Average Payoff Research Articles

Integrating the environmental factor into the strategy updating rule to promote cooperation in evolutionary games

Many previous studies have shown that the environment plays an important role for social individuals. In this paper, we integrate the environmental factor, which is defined as the average payoff of all a player's neighbours, with the standard Fermi updating rule by introducing a tunable parameter, ω. It is found that the level of cooperation increases remarkably, and that the cooperators can better resist the invasion of defection with an increase in ω. This interesting phenomenon is then explained from a microscopic view. In addition, the universality of this mechanism is also proved with the help of the small-world network and the random regular graph. This work may be helpful in understanding cooperation behaviour in species from unicellular organisms up to human beings.

Individual variation and the resolution of conflict over parental care in penduline tits

Eurasian penduline tits (Remiz pendulinus) have an unusually diverse breeding system consisting of frequent male and female polygamy, and uniparental care by the male or the female. Intriguingly, 30 to 40 per cent of all nests are deserted by both parents. To understand the evolution of this diverse breeding system and frequent clutch desertion, we use 6 years of field data to derive fitness expectations for males and females depending on whether or not they care for their offspring. The resulting payoff matrix corresponds to an asymmetric Snowdrift Game with two alternative evolutionarily stable strategies (ESSs): female-only and male-only care. This, however, does not explain the polymorphism in care strategies and frequent biparental desertion, because theory predicts that one of the two ESSs should have spread to fixation. Using a bootstrapping approach, we demonstrate that taking account of individual variation in payoffs explains the patterns of care better than a model based on the average population payoff matrix. In particular, a model incorporating differences in male attractiveness closely predicts the observed frequencies of male and female desertion. Our work highlights the need for a new generation of individual-based evolutionary game-theoretic models.

Dynamic allocation of investments promotes cooperation in spatial public goods game

In this paper, we introduce a dynamic allocation investment mechanism into the spatial public goods game. In the model, if a cooperator i collects more payoffs from one group centered on the player j in the last round, he will put more investments to the group in the next round. The intensity of investment is controlled by a tunable parameter α. Simulation results show that the emergence of cooperation is remarkably promoted, and a larger α induces a higher cooperation level. These interesting phenomena are then extensively explained by pairwise investments, average payoffs and evolution snapshots.

Bias and Overtaking Equilibria for Zero-Sum Stochastic Differential Games

This paper deals with zero-sum stochastic differential games with long-run average payoffs. Our main objective is to give conditions for existence and characterization of bias and overtaking optimal equilibria. To this end, first we characterize the family of optimal average payoff strategies. Then, within this family, we impose suitable conditions to determine the subfamilies of bias and overtaking equilibria. A key step to obtain these facts is to show the existence of solutions to the average payoff optimality equations. This is done by the usual “vanishing discount” approach. Finally, a zero-sum game associated to a certain manufacturing process illustrates our results.

Imperfect Public Monitoring with Costly Punishment: An Experimental Study

This paper experimentally investigates the effects of a costly punishment option on cooperation and social welfare in long, finitely repeated public good contribution games. In a perfect monitoring environment, increasing the severity of the potential punishment monotonically increases average net payoffs. In a more realistic imperfect monitoring environment, we find a U-shaped relationship. Access to a standard punishment technology in this setting significantly decreases net payoffs, even in the long run. Access to a severe punishment technology leads to roughly the same payoffs as with no punishment option, as the benefits of increased cooperation offset the social costs of punishing. (JEL C92, H41, K42)

Too hungry to learn? Hungry broiler breeders fail to learn a Y-maze food quantity discrimination task

AbstractChoice tests may aid determining whether qualitative dietary restriction improves the welfare of feed-restricted broiler breeder chickens(Gallus gallus domesticus). However, hunger-stress may reduce competency to choose by impairing learning. The effect of chronic feed restriction on the ability of broiler breeders to learn a hunger-relevant discrimination task was investigated using a Y-maze paradigm. The task was to associate black and white arms with large and small quantities of feed. Birds were reared to three growth curves by means of severe (n = 12), moderate (n = 12) or very mild feed restriction (n = 12). Learning the task and selecting the larger food option allowed birds to increase their feed intake. Time taken to traverse the Y-maze was also measured. Birds from all treatment groups traversed the Y-maze more quickly over time, indicating that they had learnt that running down the Y-maze arms was associated with a rewarding outcome (food). However, feed restriction significantly reduced their ability to associate the black and white cues with differences in food quantity. Consequently, average pay-offs in terms of daily feed increments disproportionately accrued to the less feed-restricted treatment groups. It is concluded that feed restriction affected the performance of broiler breeders in this task, perhaps by narrowing their attention such that they ignore potentially hunger-relevant contextual cues. However, low overall group success rates demonstrate that this task was difficult to learn even for less severely feed-restricted birds. Therefore, Y-maze choice tests may not be the most appropriate method for determining hungry broiler breeder dietary preferences.

