On Minty-variational inequalities and evolutionary stable states of generalized monotone games

In this note, we emphasize the role played by Minty variational inequalities in evolutionary game theory for studying neutrally and evolutionary stable states of nonlinear population games. This connection allows deriving new results on the sets of neutrally stable and evolutionary stable states for generalized monotone games as well as stability results for the replicator dynamics.     

The impact of other-regarding tendencies on the spatial vaccination game

Using the evolutionary game framework to describe the vaccination behaviors of individuals, this work focuses on the impact of other-regarding tendencies on collective decision-making of vaccination. With extensive computational simulations, we find that when the vaccination cost is small or moderate, the variance of vaccination coverage as well as epidemic size is monotonically depending on the other-regarding effect. However, with a high price of vaccine, a moderate deviation from being entirely self-centered results in a poor communal vaccination benefit. In addition, we also find that the other-regarding effect may lead to an optimal total social cost.     

Effect of vaccination strategies on the dynamic behavior of epidemic spreading and vaccine coverage

The transmission of infectious, yet vaccine-preventable, diseases is a typical complex social phenomenon, where the increasing level of vaccine update in the population helps to inhibit the epidemic spreading, which in turn, however, discourages more people to participate in vaccination campaigns, due to the “externality effect” raised by vaccination. We herein study the impact of vaccination strategies, pure, continuous (rather than adopt vaccination definitely, the individuals choose to taking vaccine with some probabilities), or continuous with randomly mutation, on the vaccination dynamics with a spatial susceptible-vaccinated-infected-recovered (SVIR) epidemiological model. By means of extensive Monte-Carlo simulations, we show that there is a crossover behavior of the final vaccine coverage between the pure-strategy case and the continuous-strategy case, and remarkably, both the final vaccination level and epidemic size in the continuous-strategy case are less than them in the pure-strategy case when vaccination is cheap. We explain this phenomenon by analyzing the organization process of the individuals in the continuous-strategy case in the equilibrium. Our results are robust to the SVIR dynamics defined on other spatial networks, like the Erdős–Rényi and Barabási–Albert networks.     

Does playing several competitions influence a team’s league performance? Evidence from Spanish professional football

The sporting performance of professional football teams has often been assessed considering their results in the major regular competition, namely the national league. Here, we show that evaluating league performance without controlling for extra games played in other competitions might produce misleading results. Using Data Envelopment Analysis, we assess the performance of Spanish professional football teams in the League controlling for the extra games played in the King’s Cup, Champions League and UEFA Cup. Results show that assessing performance omitting extra games underestimates teams’ true performance in the League, the more extra games played leading to greater bias. Consequently, the multioutput nature of football must be considered when assessing team performance.     

Conditional Viability for Impulse Differential Games

We introduce in this paper the concept of “impulse evolutionary game”. Examples of evolutionary games are usual differential games, differentiable games with history (path-dependent differential games), mutational differential games, etc. Impulse evolutionary systems and games cover in particular “hybrid systems” as well as “qualitative systems”. The conditional viability kernel of a constrained set (with a target) is the set of initial states such that for all strategies (regarded as continuous feedbacks) played by the second player, there exists a strategy of the first player such that the associated run starting from this initial state satisfies the constraints until it hits the target. This paper characterizes the concept of conditional viability kernel for “qualitative games” and of conditional valuation function for “qualitative games” maximinimizing an intertemporal criterion. The theorems obtained so far about viability/capturability issues for evolutionary systems, conditional viability for differential games and about impulse and hybrid systems are used to provide characterizations of conditional viability under impulse evolutionary games.     

Risk assessment for infectious disease and its impact on voluntary vaccination behavior in social networks

Achievement of the herd immunity is essential for preventing the periodic spreading of an infectious disease such as the flu. If vaccination is voluntary, as vaccination coverage approaches the critical level required for herd immunity, there is less incentive for individuals to be vaccinated; this results in an increase in the number of so-called “free-riders” who craftily avoid infection via the herd immunity and avoid paying any cost. We use a framework originating in evolutionary game theory to investigate this type of social dilemma with respect to epidemiology and the decision of whether to be vaccinated. For each individual in a population, the decision on vaccination is associated with how one assesses the risk of infection. In this study, we propose a new risk-assessment model in a vaccination game when an individual updates her strategy, she compares her own payoff to a net payoff obtained by averaging a collective payoff over individuals who adopt the same strategy as that of a randomly selected neighbor. In previous studies of vaccination games, when an individual updates her strategy, she typically compares her payoff to the payoff of a randomly selected neighbor, indicating that the risk for changing her strategy is largely based on the behavior of one other individual, i.e., this is an individual-based risk assessment. However, in our proposed model, risk assessment by any individual is based on the collective success of a strategy and not on the behavior of any one other individual. For strategy adaptation, each individual always takes a survey of the degree of success of a certain strategy that one of her neighbors has adopted, i.e., this is a strategy-based risk assessment. Using computer simulations, we determine how these two different risk-assessment methods affect the spread of an infectious disease over a social network. The proposed model is found to benefit the population, depending on the structure of the social network and cost of vaccination. Our results suggest that individuals (or governments) should understand the structure of their social networks at the regional level, and accordingly, they should adopt an appropriate risk-assessment methodology as per the demands of the situation.     

