Logical Gaps in the Approximate Solutions of the Social Learning Game and an Exact Solution

After the social learning models were proposed, finding solutions to the games becomes a well-defined mathematical question. However, almost all papers on the games and their applications are based on solutions built either upon an ad-hoc argument or a twisted Bayesian analysis of the games. Here, we present logical gaps in those solutions and offer an exact solution of our own. We also introduce a minor extension to the original game so that not only logical differences but also differences in action outcomes among those solutions become visible.     

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.     

Analysis of two-player quantum games in an EPR setting using Clifford's geometric algebra

The framework for playing quantum games in an Einstein-Podolsky-Rosen (EPR) type setting is investigated using the mathematical formalism of geometric algebra (GA). The main advantage of this framework is that the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, and hence the quantum game becomes a proper extension of the classical game, avoiding a criticism of other quantum game frameworks. We produce a general solution for two-player games, and as examples, we analyze the games of Prisoners' Dilemma and Stag Hunt in the EPR setting. The use of GA allows a quantum-mechanical analysis without the use of complex numbers or the Dirac Bra-ket notation, and hence is more accessible to the non-physicist.     

Analyzing three-player quantum games in an EPR type setup

We use the formalism of Clifford Geometric Algebra (GA) to develop an analysis of quantum versions of three-player non-cooperative games. The quantum games we explore are played in an Einstein-Podolsky-Rosen (EPR) type setting. In this setting, the players' strategy sets remain identical to the ones in the mixed-strategy version of the classical game that is obtained as a proper subset of the corresponding quantum game. Using GA we investigate the outcome of a realization of the game by players sharing GHZ state, W state, and a mixture of GHZ and W states. As a specific example, we study the game of three-player Prisoners' Dilemma.     

Physiological and skill demands of 'on-side' and 'off-side' games

This study investigated the physiological and skill demands of 'on-side' and 'off-side' games in elite rugby league players. Sixteen male rugby league players participated in 'on-side' and 'off-side' games. Both small-sided games were played in a 40- × 40-m playing area. The 'off-side' game permitted players to have 3 'plays' while in possession of the ball. Players were permitted to pass backward or forward (to an 'off-side' player). The 'on-side' game also permitted players to have 3 'plays' while in possession of the ball. However, players were only permitted to pass backward to players in an 'on-side' position. Heart rate and movement patterns (via global positioning system) were recorded continuously throughout both games. Data were collected on the distance covered, number of high-acceleration and velocity efforts, and recovery between efforts. Video footage was also taken to track the performance of the players. Post hoc inspection of the footage was undertaken to count the number of possessions and the number and quality of disposals. In comparison to 'on-side' games, 'off-side' games had a greater number of involvements ("touches"), passes, and effective passes. However, the cognitive demands of 'on-side' games were greater than 'off-side' games. 'Off-side' games resulted in a greater total distance covered, greater distance covered in mild and moderate accelerations, and greater distance covered in low, moderate, and high-velocity efforts. There were also a greater number of short duration recovery periods between efforts in 'off-side' games. The results of this study demonstrate that 'off-side' games provide greater physiological and skill demands than 'on-side' games. 'Off-side' games may provide a practical alternative to 'on-side' games for the development of skill and fitness in elite rugby league players.     

An Application of Evolutionary Game Theory to Social Dilemmas: The Traveler's Dilemma and the Minimum Effort Coordination Game

The Traveler''s Dilemma game and the Minimum Effort Coordination game are two social dilemmas that have attracted considerable attention due to the fact that the predictions of classical game theory are at odds with the results found when the games are studied experimentally. Moreover, a direct application of deterministic evolutionary game theory, as embodied in the replicator dynamics, to these games does not explain the observed behavior. In this work, we formulate natural variants of these two games as smoothed continuous-strategy games. We study the evolutionary dynamics of these continuous-strategy games, both analytically and through agent-based simulations, and show that the behavior predicted theoretically is in accord with that observed experimentally. Thus, these variants of the Traveler''s Dilemma and the Minimum Effort Coordination games provide a simple resolution of the paradoxical behavior associated with the original games.     

