Is it better to give information, receive it, or be ignorant in a two-player game?

The standard approach in a biological two-player game is toassume both players choose their actions independently of oneanother, having no information about their opponent's action(simultaneous game). However, this approach is not realisticin some circumstances. In many cases, one player chooses hisaction first and then the second player chooses her action withinformation about his action (Stackelberg game). We comparethese two games, which can be mathematically analyzed into twotypes, depending on the direction of the best response function(BRF) at the evolutionarily stable strategy in the simultaneousgame (ESS_sim). We subcategorize each type of game into two cases,depending on the change in payoff to one player, when both playersare at the ESS_sim, and the other player increases his action.Our results show that in cases where the BRF is decreasing atthe ESS_sim, the first player in the Stackelberg game receivesthe highest payoff, followed by both players in the simultaneousgame, followed by the second player in the Stackelberg game.In these cases, it is best to be the first Stackelberg player.In cases where the BRF is increasing at the ESS_sim, both Stackelbergplayers receive a higher payoff than players in a simultaneousgame. In these cases, it is better for both players to playa Stackelberg game rather than a simultaneous game. However,in some cases the first Stackelberg player receives a higherpayoff than the second Stackelberg player, and in some casesthe opposite is true.     

Evolution of Cooperation in Spatial Traveler's Dilemma Game

Traveler''s dilemma (TD) is one of social dilemmas which has been well studied in the economics community, but it is attracted little attention in the physics community. The TD game is a two-person game. Each player can select an integer value between and () as a pure strategy. If both of them select the same value, the payoff to them will be that value. If the players select different values, say and (), then the payoff to the player who chooses the small value will be and the payoff to the other player will be . We term the player who selects a large value as the cooperator, and the one who chooses a small value as the defector. The reason is that if both of them select large values, it will result in a large total payoff. The Nash equilibrium of the TD game is to choose the smallest value . However, in previous behavioral studies, players in TD game typically select values that are much larger than , and the average selected value exhibits an inverse relationship with . To explain such anomalous behavior, in this paper, we study the evolution of cooperation in spatial traveler''s dilemma game where the players are located on a square lattice and each player plays TD games with his neighbors. Players in our model can adopt their neighbors'' strategies following two standard models of spatial game dynamics. Monte-Carlo simulation is applied to our model, and the results show that the cooperation level of the system, which is proportional to the average value of the strategies, decreases with increasing until is greater than the critical value where cooperation vanishes. Our findings indicate that spatial reciprocity promotes the evolution of cooperation in TD game and the spatial TD game model can interpret the anomalous behavior observed in previous behavioral experiments.     

Fairness modulates non-conscious facial mimicry in women

In societies with high cooperation demands, implicit consensus on social norms enables successful human coexistence. Mimicking other people's actions and emotions has been proposed as a means to synchronize behaviour, thereby enhancing affiliation. Mimicry has long been thought to be reflexive, but it has recently been suggested that mimicry might also be motivationally driven. Here, we show during an economic bargaining game that automatic happy mimicry of those making unfair offers disappears. After the bargaining game, when the proposers have acquired either a fair or unfair reputation, we observe increased angry mimicry of proposers with an unfair reputation and decreased angry mimicry of fair proposers. These findings provide direct empirical evidence that non-conscious mimicry is modulated by fairness. We interpret the present results as reflecting that facial mimicry in women functions conditionally, dependent on situational demands.     

Evolutionary game dynamics in a Wright-Fisher process

Evolutionary game dynamics in finite populations can be described by a frequency dependent, stochastic Wright-Fisher process. We consider a symmetric game between two strategies, A and B. There are discrete generations. In each generation, individuals produce offspring proportional to their payoff. The next generation is sampled randomly from this pool of offspring. The total population size is constant. The resulting Markov process has two absorbing states corresponding to homogeneous populations of all A or all B. We quantify frequency dependent selection by comparing the absorption probabilities to the corresponding probabilities under random drift. We derive conditions for selection to favor one strategy or the other by using the concept of total positivity. In the limit of weak selection, we obtain the 1/3 law: if A and B are strict Nash equilibria then selection favors replacement of B by A, if the unstable equilibrium occurs at a frequency of A which is less than 1/3.     