Heterogeneous information lags and evolutionary stability

We investigate the effect of information lags in discrete time evolutionary game dynamics on symmetric games. At the end of each period, some players obtain information about the distribution of strategies among the entire population. They update their strategies according to this information. In contrast to the previous literature (e.g., Tao and Wang (1997)) where large delays lead to instability, we show that the relationship between information lags and the stability of equilibria is not “monotonic.” In anti-coordination games under smoothed best-response dynamics, a small probability of delay can stabilize the equilibrium, while a large probability can destabilize it.     

Evolutionary games and matching rules

This study considers evolutionary games with non-uniformly random matching when interaction occurs in groups of \(n\ge 2\) individuals using pure strategies from a finite strategy set. In such models, groups with different compositions of individuals generally co-exist and the reproductive success (fitness) of a specific strategy varies with the frequencies of different group types. These frequencies crucially depend on the matching process. For arbitrary matching processes (called matching rules), we study Nash equilibrium and ESS in the associated population game and show that several results that are known to hold for population games under uniform random matching carry through to our setting. In our most novel contribution, we derive results on the efficiency of the Nash equilibria of population games and show that for any (fixed) payoff structure, there always exists some matching rule leading to average fitness maximization. Finally, we provide a series of applications to commonly studied normal-form games.     

Fast Winning Strategies in Avoider-Enforcer Games

In numerous positional games the identity of the winner is easily determined. In this case one of the more interesting questions is not who wins but rather how fast can one win. These types of problems were studied earlier for Maker-Breaker games; here we initiate their study for unbiased Avoider-Enforcer games played on the edge set of the complete graph K _n on n vertices. For several games that are known to be an Enforcer’s win, we estimate quite precisely the minimum number of moves Enforcer has to play in order to win. We consider the non-planarity game, the connectivity game and the non-bipartite game.     

Order of play in strategically equivalent games in extensive form

“Can we find a pair of extensive form games that give rise to the same strategic form game such that, when played by a reasonable subject population, there is a statistically significant difference in how the games are played?” (Kreps, 1990, p. 112). And if yes, “can we organize these significant differences according to some principles that reflect recognizable differences in the extensive forms?” Both questions are answered positively by reporting results from three different experiments on public goods provision, resource dilemmas, and pure coordination games.     

Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games

The noncooperative multi-leader-follower game can be formulated as a generalized Nash equilibrium problem where each player solves a nonconvex mathematical program with equilibrium constraints. Two major deficiencies exist with such a formulation: One is that the resulting Nash equilibrium may not exist, due to the nonconvexity in each players problem; the other is that such a nonconvex Nash game is computationally intractable. In order to obtain a viable formulation that is amenable to practical solution, we introduce a class of remedial models for the multi-leader-follower game that can be formulated as generalized Nash games with convexified strategy sets. In turn, a game of the latter kind can be formulated as a quasi-variational inequality for whose solution we develop an iterative penalty method. We establish the convergence of the method, which involves solving a sequence of penalized variational inequalities, under a set of modest assumptions. We also discuss some oligopolistic competition models in electric power markets that lead to multi-leader-follower games.Jong-Shi Pang: The work of this authors research was partially supported by the National Science Foundation under grant CCR-0098013 and ECS-0080577 and by the Office of Naval Research under grant N00014-02-1-0286.Masao Fukushima: The work of this authors research was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Culture and Sports of Japan.     

Genetic learning in strategic form games

We analyze the learning behavior of a Simple Genetic Algorithm in symmetric 3 × 3 Strategic-Form-Games. In cases of contests within one population and also between two populations the behavior of the SGA is compared with the behavior of the replicator dynamics and is analyzed with respect to equilibrium concepts in evolutionary game theory. Furthermore conservative non-adaptive strings are added to the population which lead to convergence to an equilibrium even in “GA-deceptive” games where the equilibrium can not be reached by GAs using only selection and crossover.     

