Game theory

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Game theory is the study of mathematical models of strategic interaction between rational decision-makers. It has applications in all fields of social science, as well as in logic and computer science. Originally, it addressed zero-sum games, in which one person's gains result in losses for the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.

Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.

Game theory was developed extensively in the 1950s by many scholars. It was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. As of 2014, with the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole, eleven game theorists have won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology.

History

John von Neumann

John Nash

Early discussions of examples of two-person games occurred long before the rise of modern, mathematical game theory. The first known discussion of game theory occurred in a letter written by Charles Waldegrave, an active Jacobite, and uncle to James Waldegrave, a British diplomat, in 1713.^[2] In this letter, Waldegrave provides a minimax mixed strategy solution to a two-person version of the card game le Her, and the problem is now known as Waldegrave problem. James Madison made what we now recognize as a game-theoretic analysis of the ways states can be expected to behave under different systems of taxation. In his 1838 Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth), Antoine Augustin Cournot considered a duopoly and presents a solution that is a restricted version of the Nash equilibrium.

In 1913, Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels (On an Application of Set Theory to the Theory of the Game of Chess). It proved that the optimal chess strategy is strictly determined. This paved the way for more general theorems.

In 1938, the Danish mathematical economist Frederik Zeuthen proved that the mathematical model had a winning strategy by using Brouwer's fixed point theorem. In his 1938 book Applications aux Jeux de Hasard and earlier notes, Émile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix was symmetric. Borel conjectured that non-existence of mixed-strategy equilibria in two-person zero-sum games would occur, a conjecture that was proved false.

Game theory did not really exist as a unique field until John von Neumann published the paper On the Theory of Games of Strategy in 1928. Von Neumann's original proof used Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern. The second edition of this book provided an axiomatic theory of utility, which reincarnated Daniel Bernoulli's old theory of utility (of the money) as an independent discipline. Von Neumann's work in game theory culminated in this 1944 book. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. During the following time period, work on game theory was primarily focused on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.

In 1950, the first mathematical discussion of the prisoner's dilemma appeared, and an experiment was undertaken by notable mathematicians Merrill M. Flood and Melvin Dresher, as part of the RAND Corporation's investigations into game theory. RAND pursued the studies because of possible applications to global nuclear strategy. Around this same time, John Nash developed a criterion for mutual consistency of players' strategies, known as Nash equilibrium, applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern. Nash proved that every n-player, non-zero-sum (not just 2-player zero-sum) non-cooperative game has what is now known as a Nash equilibrium.

Game theory experienced a flurry of activity in the 1950s, during which time the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed. In addition, the first applications of game theory to philosophy and political science occurred during this time.

In 1979 Robert Axelrod tried setting up computer programs as players and found that in tournaments between them the winner was often a simple "tit-for-tat" program that cooperates on the first step, then on subsequent steps just does whatever its opponent did on the previous step. The same winner was also often obtained by natural selection; a fact widely taken to explain cooperation phenomena in evolutionary biology and the social sciences.

Prize-winning achievements

In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium (later he would introduce trembling hand perfection as well). In 1994 Nash, Selten and Harsanyi became Economics Nobel Laureates for their contributions to economic game theory.

In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge were introduced and analyzed.

In 2005, game theorists Thomas Schelling and Robert Aumann followed Nash, Selten and Harsanyi as Nobel Laureates. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing an equilibrium coarsening, correlated equilibrium, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences.

In 2007, Leonid Hurwicz, together with Eric Maskin and Roger Myerson, was awarded the Nobel Prize in Economics "for having laid the foundations of mechanism design theory". Myerson's contributions include the notion of proper equilibrium, and an important graduate text: Game Theory, Analysis of Conflict. Hurwicz introduced and formalized the concept of incentive compatibility.

In 2012, Alvin E. Roth and Lloyd S. Shapley were awarded the Nobel Prize in Economics "for the theory of stable allocations and the practice of market design" and, in 2014, the Nobel went to game theorist Jean Tirole.

Game types

Cooperative / Non-cooperative

A game is cooperative if the players are able to form binding commitments externally enforced (e.g. through contract law). A game is non-cooperative if players cannot form alliances or if all agreements need to be self-enforcing (e.g. through credible threats).

Cooperative games are often analysed through the framework of cooperative game theory, which focuses on predicting which coalitions will form, the joint actions that groups take and the resulting collective payoffs. It is opposed to the traditional non-cooperative game theory which focuses on predicting individual players' actions and payoffs and analyzing Nash equilibria.