Replicator dynamics for optional public good games

The public goods game represents a straightforward generalization of the prisoner's dilemma to an arbitrary number of players. Since the dominant strategy is to defect, both classical and evolutionary game theory predict the asocial outcome that no player contributes to the public goods. In contrast to the compulsory public goods game, optional participation provides a natural way to avoid deadlocks in the state of mutual defection. The three resulting strategies--collaboration or defection in the public goods game, as well as not joining at all--are studied by means of a replicator dynamics, which can be completely analysed in spite of the fact that the payoff terms are nonlinear. If cooperation is valuable enough, the dynamics exhibits a rock-scissors-paper type of cycling between the three strategies, leading to sizeable average levels of cooperation in the population. Thus, voluntary participation makes cooperation feasible. But for each strategy, the average payoff value remains equal to the earnings of those not participating in the public goods game.     

A Geometrical Perspective for the Bargaining Problem

A new treatment to determine the Pareto-optimal outcome for a non-zero-sum game is presented. An equilibrium point for any game is defined here as a set of strategy choices for the players, such that no change in the choice of any single player will increase the overall payoff of all the players. Determining equilibrium for multi-player games is a complex problem. An intuitive conceptual tool for reducing the complexity, via the idea of spatially representing strategy options in the bargaining problem is proposed. Based on this geometry, an equilibrium condition is established such that the product of their gains over what each receives is maximal. The geometrical analysis of a cooperative bargaining game provides an example for solving multi-player and non-zero-sum games efficiently.     

Statistical fluctuations in population bargaining in the ultimatum game: Static and evolutionary aspects

We explore the emergent behavior in heterogeneous populations where players negotiate via an ultimatum game: two players are offered a gift, one of them (the proposer) suggests how to divide the offer while the other player (the responder) can either accept or reject the deal. Rejection is detrimental to both players as it results in no earnings. In this context, our contribution is twofold: (i) we consider a population where the distribution of used strategies is constant over time and properties of the random payoff received by the players (average and higher moments) are reported from simple exact methods and corroborated by computer simulations; (ii) the evolution of a population is analyzed via Monte Carlo simulations where agents may change independently the proposing and accepting parameters of their strategy depending on received payoffs. Our results show that evolution leads to a stationary state in which wealth (accumulated payoff) is fairly distributed. As time evolves, an increase in average payoff and a simultaneous variance decrease is observed when we use a dynamics based on a probabilistic version of the saying: “One should not comply with small earnings, but one's greed must be limited.”     

The replicator equation on graphs

We study evolutionary games on graphs. Each player is represented by a vertex of the graph. The edges denote who meets whom. A player can use any one of n strategies. Players obtain a payoff from interaction with all their immediate neighbors. We consider three different update rules, called 'birth-death', 'death-birth' and 'imitation'. A fourth update rule, 'pairwise comparison', is shown to be equivalent to birth-death updating in our model. We use pair approximation to describe the evolutionary game dynamics on regular graphs of degree k. In the limit of weak selection, we can derive a differential equation which describes how the average frequency of each strategy on the graph changes over time. Remarkably, this equation is a replicator equation with a transformed payoff matrix. Therefore, moving a game from a well-mixed population (the complete graph) onto a regular graph simply results in a transformation of the payoff matrix. The new payoff matrix is the sum of the original payoff matrix plus another matrix, which describes the local competition of strategies. We discuss the application of our theory to four particular examples, the Prisoner's Dilemma, the Snow-Drift game, a coordination game and the Rock-Scissors-Paper game.     

Natural selection and social preferences

A large number of individuals are randomly matched into groups, where each group plays a finite symmetric game. Individuals breed true. The expected number of surviving offspring depends on own material payoff, but may also, due to cooperative breeding and/or reproductive competition, depend on the material payoffs to other group members. The induced population dynamic is equivalent with the replicator dynamic for a game with payoffs derived from those in the original game. We apply this selection dynamic to a number of examples, including prisoners' dilemma games with and without a punishment option, coordination games, and hawk-dove games. For each of these, we compare the outcomes with those obtained under the standard replicator dynamic. By way of a revealed-preference argument, our selection dynamic can explain certain "altruistic" and "spiteful" behaviors that are consistent with individuals having social preferences.     