The evolution of cooperation in spatial groups

Much of human cooperation remains an evolutionary riddle. There is evidence that individuals are often organized into groups in many social situations. Inspired by this observation, we propose a simple model of evolutionary public goods games in which individuals are organized into networked groups. Here, nodes in the network represent groups; the edges, connecting the nodes, refer to the interactions between the groups. Individuals establish public goods games with partners in the same group and migrate among neighboring groups depending on their payoffs and expectations. We show that the paradigmatic public goods social dilemma can be resolved and high cooperation levels are attained in structured groups, even in relatively harsh conditions for cooperation. Further, by means of numerical simulations and mean-field analysis, we arrive at the result: larger average group size and milder cooperation environment would lead to lower cooperation level but higher average payoffs of the entire population. Altogether, these results emphasize that our understanding of cooperation can be enhanced by investigations of how spatial groups of individuals affect the evolution dynamics, which might help in explaining the emergence and evolution of cooperation.     

Infinite Dimensional Duality Theory Applied to Investment Strategies in Environmental Policy

In this paper, we develop an infinite dimensional Lagrangian duality framework for modeling and analyzing the evolutionary pollution control problem. Specifically, we examine the situation in which different countries aim at determining the optimal investment allocation in environmental projects and the tolerable pollutant emissions, so as to maximize their welfare. We state the equilibrium conditions underlying the model, and provide an equivalent formulation in terms of an evolutionary variational inequality. Moreover, by means of infinite dimensional duality tools, we prove the existence of Lagrange multipliers that play a fundamental role in order to describe countries’ decision-making processes.     

A generalized vector-valued variational principle in Fréchet spaces

In the framework of Frechet spaces, we give a generalized vector-valued Ekeland＇s variational principle, where the perturbation involves the subadditive functions of countable generating semi-norms. By modifying and developing the method of Cammaroto and Chinni, we obtain a density theorem on extremal points of the vector-valued variational principle, which extends and improves the related known results.     

Pure strategy equilibria in symmetric two-player zero-sum games

We observe that a symmetric two-player zero-sum game has a pure strategy equilibrium if and only if it is not a generalized rock-paper-scissors matrix. Moreover, we show that every finite symmetric quasiconcave two-player zero-sum game has a pure equilibrium. Further sufficient conditions for existence are provided. Our findings extend to general two-player zero-sum games using the symmetrization of zero-sum games due to von Neumann. We point out that the class of symmetric two-player zero-sum games coincides with the class of relative payoff games associated with symmetric two-player games. This allows us to derive results on the existence of finite population evolutionary stable strategies.     

Existence of optimal strategies in Markov games with incomplete information

The existence of a value and optimal strategies is proved for the class of two-person repeated games where the state follows a Markov chain independently of players’ actions and at the beginning of each stage only Player 1 is informed about the state. The results apply to the case of standard signaling where players’ stage actions are observable, as well as to the model with general signals provided that Player 1 has a nonrevealing repeated game strategy. The proofs reduce the analysis of these repeated games to that of classical repeated games with incomplete information on one side. This research was supported in part by Israeli Science Foundation grants 382/98, 263/03, and 1123/06, and by the Zvi Hermann Shapira Research Fund.     

Non-additive anonymous games

This paper introduces a class of non-additive anonymous games where agents are assumed to be uncertain (in the sense of Knight) about opponents’ strategies and about the initial distribution over players’ characteristics in the game. We model uncertainty by non-additive measures or capacities and prove the Cournot–Nash equilibrium existence theorem for this class of games. Equilibrium distribution can be symmetrized under milder conditions than in the case of additive games. In particular, it is not required for the space characteristics to be atomless under capacities. The set-valued map of the Cournot–Nash equilibria is upper-semicontinuous as a function of initial beliefs of the players for non-additive anonymous games.     

A dynamic Cournot–Nash game: a representation of a finitely repeated feedback game

This paper studies market outcome equivalence of two dynamic production-capital investment games under uncertainty. One is played under complete information, while the other, feedback (FB) game, is played under incomplete information about the opponents’ costs and market demand. The FB game structure may be observed in some newly initiated industries, in which a homogeneous good is exchanged via an auction mechanism. In that case, the FB game setting may predict the complete information equilibrium market outcomes.      

BSDEs with two reflecting barriers : the general result

In this paper we show the existence of a solution for the BSDE with two reflecting barriers when those latter are completely separated. Neither Mokobodzkis condition nor the regularity of the barriers are supposed. The main tool is the notion of local solution of reflected BSDEs. Applications related to Dynkin games and double obstacle variational inequality are given.Mathematics Subject Classification (2000): 91A15, 60G40, 91A60