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.     

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.     

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.     

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.     

War of attrition with implicit time cost

In the game-theoretic model war of attrition, players are subject to an explicit cost proportional to the duration of contests. We construct a model where the time cost is not explicitly given, but instead depends implicitly on the strategies of the whole population. We identify and analyse the underlying mechanisms responsible for the implicit time cost. Each player participates in a series of games, where those prepared to wait longer win with higher certainty but play less frequently. The model is characterized by the ratio of the winner's score to the loser's score, in a single game. The fitness of a player is determined by the accumulated score from the games played during a generation. We derive the stationary distribution of strategies under the replicator dynamics. When the score ratio is high, we find that the stationary distribution is unstable, with respect to both evolutionary and dynamical stability, and the dynamics converge to a limit cycle. When the ratio is low, the dynamics converge to the stationary distribution. For an intermediate interval of the ratio, the distribution is dynamically but not evolutionarily stable. Finally, the implications of our results for previous models based on the war of attrition are discussed.     

Pre- and post-game body mass changes during an international rugby tournament: a practical perspective

Pre- and post-game body mass changes were recorded throughout an international rugby union tournament to determine changes in body mass when training and competing on consecutive days. A squad of 28 players with a mean (+/-SD) age of 20.0 yrs (+/-0.6), height of 1.85 m (+/-0.1) and body mass of 94.71 kg (+/-10.71) were selected to play for England and to participate in an international championship in Sydney, Australia. From the squad, 22 players played in 1 or more of 4 games over a 10-day period from the day of the first match to the day of the last match. Data were collected for a total of 84 player appearances across the 4 games. Players had their body masses recorded prior to leaving England, upon arrival in Australia, and pre- and post-game during the tournament. Players were allowed to ingest fluid ad libitum throughout each match. The mean (+/-SD) ambient temperature and relative humidity across all games was 18.5 degrees C (+/-1.63) and 39.5% (+/-17.7), respectively. Pre- and post-game comparisons of body masses for all player appearances were significantly different (p < 0.00), representing a mean loss of 0.94 kg (+/-0.94). However, there was no significant difference between games or player position (forwards vs. backs). The results demonstrate that if appropriate hydration practices are implemented, they may reduce the compound effect of fluid loss when playing multiple games in a short period of time. Strategies include daily monitoring of body mass and fluid intake and educating players about the importance of fluid intake to recovery and subsequent performance when playing and training on consecutive days.     

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.     

Body weight and tympanic temperature change in professional rugby league players during night and day games: a study in the field

This study investigated the impact of day and night games in the professional rugby league on body weight and tympanic temperature change in participants. Twenty-five players contracted to an English Super League club had their pre- and postgame body weight and tympanic temperatures recorded during 10 games played during the official professional rugby league season, representing a total of 165 player appearances. The mean (+/-SD) ambient temperature and relative humidity was 12.3 degrees C (+/-6.0) and 83.3% (+/-11.4), respectively. Body weight was recorded using a set of calibrated Soehnle digital scales with players wearing underwear only and towel-dried of all sweat (postmatch). Tympanic temperature was recorded using a Braun ThermoScan Pro LT instant thermometer. Players were allowed to ingest fluid ad libitum throughout each match. Wet and dry bulb temperatures were recorded at the commencement and completion of each match. Significant changes in pregame to postgame body weight and tympanic temperature were found, but these were not influenced by the time of day that the game was played. The mean decrease in body weight was 0.86 kg (SE 0.085, p < 0.000), and the mean increase in tympanic temperature was 0.34 degrees C (SE 0.070, p < 0.000). No significant differences in body weight or tympanic temperature change were found between forwards and backs. Participation in the English professional rugby league can produce significant decreases in body weight and increases in body temperature that may lead to impaired performance. It is important for participants, coaches, and administrators to introduce strategies that will minimize the impact of environmental conditions on thermoregulation and ultimately player performance.     