How inconsistency between attitude and behavior persists through cultural transmission

Individuals tend to conform their behavior to that of the majority. Consequently, an individual's behavior is not always consistent with his or her attitude, and such inconsistency sometimes causes mental distress. Understanding the mechanism of sustaining inconsistency between attitude and behavior is a challenging problem from the viewpoint of evolutionary theory. We constructed an evolutionary game theory model in which each player has an attitude and behavior toward a single social norm, and the players' attitudes and behaviors are affected by three types of cultural transmission: vertical, oblique, and horizontal. We assumed that strategy is a combination of attitude and behavior and that the process of learning or transmitting the social norm depends on the life stage of each player. Adults play a coordination game in which players whose behaviors match those of the majority obtain a high payoff, which is diminished by any inconsistency between attitude and behavior. The adults' strategies are passed to newborns via vertical transmission, and the frequency of a newborn's replication of strategy is proportional to the corresponding adult's payoff. Newborns imitate behaviors of unrelated adults via oblique transmission. Juveniles change their attitudes or behaviors by observing other juveniles' behaviors or inferring other juveniles' attitudes (horizontal transmission). We conclude that the key factor for sustaining inconsistency between attitude and behavior is the ability of players to infer and imitate others' attitudes, and that oblique transmission promotes inconsistency.     

Threshold Games and Cooperation on Multiplayer Graphs

ObjectiveThe study investigates the effect on cooperation in multiplayer games, when the population from which all individuals are drawn is structured—i.e. when a given individual is only competing with a small subset of the entire population.MethodTo optimize the focus on multiplayer effects, a class of games were chosen for which the payoff depends nonlinearly on the number of cooperators—this ensures that the game cannot be represented as a sum of pair-wise interactions, and increases the likelihood of observing behaviour different from that seen in two-player games. The chosen class of games are named “threshold games”, and are defined by a threshold, M > 0, which describes the minimal number of cooperators in a given match required for all the participants to receive a benefit. The model was studied primarily through numerical simulations of large populations of individuals, each with interaction neighbourhoods described by various classes of networks.ResultsWhen comparing the level of cooperation in a structured population to the mean-field model, we find that most types of structure lead to a decrease in cooperation. This is both interesting and novel, simply due to the generality and breadth of relevance of the model—it is likely that any model with similar payoff structure exhibits related behaviour. More importantly, we find that the details of the behaviour depends to a large extent on the size of the immediate neighbourhoods of the individuals, as dictated by the network structure. In effect, the players behave as if they are part of a much smaller, fully mixed, population, which we suggest an expression for.

Highlights

• Observed behaviour depends on the size of each player’s immediate interaction neighbourhood.
• When the number of players is much larger than the number of required cooperators, average payoff decreases.
• Most network structures lead to a decrease in cooperation compared to the fully mixed case.

     

Win-stay, lose-shift strategies for repeated games-memory length, aspiration levels and noise.

Win-stay, lose-shift, the principle to retain a successful action is a simple and general learning rule that can be applied to all types of repeated decision problems. In this paper I consider win-stay, lose-shift strategies with diverse memory sizes and strategies that adapt their aspiration levels, i.e. the payoff level considered as "success". I study their evolution for the Prisoner's Dilemma, as well as in a rapidly changing environment, where a randomly selected game is assigned to the players. For win-stay, lose-shift strategies with memory one the average payoffs are computed and their evolutionary stability is discussed. Using computer simulations I show that the win-stay, lose-shift strategies with longer memory are very successful both for the Prisoner's Dilemma, where cooperation dominates even for high noise levels, and the randomly assigned games, where the players achieve nearly the expected Pareto optimal payoffs. I discuss the impact of noise and show that the memory length of the players increases with the noise level. These results indicate that the win-stay, lose-shift principle is a very successful strategy in repeated games with noise.     