Asymmetric Evolutionary Games

Evolutionary game theory is a powerful framework for studying evolution in populations of interacting individuals. A common assumption in evolutionary game theory is that interactions are symmetric, which means that the players are distinguished by only their strategies. In nature, however, the microscopic interactions between players are nearly always asymmetric due to environmental effects, differing baseline characteristics, and other possible sources of heterogeneity. To model these phenomena, we introduce into evolutionary game theory two broad classes of asymmetric interactions: ecological and genotypic. Ecological asymmetry results from variation in the environments of the players, while genotypic asymmetry is a consequence of the players having differing baseline genotypes. We develop a theory of these forms of asymmetry for games in structured populations and use the classical social dilemmas, the Prisoner’s Dilemma and the Snowdrift Game, for illustrations. Interestingly, asymmetric games reveal essential differences between models of genetic evolution based on reproduction and models of cultural evolution based on imitation that are not apparent in symmetric games.     

Unpredictability induced by unfocused games in evolutionary game dynamics

Evolutionary game theory is a basis of replicator systems and has applications ranging from animal behavior and human language to ecosystems and other hierarchical network systems. Most studies in evolutionary game dynamics have focused on a single game, but, in many situations, we see that many games are played simultaneously. We construct a replicator equation with plural games by assuming that a reward of a player is a simple summation of the reward of each game. Even if the numbers of the strategies of the games are different, its dynamics can be described in one replicator equation. We here show that when players play several games at the same time, the fate of a single game cannot be determined without knowing the structures of the whole other games. The most absorbing fact is that even if a single game has a ESS (evolutionary stable strategy), the relative frequencies of strategies in the game does not always converge to the ESS point when other games are played simultaneously.     

On equilibrium properties of evolutionary multi-player games with random payoff matrices

The analysis of equilibrium points in biological dynamical systems has been of great interest in a variety of mathematical approaches to biology, such as population genetics, theoretical ecology or evolutionary game theory. The maximal number of equilibria and their classification based on stability have been the primary subjects of these studies, for example in the context of two-player games with multiple strategies. Herein, we address a different question using evolutionary game theory as a tool. If the payoff matrices are drawn randomly from an arbitrary distribution, what are the probabilities of observing a certain number of (stable) equilibria? We extend the domain of previous results for the two-player framework, which corresponds to a single diploid locus in population genetics, by addressing the full complexity of multi-player games with multiple strategies. In closing, we discuss an application and illustrate how previous results on the number of equilibria, such as the famous Feldman-Karlin conjecture on the maximal number of isolated fixed points in a viability selection model, can be obtained as special cases of our results based on multi-player evolutionary games. We also show how the probability of realizing a certain number of equilibria changes as we increase the number of players and number of strategies.     

On Nash Equilibrium and Evolutionarily Stable States That Are Not Characterised by the Folk Theorem

In evolutionary game theory, evolutionarily stable states are characterised by the folk theorem because exact solutions to the replicator equation are difficult to obtain. It is generally assumed that the folk theorem, which is the fundamental theory for non-cooperative games, defines all Nash equilibria in infinitely repeated games. Here, we prove that Nash equilibria that are not characterised by the folk theorem do exist. By adopting specific reactive strategies, a group of players can be better off by coordinating their actions in repeated games. We call it a type-k equilibrium when a group of k players coordinate their actions and they have no incentive to deviate from their strategies simultaneously. The existence and stability of the type-k equilibrium in general games is discussed. This study shows that the sets of Nash equilibria and evolutionarily stable states have greater cardinality than classic game theory has predicted in many repeated games.     

Open-Phylo: a customizable crowd-computing platform for multiple sequence alignment

Citizen science games such as Galaxy Zoo, Foldit, and Phylo aim to harness the intelligence and processing power generated by crowds of online gamers to solve scientific problems. However, the selection of the data to be analyzed through these games is under the exclusive control of the game designers, and so are the results produced by gamers. Here, we introduce Open-Phylo, a freely accessible crowd-computing platform that enables any scientist to enter our system and use crowds of gamers to assist computer programs in solving one of the most fundamental problems in genomics: the multiple sequence alignment problem.     

Action being character: a promising perspective on the solution concept of game theory

The inconsistency of predictions from solution concepts of conventional game theory with experimental observations is an enduring question. These solution concepts are based on the canonical rationality assumption that people are exclusively self-regarding utility maximizers. In this article, we think this assumption is problematic and, instead, assume that rational economic agents act as if they were maximizing their implicit utilities, which turns out to be a natural extension of the canonical rationality assumption. Implicit utility is defined by a player's character to reflect his personal weighting between cooperative, individualistic, and competitive social value orientations. The player who actually faces an implicit game chooses his strategy based on the common belief about the character distribution for a general player and the self-estimation of his own character, and he is not concerned about which strategies other players will choose and will never feel regret about his decision. It is shown by solving five paradigmatic games, the Dictator game, the Ultimatum game, the Prisoner's Dilemma game, the Public Goods game, and the Battle of the Sexes game, that the framework of implicit game and its corresponding solution concept, implicit equilibrium, based on this alternative assumption have potential for better explaining people's actual behaviors in social decision making situations.     