Neural correlates of economic game playing

The theory of games provides a mathematical formalization of strategic choices, which have been studied in both economics and neuroscience, and more recently has become the focus of neuroeconomics experiments with human and non-human actors. This paper reviews the results from a number of game experiments that establish a unitary system for forming subjective expected utility maps in the brain, and acting on these maps to produce choices. Social situations require the brain to build an understanding of the other person using neuronal mechanisms that share affective and intentional mental states. These systems allow subjects to better predict other players' choices, and allow them to modify their subjective utility maps to value pro-social strategies. New results for a trust game are presented, which show that the trust relationship includes systems common to both trusting and trustworthy behaviour, but they also show that the relative temporal positions of first and second players require computations unique to that role.     

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.     

Risk-Pooling and Herd Survival: An Agent-Based Model of a Maasai Gift-Giving System

We use agent-based modeling to study osotua, a gift giving system used by the Maasai of East Africa. Osotua’s literal meaning is “umbilical cord,” but it is used metaphorically to refer to a specific type of gift-giving relationship. Osotua relationships are characterized by respect, responsibility and restraint. Osotua partners ask each other for help only if they are in need and provide help only when asked and only if they are able. We hypothesize that under the ecologically volatile conditions in which Maasai pastoralists have traditionally lived, such a system is particularly suited to risk pooling. Here we explore whether osotua increases the viability of herds by comparing herd survivorship and stability under osotua rules to a) no exchange and b) probabilistic rules for requesting and giving livestock. Results from this model suggest that this gift-giving system can dramatically increase herd longevity through a limited pooling of risk.     

Global mutations and local mutations have very different effects on evolution, illustrated by mixed strategies of asymmetric binary games

We study the evolutionary effect of rare mutations causing global changes in traits. We consider asymmetric binary games between two players. The first player takes two alternative options with probability x and 1−x; and the second player takes options with probability y and 1−y. Due to natural selection and recurrent mutation, the population evolves to have broad distributions of x and y. We analyze three cases showing qualitatively different dynamics, exemplified by (1) vigilance-intrusion game, (2) asymmetric hawk-dove game and (3) cleaner-client game. We found that the evolutionary outcome is strongly dependent upon the distribution of mutants’ traits, more than the mutation rates. For example in the vigilance-intrusion game, the evolutionary dynamics show a perpetual stable oscillation if mutants are always close to the parent (local-mutation mode), whilst the population converges to a stable equilibrium distribution if mutants can be quite different from the parent (global-mutation mode), even for extremely low mutation rate. When common local mutations and rare global mutations occur simultaneously, the evolutionary outcome is controlled by the latter.     

Evolutionary Stability in the Asymmetric Volunteer's Dilemma

It is often assumed that in public goods games, contributors are either strong or weak players and each individual has an equal probability of exhibiting cooperation. It is difficult to explain why the public good is produced by strong individuals in some cooperation systems, and by weak individuals in others. Viewing the asymmetric volunteer''s dilemma game as an evolutionary game, we find that whether the strong or the weak players produce the public good depends on the initial condition (i.e., phenotype or initial strategy of individuals). These different evolutionarily stable strategies (ESS) associated with different initial conditions, can be interpreted as the production modes of public goods of different cooperation systems. A further analysis revealed that the strong player adopts a pure strategy but mixed strategies for the weak players to produce the public good, and that the probability of volunteering by weak players decreases with increasing group size or decreasing cost-benefit ratio. Our model shows that the defection probability of a “strong” player is greater than the “weak” players in the model of Diekmann (1993). This contradicts Selten''s (1980) model that public goods can only be produced by a strong player, is not an evolutionarily stable strategy, and will therefore disappear over evolutionary time. Our public good model with ESS has thus extended previous interpretations that the public good can only be produced by strong players in an asymmetric game.     

Can transitive inference evolve in animals playing the hawk-dove game?