Leading the Game,Losing the Competition: Identifying Leaders and Followers in a Repeated Game

We explore a new method for identifying leaders and followers, LF, in repeated games by analyzing an experimental, repeated (50 rounds) game where Row player shifts the payoff between small and large values–a type of “investor” and Column player determines who gets the payoff–a type of “manager”. We found that i) the Investor (Row) most often is a leading player and the manager (Column) a follower. The longer the Investor leads the game, the higher is both player’s payoff. Surprisingly however, it is always the Manager that achieves the largest payoff. ii) The game has an efficient cooperative strategy where the players alternate in receiving a high payoff, but the players never identify, or accept, that strategy. iii) Under the assumption that the information used by the players is closely associated with the leader- follower sequence, and that information is available before the player’s decisions are made, the players switched LF- strategy primarily as a function of information on the Investor’s investment and moves and secondly as a function of the Manager’s payoff.     

An asymmetric parental investment conflict with continuous strategy sets

In the parental investment conflict each of the sexes decides how much to invest in its brood, where its decision influences both sexes' fitness. In nature, each species is usually characterized by a common parental care pattern, male-only care, female-only care or biparental care. A possible way for understanding the factors that have led each species to adopt its unique parental care pattern is to analyse a male's and a female's decision process using a game-theoretical model. This paper suggests a two-stage game-theoretical model with two types of players, male and female. During the game each parent makes three decisions. The interval between the beginning of the game, i.e. after mating and having offspring, and the moment a parent starts to care for them is a random variable. Thus, in the first stage a parent chooses the cumulative probability distribution of this interval, and its amount of parental care. In the second stage the other parent chooses its probability for cooperation. It is assumed that as long as parental care is not provided the offspring are at risk, and that parental caring accrues a different cost for each sex. We compute the Evolutionary Stable Strategies (ESS) under payoff-relevant asymmetry, and show that uniparental and biparental care are possible ESS. We also characterize cases where the sex having the lower cost "forces" the sex having the higher cost to care and vice versa.     

Evolution of interactions and cooperation in the spatial prisoner's dilemma game

We study the evolution of cooperation in the spatial prisoner's dilemma game where players are allowed to establish new interactions with others. By employing a simple coevolutionary rule entailing only two crucial parameters, we find that different selection criteria for the new interaction partners as well as their number vitally affect the outcome of the game. The resolution of the social dilemma is most probable if the selection favors more successful players and if their maximally attainable number is restricted. While the preferential selection of the best players promotes cooperation irrespective of game parametrization, the optimal number of new interactions depends somewhat on the temptation to defect. Our findings reveal that the "making of new friends" may be an important activity for the successful evolution of cooperation, but also that partners must be selected carefully and their number limited.     

N-player quantum games in an EPR setting

The N-player quantum games are analyzed that use an Einstein-Podolsky-Rosen (EPR) experiment, as the underlying physical setup. In this setup, a player's strategies are not unitary transformations as in alternate quantum game-theoretic frameworks, but a classical choice between two directions along which spin or polarization measurements are made. The players' strategies thus remain identical to their strategies in the mixed-strategy version of the classical game. In the EPR setting the quantum game reduces itself to the corresponding classical game when the shared quantum state reaches zero entanglement. We find the relations for the probability distribution for N-qubit GHZ and W-type states, subject to general measurement directions, from which the expressions for the players' payoffs and mixed Nash equilibrium are determined. Players' N x N payoff matrices are then defined using linear functions so that common two-player games can be easily extended to the N-player case and permit analytic expressions for the Nash equilibrium. As a specific example, we solve the Prisoners' Dilemma game for general N ≥ 2. We find a new property for the game that for an even number of players the payoffs at the Nash equilibrium are equal, whereas for an odd number of players the cooperating players receive higher payoffs. By dispensing with the standard unitary transformations on state vectors in Hilbert space and using instead rotors and multivectors, based on Clifford's geometric algebra (GA), it is shown how the N-player case becomes tractable. The new mathematical approach presented here has wide implications in the areas of quantum information and quantum complexity, as it opens up a powerful way to tractably analyze N-partite qubit interactions.     

