Asymmetric evaluation of fitness enhances spatial reciprocity in social dilemmas

Evolutionary game theory has been a powerful tool to understand the ubiquity of cooperation in many real-world systems which ranges from biological to economical and social sciences. In this paper, we propose a novel game model, which considers different fitness evaluation means for focal agent and his randomly chosen neighbor, to characterize the asymmetry of information during the game playing. When playing the game, the focal agent obtains its own fitness which includes its own payoff and the average payoff of his neighbors. However, due to the limited information for the chosen neighbor, his fitness can not be evaluated as the focal agent but just the traditional way, namely his fitness equals to his payoff. Large-scale numerical simulations indicate that this kind of asymmetric fitness computation can have substantial effects on the cooperative behaviors on the regular lattice. Interestingly, the larger the asymmetric factor α, the higher the cooperation level ρ_C. Meanwhile, introducing the asymmetric fitness evaluation can induce the cooperative clusters to become larger and larger as α increases. The phase space between b and α can be enlarged, and thereby the ranges for cooperators to survive in the sea of defectors can be widely extended. Our findings can greatly be conducive to interpret the emergence of cooperation within the population.     

Considering individual satisfaction levels enhances cooperation in a spatial prisoner’s dilemma game

Based on the observance in human society, the satisfaction level of an individual as a result of an obtained payoff depends on personal tendency to some extent; we establish a new model for spatial prisoner’s dilemma games. We describe individual satisfaction as a stochastically deviated value around each of the four payoffs stipulated by a payoff matrix, which is maintained throughout the life of a certain agent. When strategy updating, an agent who refers to his own satisfaction level cannot see neighbors’ satisfaction levels but can only observe neighbors’ accumulated payoffs. By varying the update rule and underlying topology, we perform numerical simulations that reveal cooperation is significantly enhanced by this change. We argue that this enhancement of cooperation is analogous to a stochastic resonance effect, like the payoff noise effects Perc (2006).     

Cooperation in spatial prisoner’s dilemma game with delayed decisions

Phenomena that time delays of information lead to delayed decisions are extensive in reality. The effect of delayed decisions on the evolution of cooperation in the spatial prisoner’s dilemma game is explored in this work. Players with memory are located on a two dimensional square lattice, and they can keep the payoff information of his neighbors and his own in every historic generation in memory. Every player uses the payoff information in some generation from his memory and the strategy information in current generation to determine which strategy to choose in next generation. The time interval between two generations is set by the parameter m. For the payoff information is used to determine the role model for the focal player when changing strategies, the focal player’s decision to learn from which neighbor is delayed by m generations. Simulations show that cooperation can be enhanced with the increase of m. In addition, just like the original evolutionary game model (m = 0), pretty dynamic fractal patterns featuring symmetry can be obtained when m > 0 if we simulate the invasion of a single defector in world of cooperators on square lattice.     

Impact of neighborhood separation on the spatial reciprocity in the prisoner’s dilemma game

The evolutionary game theory is a very powerful tool to understand the collective cooperation behavior in many real-world systems. In the spatial game model, the payoff is often first obtained within a specific neighborhood (i.e., interaction neighborhood) and then the focal player imitates or learns the behavior of a randomly selected one inside another neighborhood which is named after the learning neighborhood. However, most studies often assume that the interaction neighborhood is identical with the learning neighborhood. Beyond this assumption, we present a spatial prisoner’s dilemma game model to discuss the impact of separation between interaction neighborhood and learning neighborhood on the cooperative behaviors among players on the square lattice. Extensive numerical simulations demonstrate that separating the interaction neighborhood from the learning neighborhood can dramatically affect the density of cooperators (ρ_C) in the population at the stationary state. In particular, compared to the standard case, we find that the medium-sized learning (interaction) neighborhood allows the cooperators to thrive and substantially favors the evolution of cooperation and ρ_C can be greatly elevated when the interaction (learning) neighborhood is fixed, that is, too little or much information is not beneficial for players to make the contributions for the collective cooperation. Current results are conducive to further analyzing and understanding the emergence of cooperation in many natural, economic and social systems.     