Cooperative game theory provides a high-level approach as it only describes the structure, strategies and payoffs of coalitions, whereas non-cooperative game theory also looks at how bargaining procedures will affect the distribution of payoffs within each coalition. As non-cooperative game theory is more general, cooperative games can be analyzed through the approach of non-cooperative game theory (the converse does not hold) provided that sufficient assumptions are made to encompass all the possible strategies available to players due to the possibility of external enforcement of cooperation. While it would thus be optimal to have all games expressed under a non-cooperative framework, in many instances insufficient information is available to accurately model the formal procedures available to the players during the strategic bargaining process, or the resulting model would be of too high complexity to offer a practical tool in the real world. In such cases, cooperative game theory provides a simplified approach that allows analysis of the game at large without having to make any assumption about bargaining powers.

Symmetric / Asymmetric

	E	F
E	1, 2	0, 0
F	0, 0	1, 2
An asymmetric game

A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric. Many of the commonly studied 2×2 games are symmetric. The standard representations of chicken, the prisoner's dilemma, and the stag hunt are all symmetric games. Some scholars would consider certain asymmetric games as examples of these games as well. However, the most common payoffs for each of these games are symmetric. Most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the ultimatum game and similarly the dictator game have different strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players.

Zero-sum / Non-zero-sum

	A	B
A	–1, 1	3, –3
B	0, 0	–2, 2
A zero-sum game

Zero-sum games are a special case of constant-sum games, in which choices by players can neither increase nor decrease the available resources. In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others). Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose. Other zero-sum games include matching pennies and most classical board games including Go and chess.

Many games studied by game theorists (including the famed prisoner's dilemma) are non-zero-sum games, because the outcome has net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.

Constant-sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade. It is possible to transform any game into a (possibly asymmetric) zero-sum game by adding a dummy player (often called "the board") whose losses compensate the players' net winnings.

Simultaneous / Sequential

Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players' actions (making them effectively simultaneous). Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions. This need not be perfect information about every action of earlier players; it might be very little knowledge. For instance, a player may know that an earlier player did not perform one particular action, while s/he does not know which of the other available actions the first player actually performed.

The difference between simultaneous and sequential games is captured in the different representations discussed above. Often, normal form is used to represent simultaneous games, while extensive form is used to represent sequential ones. The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form. Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see subgame perfection.

In short, the differences between sequential and simultaneous games are as follows:

	Sequential	Simultaneous
Normally denoted by	Decision trees	Payoff matrices
Prior knowledge of opponent's move?	Yes	No
Time axis?	Yes	No
Also known as	Extensive-form game Extensive game	Strategy game Strategic game

Perfect information and imperfect information

A game of imperfect information (the dotted line represents ignorance on the part of player 2, formally called an information set)

An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players know the moves previously made by all other players. Most games studied in game theory are imperfect-information games. Examples of perfect-information games include tic-tac-toe, checkers, infinite chess, and Go.

Many card games are games of imperfect information, such as poker and bridge. Perfect information is often confused with complete information, which is a similar concept. Complete information requires that every player know the strategies and payoffs available to the other players but not necessarily the actions taken. Games of incomplete information can be reduced, however, to games of imperfect information by introducing "moves by nature".

Combinatorial games

Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games. Examples include chess and go. Games that involve imperfect information may also have a strong combinatorial character, for instance backgammon. There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve particular problems and answer general questions.

Games of perfect information have been studied in combinatorial game theory, which has developed novel representations, e.g. surreal numbers, as well as combinatorial and algebraic (and sometimes non-constructive) proof methods to solve games of certain types, including "loopy" games that may result in infinitely long sequences of moves. These methods address games with higher combinatorial complexity than those usually considered in traditional (or "economic") game theory. A typical game that has been solved this way is hex. A related field of study, drawing from computational complexity theory, is game complexity, which is concerned with estimating the computational difficulty of finding optimal strategies.

Research in artificial intelligence has addressed both perfect and imperfect information games that have very complex combinatorial structures (like chess, go, or backgammon) for which no provable optimal strategies have been found. The practical solutions involve computational heuristics, like alpha-beta pruning or use of artificial neural networks trained by reinforcement learning, which make games more tractable in computing practice.

Infinitely long games

Games, as studied by economists and real-world game players, are generally finished in finitely many moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner (or other payoff) not known until after all those moves are completed.

The focus of attention is usually not so much on the best way to play such a game, but whether one player has a winning strategy. (It can be proven, using the axiom of choice, that there are games – even with perfect information and where the only outcomes are "win" or "lose" – for which neither player has a winning strategy.) The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.