Influence of Opinion Dynamics on the Evolution of Games

Under certain circumstances such as lack of information or bounded rationality, human players can take decisions on which strategy to choose in a game on the basis of simple opinions. These opinions can be modified after each round by observing own or others payoff results but can be also modified after interchanging impressions with other players. In this way, the update of the strategies can become a question that goes beyond simple evolutionary rules based on fitness and become a social issue. In this work, we explore this scenario by coupling a game with an opinion dynamics model. The opinion is represented by a continuous variable that corresponds to the certainty of the agents respect to which strategy is best. The opinions transform into actions by making the selection of an strategy a stochastic event with a probability regulated by the opinion. A certain regard for the previous round payoff is included but the main update rules of the opinion are given by a model inspired in social interchanges. We find that the fixed points of the dynamics of the coupled model are different from those of the evolutionary game or the opinion models alone. Furthermore, new features emerge such as the independence of the fraction of cooperators with respect to the topology of the social interaction network or the presence of a small fraction of extremist players.     

Heterogeneous aspirations promote cooperation in the prisoner's dilemma game

To be the fittest is central to proliferation in evolutionary games. Individuals thus adopt the strategies of better performing players in the hope of successful reproduction. In structured populations the array of those that are eligible to act as strategy sources is bounded to the immediate neighbors of each individual. But which one of these strategy sources should potentially be copied? Previous research dealt with this question either by selecting the fittest or by selecting one player uniformly at random. Here we introduce a parameter that interpolates between these two extreme options. Setting equal to zero returns the random selection of the opponent, while positive favor the fitter players. In addition, we divide the population into two groups. Players from group select their opponents as dictated by the parameter , while players from group do so randomly irrespective of . We denote the fraction of players contained in groups and by and , respectively. The two parameters and allow us to analyze in detail how aspirations in the context of the prisoner''s dilemma game influence the evolution of cooperation. We find that for sufficiently positive values of there exist a robust intermediate for which cooperation thrives best. The robustness of this observation is tested against different levels of uncertainty in the strategy adoption process and for different interaction networks. We also provide complete phase diagrams depicting the dependence of the impact of and for different values of , and contrast the validity of our conclusions by means of an alternative model where individual aspiration levels are subject to evolution as well. Our study indicates that heterogeneity in aspirations may be key for the sustainability of cooperation in structured populations.     

Optimal Cooperation-Trap Strategies for the Iterated Rock-Paper-Scissors Game

In an iterated non-cooperative game, if all the players act to maximize their individual accumulated payoff, the system as a whole usually converges to a Nash equilibrium that poorly benefits any player. Here we show that such an undesirable destiny is avoidable in an iterated Rock-Paper-Scissors (RPS) game involving two rational players, X and Y. Player X has the option of proactively adopting a cooperation-trap strategy, which enforces complete cooperation from the rational player Y and leads to a highly beneficial and maximally fair situation to both players. That maximal degree of cooperation is achievable in such a competitive system with cyclic dominance of actions may stimulate further theoretical and empirical studies on how to resolve conflicts and enhance cooperation in human societies.     

The donation game with roles played between relatives

We consider a social game with two choices, played between two relatives, where roles are assigned to individuals so that the interaction is asymmetric. Behaviour in each of the two roles is determined by a separate genetic locus. Such asymmetric interactions between relatives, in which individuals occupy different behavioural contexts, may occur in nature, for example between adult parents and juvenile offspring. The social game considered is known to be equivalent to a donation game with non-additive payoffs, and has previously been analysed for the single locus case, both for discrete and continuous strategy traits. We present an inclusive fitness analysis of the discrete trait game with roles and recover equilibrium conditions including fixation of selfish or altruistic behaviour under both behavioural contexts, or fixation of selfish behaviour under one context and altruistic behaviour under the other context. These equilibrium solutions assume that the payoff matrices under each behavioural context are identical. The equilibria possible do depend crucially, however, on the deviation from payoff additivity that occurs when both interacting individuals act altruistically.