Utility maximization, risk aversion, and stochastic dominance

Consider an investor trading dynamically to maximize expected utility from terminal wealth. Our aim is to study the dependence between her risk aversion and the distribution of the optimal terminal payoff. Economic intuition suggests that high risk aversion leads to a rather concentrated distribution, whereas lower risk aversion results in a higher average payoff at the expense of a more widespread distribution. Dybvig and Wang (J. Econ. Theory, 2011, to appear) find that this idea can indeed be turned into a rigorous mathematical statement in one-period models. More specifically, they show that lower risk aversion leads to a payoff which is larger in terms of second order stochastic dominance. In the present study, we extend their results to (weakly) complete continuous-time models. We also complement an ad-hoc counterexample of Dybvig and Wang, by showing that these results are “fragile”, in the sense that they fail in essentially any model, if the latter is perturbed on a set of arbitrarily small probability. On the other hand, we establish that they hold for power investors in models with (conditionally) independent increments.

Optional contributions have positive effects for volunteering public goods games

Public goods (PG) games with the volunteering mechanism are referred to as volunteering public goods (VPG) games, in which loners are introduced to the PG games, and a loner obtains a constant payoff but not participating the game. Considering that small contributions may have positive effects to encourage more players with bounded rationality to contribute, this paper introduces optional contributions (high value or low value) to these typical VPG games—a cooperator can contribute a high or low payoff to the public pools. With the low contribution, the logit dynamics show that cooperation can be promoted in a well mixed population comparing to the typical VPG games, furthermore, as the multiplication factor is greater than a threshold, the average payoff of the population is also enhanced. In spatial VPG games, we introduce a new adjusting mechanism that is an approximation to best response. Some results in agreement with the prediction of the logit dynamics are found. These simulation results reveal that for VPG games the option of low contributions may be a better method to stimulate the growth of cooperation frequency and the average payoff of the population.

On the effectiveness of service differentiation based resource-provision incentive mechanisms in dynamic and autonomous P2P networks

Intrinsically, P2P (Peer-to-Peer) networks are anonymous, dynamic and autonomous, which has the following implications: users can change their identities with near zero cost (cheap pseudonyms); most interactions should be one-time (that is, each peer has no idea about other peers’ behavior history, except their current behaviors); and all behaviors and actions are all endogenous, voluntarily chosen and determined by independent and rational peers. On the other hand, service differentiation based incentive mechanisms are proposed in P2P networks, which basically could be provided with two ways: punish defect behavior (punishment-based scheme), or reward cooperative behavior (reward-based scheme). Then the naturally resulted question is that: under the above P2P networking environment, how to effectively design service differentiation based resource-provision incentive mechanisms? Our contributions are threefold. First, we found that the traditional service differentiation based incentive schemes cannot successfully encourage peers to contribute resource to the whole system, irrespective of punishment-based and reward-based schemes. Then, if peers can voluntarily join the system, and small entry fee is set for participation, we obtained that the performance of punishment-based scheme (first providing high-level service plus punishment) is always better than that of reward-based scheme (first providing low-level service plus reward). Finally, unlike the existing result that was based on persistent users’ identities and truly repeated interactions, we illustrate that punisher’s average payoff in punishment-based scheme is almost same as the ideal but unfeasible case of reward-based scheme: rewarder could selectively reward other rewarders, and reward cost is zero.

Environmental options and technological innovation: an evolutionary game model

This paper analyzes the effects on economic agents’ behavior of an innovative environmental protection mechanism that the public administration of a tourist region may adopt to attract visitors while protecting the environment. On the one hand, the public administration sells to the tourists an environmental call option that gives them the possibility of being (partially or totally) reimbursed if the environmental quality in the region turns out to be unsatisfactory. On the other hand, it offers the firms that adopt an innovative, non-polluting technology an environmental put option that allows them to get a reimbursement for the additional costs imposed by the new technology if the environmental quality is sufficiently good. The aim of the paper is to study the dynamics that arise with this financial mechanism from the interaction between the economic agents and the public administration in an evolutionary game context. The evolution of visitors’ and firms’ behavior is modeled in the paper using the so-called replicator dynamics, according to which a given choice spreads across the population as long as its expected payoff is greater than the average payoff. From the model it emerges that such dynamics may lead either to a welfare-improving attractive Nash equilibrium, in which all firms adopt the environmental-friendly technology, or to a Pareto-dominated equilibrium with no technological innovation and no tourism. As shown in the paper, the attraction basin of the virtuous equilibrium will be maximum if total reimbursement is offered by the public administration to the visitors, and will be minimum if a simple entrance ticket is imposed on the tourists with no chance of reimbursement.