Discrete and continuous games

Much of game theory is concerned with finite, discrete games, that have a finite number of players, moves, events, outcomes, etc. Many concepts can be extended, however. Continuous games allow players to choose a strategy from a continuous strategy set. For instance, Cournot competition is typically modeled with players' strategies being any non-negative quantities, including fractional quantities.

Differential games

Differential games such as the continuous pursuit and evasion game are continuous games where the evolution of the players' state variables is governed by differential equations. The problem of finding an optimal strategy in a differential game is closely related to the optimal control theory. In particular, there are two types of strategies: the open-loop strategies are found using the Pontryagin maximum principle while the closed-loop strategies are found using Bellman's Dynamic Programming method.
A particular case of differential games are the games with a random time horizon. In such games, the terminal time is a random variable with a given probability distribution function. Therefore, the players maximize the mathematical expectation of the cost function. It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval.

Evolutionary game theory

Evolutionary game theory studies players who adjust their strategies over time according to rules that are not necessarily rational or farsighted. In general, the evolution of strategies over time according to such rules is modeled as a Markov chain with a state variable such as the current strategy profile or how the game has been played in the recent past. Such rules may feature imitation, optimization or survival of the fittest.

In biology, such models can represent (biological) evolution, in which offspring adopt their parents' strategies and parents who play more successful strategies (i.e. corresponding to higher payoffs) have a greater number of offspring. In the social sciences, such models typically represent strategic adjustment by players who play a game many times within their lifetime and, consciously or unconsciously, occasionally adjust their strategies.

Stochastic outcomes (and relation to other fields)

Individual decision problems with stochastic outcomes are sometimes considered "one-player games". These situations are not considered game theoretical by some authors. They may be modeled using similar tools within the related disciplines of decision theory, operations research, and areas of artificial intelligence, particularly AI planning (with uncertainty) and multi-agent system. Although these fields may have different motivators, the mathematics involved are substantially the same, e.g. using Markov decision processes (MDP).

Stochastic outcomes can also be modeled in terms of game theory by adding a randomly acting player who makes "chance moves" ("moves by nature"). This player is not typically considered a third player in what is otherwise a two-player game, but merely serves to provide a roll of the dice where required by the game.

For some problems, different approaches to modeling stochastic outcomes may lead to different solutions. For example, the difference in approach between MDPs and the minimax solution is that the latter considers the worst-case over a set of adversarial moves, rather than reasoning in expectation about these moves given a fixed probability distribution. The minimax approach may be advantageous where stochastic models of uncertainty are not available, but may also be overestimating extremely unlikely (but costly) events, dramatically swaying the strategy in such scenarios if it is assumed that an adversary can force such an event to happen.

General models that include all elements of stochastic outcomes, adversaries, and partial or noisy observability (of moves by other players) have also been studied. The "gold standard" is considered to be partially observable stochastic game (POSG), but few realistic problems are computationally feasible in POSG representation.

Metagames

These are games the play of which is the development of the rules for another game, the target or subject game. Metagames seek to maximize the utility value of the rule set developed. The theory of metagames is related to mechanism design theory.

The term metagame analysis is also used to refer to a practical approach developed by Nigel Howard. whereby a situation is framed as a strategic game in which stakeholders try to realise their objectives by means of the options available to them. Subsequent developments have led to the formulation of confrontation analysis.

Pooling games

These are games prevailing over all forms of society. Pooling games are repeated plays with changing payoff table in general over an experienced path and their equilibrium strategies usually take a form of evolutionary social convention and economic convention. Pooling game theory emerges to formally recognize the interaction between optimal choice in one play and the emergence of forthcoming payoff table update path, identify the invariance existence and robustness, and predict variance over time. The theory is based upon topological transformation classification of payoff table update over time to predict variance and invariance, and is also within the jurisdiction of the computational law of reachable optimality for ordered system.

Mean field game theory

Mean field game theory is the study of strategic decision making in very large populations of small interacting agents. This class of problems was considered in the economics literature by Boyan Jovanovic and Robert W. Rosenthal, in the engineering literature by Peter E. Caines and by mathematician Pierre-Louis Lions and Jean-Michel Lasry.

Representation of games

The games studied in game theory are well-defined mathematical objects. To be fully defined, a game must specify the following elements: the players of the game, the information and actions available to each player at each decision point, and the payoffs for each outcome. (Eric Rasmusen refers to these four "essential elements" by the acronym "PAPI".) A game theorist typically uses these elements, along with a solution concept of their choosing, to deduce a set of equilibrium strategies for each player such that, when these strategies are employed, no player can profit by unilaterally deviating from their strategy. These equilibrium strategies determine an equilibrium to the game—a stable state in which either one outcome occurs or a set of outcomes occur with known probability. Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.