What should an individual do if there are no reliable cues to the strength of a competitor when fighting with it for resources? We herein examine the evolutionarily stable strategy (ESS) in the hawk-dove game, if the opponent's resource-holding potential (RHP) can only indirectly be inferred from the outcome of past interactions in the population. The strategies we examined include the classical mixed strategy in which no information on past games is utilized, the 'imprinting' strategy in which a player increases/decreases its aggressiveness if it wins/loses a game, the 'immediate inference' strategy in which a player can infer the strength of those opponents it fought before, and the 'transitive inference' strategy in which a player can infer the strength of a new opponent through a third party with which both players have fought before. Invasibility analysis for each pair of strategies revealed that (i) the transitive-inference strategy can always invade the mixed strategy and the imprinting strategy, and itself refuses invasion by these strategies; (ii) the largest advantage for transitive inference is achieved when the number of games played per individual in one generation is small and when the cost of losing an escalated game is large; (iii) the immediate inference, rather than the transitive inference, can be an ESS if the cost of fighting is small; (iv) a strong linear ranking is established in the population of transitive-inference strategists, though it does not perfectly correlate to the ranking by actual RHPs. We found that the advantage of the transitive inference is not in its ability to correct a misassessment (it is actually the worst in doing so), but in the ability of quickly lining up either incorrect or correct assessments to form a linear dominance hierarchy.     

Match analysis of elite adolescent team handball players

The purposes of this study were to examine the activity profile of elite adolescent players during regular team handball games and to compare the physical and motor performance of players between the first and second halves of a match. Activity patterns (video analysis) and heart-rate (HR) responses (telemetry) were monitored in top national-division adolescent players (18 men, aged 15.1 ± 0.6 years) throughout 6 regulation games (25-minute halves with a 10-minute interval). The total distance covered averaged 1,777 ± 264 m per game (7.4% less in the second than in the first half, p > 0.05). Players ran 170 ± 24 m at high intensity and 86 ± 12 m at maximal speed, with 32 ± 6 bouts of running (duration 2.3 ± 0.3 seconds) at speeds > 18 km·h(-1); they stood still for 16% of the playing time. The mean HR during play was 172 ± 2 b·min(-1) (82 ± 3% of maximal HR). Blood lactate concentrations at the end of the first and second halves were 9.7 ± 1.1 and 8.3 ± 0.9 mmol·L(-1), respectively (difference p < 0.05). We conclude that adolescent handball players cover less distance and engage in fewer technical actions in the second half of a match. This indicates that team handball is physiologically very demanding. The practical implication is that coaches should seek to sustain performance in the second period of a game by modifying playing tactics and maximizing both aerobic and anaerobic fitness during training sessions.     

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.     

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.     

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.     

Tackling in a professional rugby league

During the course of a rugby league match, players are involved in multiple physical collisions, predominantly in the form of tackles. The purpose of this study was to describe the nature of, and circumstances relating to the various types of tackles completed by various playing positions in professional rugby league competition. Time-motion analysis was used during 5 competition matches; 1 player from 3 positional groups (hit-up forwards, adjustables, and outside backs) was analyzed in each match. Tackles were assessed by recording the sequence of involvement (e.g., whether a player was the first, second, or third player to engage in the tackle), the area of initial body contact on the player being tackled (e.g., high-above waist or low-below waist) and the type of tackle (e.g., front-on tackle, side-on tackle, and tackle from behind). The hit-up forwards, adjustables and outside backs averaged 166, 89, and 41 tackles, respectively, a game; the majority (46%) involved the observed defender being the first physical contact in the tackle. The present data show that the first defender generally makes a front-on tackle, either low or high, whereas the second player performs a front-on high tackle. If a third player is involved in a tackle, he or she makes contact with the player from the side and above the waist. The most frequent activity immediately before tackling is striding, followed by sprinting. The development of strength-based wrestling for individual playing positions should be an integral part of physical conditioning for rugby league players. The development of tackling skills at various movement intensities should also be considered.     

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.