap with our model, and we consider how one can generalize our model even further.

Emergence of cooperation in public goods games

Evolution of cooperation has been a major issue in evolutionary biology. Cooperation is observed not only in dyadic interactions, but also in social interactions involving more than two individuals. It has been argued that direct reciprocity cannot explain the emergence of cooperation in large groups because the basin of attraction for the 'cooperative' equilibrium state shrinks rapidly as the group size increases. However, this argument is based on the analysis of models that consider the deterministic process. More recently, stochastic models of two-player games have been developed and the conditions for natural selection to favour the emergence of cooperation in finite populations have been specified. These conditions have been given as a mathematically simple expression, which is called the one-third law. In this paper, we investigate a stochastic model of n-player games and show that natural selection can favour a reciprocator replacing a population of defectors in the n-player repeated Prisoner's Dilemma game. We also derive a generalized version of the one-third law (the {2/[n(n+1)]}1/(n-1) law). Additionally, contrary to previous studies, the model suggests that the evolution of cooperation in public goods game can be facilitated by larger group size under certain conditions.

The Evolution of Cooperation in the Centipede Game with Finite Populations

The partial cooperation displayed by subjects in the Centipede Game deviates radically from the predictions of traditional game theory. Even standard, infinite population, evolutionary settings have failed to provide an explanation for this behavior. However, recent work in finite population evolutionary models has shown that such settings can produce radically different results from the standard models. This paper examines the evolution of partial cooperation in finite populations. The results reveal a new possible explanation that is not open to the standard models and gives us reason to be cautious when employing these otherwise helpful idealizations.

Chromodynamics of Cooperation in Finite Populations

BackgroundThe basic idea of tag-based models for cooperation is that individuals recognize each other via arbitrary signals, so-called tags. If there are tags of different colors, then cooperators can always establish new signals of recognition. The resulting “chromodynamics” is a mechanism for the evolution of cooperation. Cooperators use a secret tag until they are discovered by defectors who then destroy cooperation based on this tag. Subsequently, a fraction of the population manages to establish cooperation based on a new tag.Methodology/Principal FindingsWe derive a mathematical description of stochastic evolutionary dynamics of tag-based cooperation in populations of finite size. Benefit and cost of cooperation are given by b and c. We find that cooperators are more abundant than defectors if b/c > 1+2u/v, where u is the mutation rate changing only the strategy and v is the mutation rate changing strategy and tag. We study specific assumptions for u and v in two genetic models and one cultural model.Conclusions/SignificanceIn a genetic model, tag-based cooperation only evolves if a gene encodes both strategy and tag. In a cultural model with equal mutation rates between all possible phenotypes (tags and behaviors), the crucial condition is b/c > (K+1)/(K−1), where K is the number of tags. A larger number of tags requires a smaller benefit-to-cost ratio. In the limit of many different tags, the condition for cooperators to have a higher average abundance than defectors becomes b > c.

Stochastic dynamics of invasion and fixation

We study evolutionary game dynamics in finite populations. We analyze an evolutionary process, which we call pairwise comparison, for which we adopt the ubiquitous Fermi distribution function from statistical mechanics. The inverse temperature in this process controls the intensity of selection, leading to a unified framework for evolutionary dynamics at all intensities of selection, from random drift to imitation dynamics. We derive a simple closed formula that determines the feasibility of cooperation in finite populations, whenever cooperation is modeled in terms of any symmetric two-person game. In contrast with previous results, the present formula is valid at all intensities of selection and for any initial condition. We investigate the evolutionary dynamics of cooperators in finite populations, and study the interplay between intensity of selection and the remnants of interior fixed points in infinite populations, as a function of a given initial number of cooperators, showing how this interplay strongly affects the approach to fixation of a given trait in finite populations, leading to counterintuitive results at different intensities of selection.

有限集団における協力の進化：１／３則

有限集団における協力の進化：１／３則