Extensive form

An extensive form game

The extensive form can be used to formalize games with a time sequencing of moves. Games here are played on trees (as pictured here). Here each vertex (or node) represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree. The extensive form can be viewed as a multi-player generalization of a decision tree. To solve any extensive form game, backward induction must be used. It involves working backwards up the game tree to determine what a rational player would do at the last vertex of the tree, what the player with the previous move would do given that the player with the last move is rational, and so on until the first vertex of the tree is reached.

The game pictured consists of two players. The way this particular game is structured (i.e., with sequential decision making and perfect information), Player 1 "moves" first by choosing either F or U (Fair or Unfair). Next in the sequence, Player 2, who has now seen Player 1's move, chooses to play either A or R. Once Player 2 has made his/ her choice, the game is considered finished and each player gets their respective payoff. Suppose that Player 1 chooses U and then Player 2 chooses A: Player 1 then gets a payoff of "eight" (which in real-world terms can be interpreted in many ways, the simplest of which is in terms of money but could mean things such as eight days of vacation or eight countries conquered or even eight more opportunities to play the same game against other players) and Player 2 gets a payoff of "two".

The extensive form can also capture simultaneous-move games and games with imperfect information. To represent it, either a dotted line connects different vertices to represent them as being part of the same information set (i.e. the players do not know at which point they are), or a closed line is drawn around them. (See example in the imperfect information section.)

Normal form

	Player 2 chooses Left	Player 2 chooses Right
Player 1 chooses Up	4, 3	–1, –1
Player 1 chooses Down	0, 0	3, 4
Normal form or payoff matrix of a 2-player, 2-strategy game

The normal (or strategic form) game is usually represented by a matrix which shows the players, strategies, and payoffs (see the example to the right). More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions. In the accompanying example there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3.

When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually presented in extensive form.

Every extensive-form game has an equivalent normal-form game, however the transformation to normal form may result in an exponential blowup in the size of the representation, making it computationally impractical.

Characteristic function form

In games that possess removable utility, separate rewards are not given; rather, the characteristic function decides the payoff of each unity. The idea is that the unity that is 'empty', so to speak, does not receive a reward at all.

The origin of this form is to be found in John von Neumann and Oskar Morgenstern's book; when looking at these instances, they guessed that when a union ${\displaystyle \mathbf {C} }$ appears, it works against the fraction ${\displaystyle \left({\frac {\mathbf {N} }{\mathbf {C} }}\right)}$ as if two individuals were playing a normal game. The balanced payoff of C is a basic function. Although there are differing examples that help determine coalitional amounts from normal games, not all appear that in their function form can be derived from such.

Formally, a characteristic function is seen as: (N,v), where N represents the group of people and ${\displaystyle v:2^{N}\to \mathbf {R} }$ is a normal utility.

Such characteristic functions have expanded to describe games where there is no removable utility.

General and applied uses

As a method of applied mathematics, game theory has been used to study a wide variety of human and animal behaviors. It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers. The first use of game-theoretic analysis was by Antoine Augustin Cournot in 1838 with his solution of the Cournot duopoly. The use of game theory in the social sciences has expanded, and game theory has been applied to political, sociological, and psychological behaviors as well.

Although pre-twentieth century naturalists such as Charles Darwin made game-theoretic kinds of statements, the use of game-theoretic analysis in biology began with Ronald Fisher's studies of animal behavior during the 1930s. This work predates the name "game theory", but it shares many important features with this field. The developments in economics were later applied to biology largely by John Maynard Smith in his book Evolution and the Theory of Games.

In addition to being used to describe, predict, and explain behavior, game theory has also been used to develop theories of ethical or normative behavior and to prescribe such behavior. In economics and philosophy, scholars have applied game theory to help in the understanding of good or proper behavior. Game-theoretic arguments of this type can be found as far back as Plato.

Description and modeling

A four-stage centipede game

The primary use of game theory is to describe and model how human populations behave. Some scholars believe that by finding the equilibria of games they can predict how actual human populations will behave when confronted with situations analogous to the game being studied. This particular view of game theory has been criticized. It is argued that the assumptions made by game theorists are often violated when applied to real world situations. Game theorists usually assume players act rationally, but in practice, human behavior often deviates from this model. Game theorists respond by comparing their assumptions to those used in physics. Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists. However, empirical work has shown that in some classic games, such as the centipede game, guess 2/3 of the average game, and the dictator game, people regularly do not play Nash equilibria. There is an ongoing debate regarding the importance of these experiments and whether the analysis of the experiments fully captures all aspects of the relevant situation.