The Bounded Strength of Weak Expectations

The rational price of the Pasadena and Altadena games, introduced by Nover and Hajek (2004), has been the subject of considerable discussion. Easwaran (2008) has suggested that weak expectations — the value to which the average payoffs converge in probability — can give the rational price of such games. We argue against the normative force of weak expectations in the standard framework. Furthermore, we propose to replace this framework by a bounded utility perspective: this shift renders the problem more realistic and accounts for the role of weak expectations. In particular, we demonstrate that in a bounded utility framework, all agents, even if they have different value functions and disagree on the price of an individual game, will finally agree on the rational price of a repeated, averaged game. Thus, we explain the intuitive appeal of weak expectations, while avoiding both trivialization of the original paradox and the drawbacks of previous approaches.

An Adjusted Payoff-Based Procedure for Normal Form Games

We study a simple adaptive model in the framework of an N-player normal form game. The model consists of a repeated game where the players only know their own action space and their own payoff scored at each stage, not those of the other agents. Each player, in order to update her mixed action, computes the average vector payoff she has obtained by using the number of times she has played each pure action. The resulting stochastic process is analyzed via the ODE method from stochastic approximation theory. We are interested in the convergence of the process to rest points of the related continuous dynamics. Results concerning almost sure convergence and convergence with positive probability are obtained and applied to a traffic game. We also provide some examples where convergence occurs with probability zero.

Distributed Coding of Actual and Hypothetical Outcomes in the Orbital and Dorsolateral Prefrontal Cortex

Knowledge about hypothetical outcomes from unchosen actions is beneficial only when such outcomes can be correctly attributed to specific actions. Here we show that during a simulated rock-paper-scissors game, rhesus monkeys can adjust their choice behaviors according to both actual and hypothetical outcomes from their chosen and unchosen actions, respectively. In addition, neurons in both dorsolateral prefrontal cortex and orbitofrontal cortex encoded the signals related to actual and hypothetical outcomes immediately after they were revealed to the animal. Moreover, compared to the neurons in the orbitofrontal cortex, those in the dorsolateral prefrontal cortex were more likely to change their activity according to the hypothetical outcomes from specific actions. Conjunctive and parallel coding of multiple actions and their outcomes in the prefrontal cortex might enhance the efficiency of reinforcement learning and also contribute to their context-dependent memory.

Stochastic Games with Unbounded Payoffs: Applications to Robust Control in Economics

We study a discounted maxmin control problem with general state space. The controller is unsure about his model in the sense that he also considers a class of approximate models as possibly true. The objective is to choose a maxmin strategy that will work under a range of different model specifications. This is done by dynamic programming techniques. Under relatively weak conditions, we show that there is a solution to the optimality equation for the maxmin control problem as well as an optimal strategy for the controller. These results are applied to the theory of optimal growth and the Hansen–Sargent robust control model in macroeconomics. We also study a class of zero-sum discounted stochastic games with unbounded payoffs and simultaneous moves and give a brief overview of recent results on stochastic games with weakly continuous transitions and the limiting average payoffs.

Bounded Memory, Inertia, Sampling and Weighting Model for Market Entry Games

This paper describes the “Bounded Memory, Inertia, Sampling and Weighting” (BI-SAW) model, which won the http://sites.google.com/site/gpredcomp/Market Entry Prediction Competition in 2010. The BI-SAW model refines the I-SAW Model (Erev et al. [1]) by adding the assumption of limited memory span. In particular, we assume when players draw a small sample to weight against the average payoff of all past experience, they can only recall 6 trials of past experience. On the other hand, we keep all other key features of the I-SAW model: (1) Reliance on a small sample of past experiences, (2) Strong inertia and recency effects, and (3) Surprise triggers change. We estimate this model using the first set of experimental results run by the competition organizers, and use it to predict results of a second set of similar experiments later ran by the organizers. We find significant improvement in out-of-sample predictability (against the I-SAW model) in terms of smaller mean normalized MSD, and such result is robust to resampling the predicted game set and reversing the role of the sets of experimental results. Our model’s performance is the best among all the participants.