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.     

Habitual action video game playing is associated with caudate nucleus-dependent navigational strategies

The habitual playing of video games is associated with increased grey matter and activity in the striatum. Studies in humans and rodents have shown an inverse relationship between grey matter in the striatum and hippocampus. We investigated whether action video game playing is also associated with increased use of response learning strategies during navigation, known to be dependent on the caudate nucleus of the striatum, when presented in a dual solution task. We tested 26 action video game players (actionVGPs) and 33 non-action video game players (nonVGPs) on the 4-on-8 virtual maze and a visual attention event-related potential (ERP) task, which elicits a robust N-2-posterior-controlateral (N2pc) component. We found that actionVGPs had a significantly higher likelihood of using a response learning strategy (80.76%) compared to nonVGPs (42.42%). Consistent with previous evidence, actionVGPs and nonVGPs differed in the way they deployed visual attention to central and peripheral targets as observed in the elicited N2pc component during an ERP visual attention task. Increased use of the response strategy in actionVGPs is consistent with previously observed increases in striatal volume in video game players (VGPs). Using response strategies is associated with decreased grey matter in the hippocampus. Previous studies have shown that decreased volume in the hippocampus precedes the onset of many neurological and psychiatric disorders. If actionVGPs have lower grey matter in the hippocampus, as response learners normally do, then these individuals could be at increased risk of developing neurological and psychiatric disorders during their lifetime.     

Influence of field size on the physiological and skill demands of small-sided games in junior and senior rugby league players

The purpose of this study was to investigate the effect of changes in field size on the physiological and skill demands of small-sided games in elite junior and senior rugby league players. Sixteen elite senior rugby league players ([mean ± SE] age, 23.6 ± 0.5 years) and 16 elite junior rugby league players ([mean ± SE] age, 17.3 ± 0.3 years) participated in this study. On day 1, 2 teams played an 8-minute small-sided game on a small field (10-m width × 40-m length), whereas the remaining 2 teams played the small-sided game on a larger sized field (40-m width × 70-m length). On day 2, the groups were crossed over. Movement was recorded by a global positioning system unit sampling at 5 Hz. Games were filmed to count the number of possessions and the number and quality of disposals. The games played on a larger field resulted in a greater (p < 0.05) total distance covered, and distances covered in moderate, high, and very-high velocity movement intensities. Senior players covered more distance at moderate, high, and very-high intensities, and less distance at low and very-low intensities during small-sided games than junior players. Although increasing field size had no significant influence (p > 0.05) over the duration of recovery periods for junior players, larger field size significantly reduced (p < 0.05) the amount of short-, moderate-, and long-duration recovery periods in senior players. No significant between-group differences (p > 0.05) were detected for games played on a small or large field for the number or quality of skill involvements. These results suggest that increases in field size serve to increase the physiological demands of small-sided games but have minimal influence over the volume or quality of skill executions in elite rugby league players.     

The Effect of Power Asymmetries on Cooperation and Punishment in a Prisoner’s Dilemma Game

Recent work has suggested that punishment is detrimental because punishment provokes retaliation, not cooperation, resulting in lower overall payoffs. These findings may stem from the unrealistic assumption that all players are equal: in reality individuals are expected to vary in the power with which they can punish defectors. Here, we allowed strong players to interact with weak players in an iterated prisoner''s dilemma game with punishment. Defecting players were most likely to switch to cooperation if the partner cooperated: adding punishment yielded no additional benefit and, under some circumstances, increased the chance that the partner would both defect and retaliate against the punisher. Our findings show that, in a two-player game, cooperation begets cooperation and that punishment does not seem to yield any additional benefits. Further work should explore whether strong punishers might prevail in multi-player games.     

Emergence of Super Cooperation of Prisoner’s Dilemma Games on Scale-Free Networks

Recently, the authors proposed a quantum prisoner’s dilemma game based on the spatial game of Nowak and May, and showed that the game can be played classically. By using this idea, we proposed three generalized prisoner’s dilemma (GPD, for short) games based on the weak Prisoner’s dilemma game, the full prisoner’s dilemma game and the normalized Prisoner’s dilemma game, written by GPD_W, GPD_F and GPD_N respectively. Our games consist of two players, each of which has three strategies: cooperator (C), defector (D) and super cooperator (denoted by Q), and have a parameter γ to measure the entangled relationship between the two players. We found that our generalised prisoner’s dilemma games have new Nash equilibrium principles, that entanglement is the principle of emergence and convergence (i.e., guaranteed emergence) of super cooperation in evolutions of our generalised prisoner’s dilemma games on scale-free networks, that entanglement provides a threshold for a phase transition of super cooperation in evolutions of our generalised prisoner’s dilemma games on scale-free networks, that the role of heterogeneity of the scale-free networks in cooperations and super cooperations is very limited, and that well-defined structures of scale-free networks allow coexistence of cooperators and super cooperators in the evolutions of the weak version of our generalised prisoner’s dilemma games.     