Sequentially compatible payoffs and the core in TU-games

In the context of cooperative TU-games, we introduce a recursive procedure to distribute the surplus of cooperation when there is an exogenous ordering among the set of players N. In each step of the process, using a given notion of reduced games, an upper and a lower bound for the payoff to the player at issue are required. Sequentially compatible payoffs are defined as those allocation vectors that meet these recursive bounds. For a family of reduction operations, the behavior of this new solution concept is analyzed. For any ordering of N, the core of the game turns out to be the set of sequentially compatible payoffs when the Davis–Maschler reduced games are used. Finally, we study which reduction operations give an advantage to the first player in the ordering.     

Potentials and reduced games for share functions

A value function for cooperative games with transferable utility assigns to every game a distribution of the payoffs. A value function is efficient if for every such a game it exactly distributes the worth that can be obtained by all players cooperating together. An approach to efficiently allocate the worth of the ‘grand coalition’ is using share functions which assign to every game a vector whose components sum up to one. Every component of this vector is the corresponding players’ share in the total payoff that is to be distributed. In this paper we give characterizations of a class of share functions containing the Shapley share function and the Banzhaf share function using generalizations of potentials and of Hart and Mas-Colell's reduced game property.     

Contests with productive effort

I consider competitions in which, conditional on winning or losing, the effort exerted by a competitor does not necessarily decrease his payoff. This happens, for example, in competitions for promotions in which workers are intrinsically motivated, and in research and development races in which better performance implies a higher payoff from winning. I characterize players’ equilibrium payoffs in closed form, thereby generalizing Siegel (Econometrica 77(1):71–92, 2009) payoff result to contests in which players’ payoff functions are non-monotonic.     

Variational inequality formulation for the games with random payoffs

We consider an n-player non-cooperative game with random payoffs and continuous strategy set for each player. The random payoffs of each player are defined using a finite dimensional random vector. We formulate this problem as a chance-constrained game by defining the payoff function of each player using a chance constraint. We first consider the case where the continuous strategy set of each player does not depend on the strategies of other players. If a random vector defining the payoffs of each player follows a multivariate elliptically symmetric distribution, we show that there exists a Nash equilibrium. We characterize the set of Nash equilibria using the solution set of a variational inequality (VI) problem. Next, we consider the case where the continuous strategy set of each player is defined by a shared constraint set. In this case, we show that there exists a generalized Nash equilibrium for elliptically symmetric distributed payoffs. Under certain conditions, we characterize the set of a generalized Nash equilibria using the solution set of a VI problem. As an application, the random payoff games arising from electricity market are studied under chance-constrained game framework.     

Stochastic games with non-observable actions

We examine n-player stochastic games. These are dynamic games where a play evolves in stages along a finite set of states; at each stage players independently have to choose actions in the present state and these choices determine a stage payoff to each player as well as a transition to a new state where actions have to be chosen at the next stage. For each player the infinite sequence of his stage payoffs is evaluated by taking the limiting average. Normally stochastic games are examined under the condition of full monitoring, i.e. at any stage each player observes the present state and the actions chosen by all players. This paper is a first attempt towards understanding under what circumstances equilibria could exist in n-player stochastic games without full monitoring. We demonstrate the non-existence of -equilibria in n-player stochastic games, with respect to the average reward, when at each stage each player is able to observe the present state, his own action, his own payoff, and the payoffs of the other players, but is unable to observe the actions of them. For this purpose, we present and examine a counterexample with 3 players. If we further drop the assumption that the players can observe the payoffs of the others, then counterexamples already exist in games with only 2 players.     