Graph-based quadratic optimization: A fast evolutionary approach

Quadratic optimization lies at the very heart of many structural pattern recognition and computer vision problems, such as graph matching, object recognition, image segmentation, etc., and it is therefore of crucial importance to devise algorithmic solutions that are both efficient and effective. As it turns out, a large class of quadratic optimization problems can be formulated in terms of so-called “standard quadratic programs” (StQPs), which ask for finding the extrema of a quadratic polynomial over the standard simplex. Computationally, the standard approach for attacking this class of problems is to use replicator dynamics, a well-known family of algorithms from evolutionary game theory inspired by Darwinian selection processes. Despite their effectiveness in finding good solutions in a variety of applications, however, replicator dynamics suffer from being computationally expensive, as they require a number of operations per step which grows quadratically with the dimensionality of the problem being solved. In order to avoid this drawback, in this paper we propose a new population game dynamics (I nI mD yn) which is motivated by the analogy with infection and immunization processes within a population of “players.” We prove that the evolution of our dynamics is governed by a quadratic Lyapunov function, representing the average population payoff, which strictly increases along non-constant trajectories and that local solutions of StQPs are asymptotically stable (i.e., attractive) points. Each step of I nI mD yn is shown to have a linear time/space complexity, thereby allowing us to use it as a more efficient alternative to standard approaches for solving StQPs and related optimization problems. Indeed, we demonstrate experimentally that I nI mD yn is orders of magnitude faster than, and as accurate as, replicator dynamics on various applications ranging from tree matching to image registration, matching and segmentation.

A MODEL FOR THE LONG-TERM OPTIMAL CAPACITY LEVEL OF AN INVESTMENT PROJECT

We consider an investment project that produces a single commodity. The project's operation yields payoff at a rate that depends on the project's installed capacity level and on an underlying economic indicator such as the output commodity's price or demand, which we model by an ergodic, one-dimensional Itô diffusion. The project's capacity level can be increased dynamically over time. The objective is to determine a capacity expansion strategy that maximizes the ergodic or long-term average payoff resulting from the project's management. We prove that it is optimal to increase the project's capacity level to a certain value and then take no further actions. The optimal capacity level depends on both the long-term average and the volatility of the underlying diffusion.

Coveting thy neighbors fitness as a means to resolve social dilemmas

In spatial evolutionary games the fitness of each individual is traditionally determined by the payoffs it obtains upon playing the game with its neighbors. Since defection yields the highest individual benefits, the outlook for cooperators is gloomy. While network reciprocity promotes collaborative efforts, chances of averting the impending social decline are slim if the temptation to defect is strong. It is, therefore, of interest to identify viable mechanisms that provide additional support for the evolution of cooperation. Inspired by the fact that the environment may be just as important as inheritance for individual development, we introduce a simple switch that allows a player to either keep its original payoff or use the average payoff of all its neighbors. Depending on which payoff is higher, the influence of either option can be tuned by means of a single parameter. We show that, in general, taking into account the environment promotes cooperation. Yet coveting the fitness of one's neighbors too strongly is not optimal. In fact, cooperation thrives best only if the influence of payoffs obtained in the traditional way is equal to that of the average payoff of the neighborhood. We present results for the prisoner's dilemma and the snowdrift game, for different levels of uncertainty governing the strategy adoption process, and for different neighborhood sizes. Our approach outlines a viable route to increased levels of cooperative behavior in structured populations, but one that requires a thoughtful implementation.

Blowing in the Wind: Vanishing Payoffs of a Tolling Agreement for Natural-gas-fired Generation of Electricity in Texas

We use a large Texas database to quantify the effect of rising wind generation on the payoffs of a tolling agreement for natural-gas-fired generation of electricity. We find that while a 20% increase in wind generation may not have a statistically-significant effect, a 40% increase can reduce the agreement’s average payoff by 8% to 13%. Since natural-gas-fired generation is necessary for integrating large amounts of intermittent wind energy into an electric grid, our finding contributes to the policy debate of capacity adequacy and system reliability in a restructured electricity market that will see large-scale wind-generation development. doi: 10.5547/ISSN0195-6574-EJ-Vol33-No1-9

Are Monkeys Sensitive to the Regularity of Pay-off?

Animals commonly face fluctuations in their environment and resources. To maximize their benefits,they need to integrate the risks attached to potential pay-offs. We do not know, however, to what extent individuals account for irregularity in the latter. We tested the sensitivity of monkeys (Cebusapella, Macaca tonkeana, M. fascicularis) to the irregularity of pay-offs in two different tasks. In a first experiment, the subjects were given an exchange task where the reward probability varied between different conditions, but yielded the same average pay-off. There was no evidence of subjects favoring either condition, meaning that they behaved in accordance with the predictions ofthe classical decision theory (Expected Utility Theory). In a second experiment, we offered to subjects a choice between two options involving different pay-off regularity. In this case, a wide range of inter-individual variation was found in the choices of individuals. Whereas monkeys accepted irregular pay-off in a rational way, there were individual biases in their preferences. These results indicate that the preferences of animals in a risky situation were not unequivocally shaped by the environment in which species have evolved.