The influence of cultural framing on play in the trust game: a Maasai example

The effects of cultural framing on behavior in experimental games were explored with a trust game and the Maasai concept of osotua. Maasai use the term osotua to refer to gift-giving relationships based on obligation, need, respect, and restraint. In the trust game, the first player is given money and an opportunity to give any portion of it to the second player. The amount given is then multiplied by the experimenter, and the second player has an opportunity to give any amount back to the first player. Fifty trust games were played by Maasai men at a field site in north central Kenya. Half of the games were played without deliberate framing, and half were framed with the statement, “This is an osotua game.” Compared to games with no deliberate framing, those played within the osotua rhetorical frame were associated with lower transfers by both players and with lower expected returns on the part of the first players. Osotua rhetorical framing is also associated with a negative correlation between amounts given by the first player and amounts returned by the second. These results have implications both for the experimental game method and for our understanding of the relationship between culture and behavior.     

The effect of players' standard and tactical strategy on game demands in men's basketball

The purpose of the present study was to examine the effects of competitive level and team tactic on game demands in men's basketball. Sixteen international-level male basketball players (INPs) and 22 national-level male basketball players (NLPs) were studied during 6 games. Time-motion analysis was performed to track game activities. Game physiological demands were assessed by monitoring heart rate (HR) and blood-lactate concentration. Results showed that INPs sprinted significantly more and performed more high-intensity shuffling than did NLPs (p < 0.05). Game-activity changes and frequency of high-intensity bouts were similar in man-to-man and zone-marking games (1,053 vs. 1,056 and 253 vs. 224, respectively, p > 0.05). Time spent in the maximal (>95% of HRmax) and high-intensity zone (85-95% of HRmax) was greater in the INPs than in the NLPs (17.8 vs. 15.2%, p < 0.01 and 59.1 vs. 54.4%, p < 0.05, respectively). No significant differences in mean HR were evident between man-to-man and zone-marking games (93.3 ± 2.1 vs. 92.8 ± 1.8% of HRmax, p > 0.05). Blood-lactate concentration was higher in the INPs than in the NLPs (6.60 ± 1.22 vs. 5.66 ± 1.19 mmol·L?1 at halftime and 5.65 ± 1.21 vs. 4.43 ± 1.43 mmol·L?1 at full time, p < 0.05). No mean or peak blood-lactate concentration differences resulted between man-to-man and zone-marking games (5.15 ± 1.32 vs. 5.83 ± 1.10 and 5.90 ± 1.25 vs. 6.30 ± 1.27 mmol·L?1, respectively, p > 0.05). These results suggest an effect of competitive level over game demands in men's basketball. No marking strategy effect was evident. Basketball coaches and fitness trainers should develop the ability to repeatedly perform high-intensity activity during the game. Repeated sprinting and high-intensity shuffling ability should be trained to successfully play man-to-man and zone defense, respectively.     

Endogenous timing in competitive interactions among relatives

Most evolutionary game theory models solve for equilibrium levels of some behaviour on the restrictive assumptions that players choose their actions simultaneously, and that a player cannot change its action after observing that of its opponent. An alternative framework is provided by sequential or 'Stackelberg' games in which one player commits to a 'first move' and the other has an opportunity to observe this move before choosing its response. Recent interest in the economic literature has focused on Stackelberg games which exhibit 'endogenous timing', i.e. games in which a leader and a follower arise spontaneously as a consequence of each player attempting to maximize its reward. Here, we provide the first demonstration of endogenous timing in an evolutionary context using a simple model of resource competition (the 'tug-of-war' model). We show that whenever two related individuals compete for a share of communal resources, both do best to adopt distinct roles in a sequential game rather than engage in simultaneous competition. Somewhat counterintuitively, the stable solution is for the weaker individual to act as leader and commit to a first move, because this arrangement leads to a lower total effort invested in competition. Endogenous timing offers a new explanation for the spontaneous emergence of leaders and followers in social groups, and highlights the benefits of commitment in social interaction.