Role of update dynamics in the collective cooperation on the spatial snowdrift games: Beyond unconditional imitation and replicator dynamics

In this paper, we investigate the role of update or imitation rules in the spatial snowdrift game on regular lattices. Three different update rules, including unconditional imitation (UI), replicator dynamics (RD) and the Moran process, are utilized to update the strategies of focal players during the game process in the spatial snowdrift on the lattice. We observe that the aggregate cooperation level between players is largely elevated by using the Moran process in the spatial snowdrift game, when compared to the UI or replicator dynamics. Meanwhile, we carefully explore the dynamical evolution of frequency of cooperators and the cluster formation pattern for these three update rules. Moreover, it is also shown that the evolutionary behavior under the Moran update is independent of and insensitive to the randomly initial configurations of cooperators and defectors. The current results clearly indicate that the introduction of moderate randomness in the strategy update will highly promote the maintenance and persistence of cooperation among selfish individuals, which will be greatly instrumental to deeply understand the evolution of cooperation within many natural, biological and social systems.     

Monotone and bounded interval equilibria in a coordination game with information aggregation

We analyze how private learning in a class of games with common stochastic payoffs affects the form of equilibria, and how properties such as player welfare and the extent of strategic miscoordination relate across monotone and non-monotone equilibria. Researchers typically focus on monotone equilibria. We provide conditions under which non-monotone equilibria also exist, where players attempt to coordinate to obtain the stochastic payoff whenever signals are in a bounded interval. In bounded interval equilibria (BIE), an endogenous fear of miscoordination discourages players from coordinating to obtain the stochastic payoff when their signals suggest coordination is most beneficial. In contrast to monotone equilibria, expected payoffs from successful coordination in BIE are lower than the ex-ante expected payoff from ignoring signals and always trying to coordinate to obtain the stochastic payoff. We show that BIE only exist when, absent private information, the game would be a coordination game.     

On the Structural Stability of Values for Cooperative Games

It is generally assumed that any set of players can form a feasible coalition for classical cooperative games. But, in fact, some players may withdraw from the current game and form a union, if this makes them better paid than proposed. Based on the principle of coalition split, this paper presents an endogenous procedure of coalition formation by levels and bargaining for payoffs simultaneously, where the unions formed in the previous step continue to negotiate with others in the next step as “individuals,” looking for maximum share of surplus by organizing themselves as a partition. The structural stability of the induced payoff configuration is discussed, using two stability criteria of core notion for cooperative games and strong equilibrium notion for noncooperative games.

     

An <Emphasis Type="Italic">m</Emphasis><Superscript>th</Superscript> Generation Model for Weak Ranking of Players in a Tournament

The problem of ranking players in a tournament has been studied by a number of authors. The methods developed for ranking players fall under two general headings—direct methods and rating methods. The present paper extends the tournament ranking problem in two directions. First, the usual definition of a tournament is broadened to include ties or draws. Thus, our model determines the best weak ranking of the players. Second, the ranking method presented takes account of player strength in that wins over strong players are valued higher than wins over weak players. To account for player strength, we evaluate both direct or first-order wins of players over opponents (i defeats j) and indirect or higher-order wins (i defeats k, who defeats j). A model which derives a composite score for each player, combining both direct and indirect wins, is used to obtain an overall ranking of the competitors.     

On coalition formation: a game-theoretical approach

This paper deals with the question of coalition formation inn-person cooperative games. Two abstract game models of coalition formation are proposed. We then study the core and the dynamic solution of these abstract games. These models assume that there is a rule governing the allocation of payoffs to each player in each coalition structure called a payoff solution concept. The predictions of these models are characterized for the special case of games with side payments using various payoff solution concepts such as the individually rational payoffs, the core, the Shapley value and the bargaining set M₁ ⁽ⁱ⁾. Some modifications of these models are also discussed.     

Internal correlation in repeated games

This paper characterizes the set of all the Nash equilibrium payoffs in two player repeated games where the signal that the players get after each stage is either trivial (does not reveal any information) or standard (the signal is the pair of actions played). It turns out that if the information is not always trivial then the set of all the Nash equilibrium payoffs coincides with the set of the correlated equilibrium payoffs. In particular, any correlated equilibrium payoff of the one shot game is also a Nash equilibrium payoff of the repeated game.For the proof we develop a scheme by which two players can generate any correlation device, using the signaling structure of the game. We present strategies with which the players internally correlate their actions without the need of an exogenous mediator.     

Unequal connections

Empirical work suggests that social and economic networks are characterized by an unequal distribution of connections across individuals. This paper explores the circumstances under which networks will or will not exhibit inequality. Two specific models of network formation are explored. The first is a playing the field game in which the aggregate payoffs of an individual depend only on the number of his links and the aggregate number of links of the rest of the population. The second is a local spillovers game in which the aggregate payoffs of an individual depend on the distribution of links of all players and the identity of neighbors. For both class of games we develop results on existence and characterize equilibrium networks under different combinations of externalities/spillovers. We also examine conditions under which having more connections implies a higher payoff.     

An improved public goods game model with reputation effect on the spatial lattices

How to model the evolution of cooperation within the population is an important and interdisciplinary issue across the academia. In this paper, we propose an improved public goods game model with reputation effect on spatial lattices to investigate the evolution of cooperation regarding the allocation of public resources. In our model, we modify the individual utility or fitness as a product of the present payoff and reputation-related power function, and strategy update adopts a Fermi-like probability function during the game evolution. Meanwhile, for an interaction between a pair of partners, the reputation of a cooperative agent will be accrued beyond two units, but the defective player will decrease his reputation by one unit. Extensive Monte Carlo numerical simulations indicate the introduction of reputation will foster the formation of cooperative clusters, and greatly enhance the level of public cooperation on the spatial lattices. The larger reputation factor leads to the higher cooperation level since the reputation effect will be enormously embedded into the utility evaluation under this scenario. The current results are vastly beneficial to understand the persistence and emergence of cooperation among many natural, social and synthetic systems, and also provide some useful suggestions to devise the feasible social governance measures and modes for the public resources or affairs.     

Evolutionary Selection of Socially Sensitive Preferences in Random Matching Environments

In this paper we study the evolutionary selection of socially sensitive preferences in the context of reference interaction settings such as coordination failure and cooperation. We refer to a specific class of socially sensitive preferences in which players weigh additively their own material payoff against the opponent with either a positive or negative coefficient (λ-players). Preference evolution is guided by replicator dynamics in a context of perfect observability of preferences types and stochastic pairwise matching. We take an indirect evolutionary approach, that is, the selection mechanism operates on the actual material payoffs earned by players, so that any instance of socially sensitive preference can be thought of as instrumentally maintained. We find that the evolutionary viability of socially sensitive preferences basically depends on whether or not they cause a substantial improvement in the achievement of socially efficient outcomes with respect to the case where only self-serving or unconditionally focused preference orientations are observed. Our results suggest that moderate pro-social preference orientations are likely to emerge from social selection even in the absence of an intrinsic motivational drive, whereas extremely pro-social orientations as well as competitive and anti-social ones may need a stronger motivational base.     

A dynamic solution in N-person cooperative game theory

We formulate a cooperative game as an extended form game in which each player in turn proposes payoffs to a coalition over M steps. Payoffs at time t are discounted by a penalty function f(t). If all players in a coalition agree to their payoffs, they receive them. Under a convergence hypothesis verified by computer for three players in many cases, we compute the payoffs resulting from a coalition pattern and give necessary conditions for particular patterns. The resulting solution is related to the Nash bargaining solution and the competitive solution.     

Average-discounted equilibria in stochastic games

In stochastic games with finite state and action spaces, we examine existence of equilibria where player 1 uses the limiting average reward and player 2 a discounted reward for the evaluations of the respective payoff sequences. By the nature of these rewards, the far future determines player 1's reward, while player 2 is rather interested in the near future. This gives rise to a natural cooperation between the players along the course of the play. First we show the existence of stationary ε-equilibria, for all ε>0, in these games. However, besides these stationary ε-equilibria, there also exist ε-equilibria, in terms of only slightly more complex ultimately stationary strategies, which are rather in the spirit of these games because, after a large stage when the discounted game is not interesting any longer, the players cooperate to guarantee the highest feasible reward to player 1. Moreover, we analyze an interesting example demonstrating that 0-equilibria do not necessarily exist in these games, not even in terms of history dependent strategies. Finally, we examine special classes of stochastic games with specific conditions on the transition and payoff structures. Several examples are given to clarify all these issues.