Some game theorists, following the work of John Maynard Smith and George R. Price, have turned to evolutionary game theory in order to resolve these issues. These models presume either no rationality or bounded rationality on the part of players. Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning (for example, fictitious play dynamics).

Prescriptive or normative analysis

	Cooperate	Defect
Cooperate	-1, -1	-10, 0
Defect	0, -10	-5, -5
The Prisoner's Dilemma

Some scholars, like Leonard Savage, see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave. Since a strategy, corresponding to a Nash equilibrium of a game constitutes one's best response to the actions of the other players – provided they are in (the same) Nash equilibrium – playing a strategy that is part of a Nash equilibrium seems appropriate. This normative use of game theory has also come under criticism.

Economics and business

Game theory is a major method used in mathematical economics and business for modeling competing behaviors of interacting agents. Applications include a wide array of economic phenomena and approaches, such as auctions, bargaining, mergers & acquisitions pricing, fair division, duopolies, oligopolies, social network formation, agent-based computational economics, general equilibrium, mechanism design, and voting systems; and across such broad areas as experimental economics, behavioral economics, information economics, industrial organization, and political economy.

This research usually focuses on particular sets of strategies known as "solution concepts" or "equilibria". A common assumption is that players act rationally. In non-cooperative games, the most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. If all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing.

The payoffs of the game are generally taken to represent the utility of individual players.
A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of a particular economic situation. One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type. Naturally one might wonder to what use this information should be put. Economists and business professors suggest two primary uses (noted above): descriptive and prescriptive.

Political science

The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice, war bargaining, positive political theory, and social choice theory. In each of these areas, researchers have developed game-theoretic models in which the players are often voters, states, special interest groups, and politicians.

Early examples of game theory applied to political science are provided by Anthony Downs. In his book An Economic Theory of Democracy, he applies the Hotelling firm location model to the political process. In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. Downs first shows how the political candidates will converge to the ideology preferred by the median voter if voters are fully informed, but then argues that voters choose to remain rationally ignorant which allows for candidate divergence. Game Theory was applied in 1962 to the Cuban missile crisis during the presidency of John F. Kennedy.

It has also been proposed that game theory explains the stability of any form of political government. Taking the simplest case of a monarchy, for example, the king, being only one person, does not and cannot maintain his authority by personally exercising physical control over all or even any significant number of his subjects. Sovereign control is instead explained by the recognition by each citizen that all other citizens expect each other to view the king (or other established government) as the person whose orders will be followed. Coordinating communication among citizens to replace the sovereign is effectively barred, since conspiracy to replace the sovereign is generally punishable as a crime. Thus, in a process that can be modeled by variants of the prisoner's dilemma, during periods of stability no citizen will find it rational to move to replace the sovereign, even if all the citizens know they would be better off if they were all to act collectively.

A game-theoretic explanation for democratic peace is that public and open debate in democracies send clear and reliable information regarding their intentions to other states. In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a non-democracy.

On the other hand, game theory predicts that two countries may still go to war even if their leaders are cognizant of the costs of fighting. War may result from asymmetric information; two countries may have incentives to mis-represent the amount of military resources they have on hand, rendering them unable to settle disputes agreeably without resorting to fighting. Moreover, war may arise because of commitment problems: if two countries wish to settle a dispute via peaceful means, but each wishes to go back on the terms of that settlement, they may have no choice but to resort to warfare. Finally, war may result from issue indivisibilities.

Game theory could also help predict a nation's responses when there is a new rule or law to be applied to that nation. One example would be Peter John Wood's (2013) research when he looked into what nations could do to help reduce climate change. Wood thought this could be accomplished by making treaties with other nations to reduce green house gas emissions. However, he concluded that this idea could not work because it would create a prisoner's dilemma to the nations.

Biology

	Hawk	Dove
Hawk	20, 20	80, 40
Dove	40, 80	60, 60
The hawk-dove game

Unlike those in economics, the payoffs for games in biology are often interpreted as corresponding to fitness. In addition, the focus has been less on equilibria that correspond to a notion of rationality and more on ones that would be maintained by evolutionary forces. The best known equilibrium in biology is known as the evolutionarily stable strategy (ESS), first introduced in (Smith & Price 1973). Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium.

In biology, game theory has been used as a model to understand many different phenomena. It was first used to explain the evolution (and stability) of the approximate 1:1 sex ratios. (Fisher 1930) suggested that the 1:1 sex ratios are a result of evolutionary forces acting on individuals who could be seen as trying to maximize their number of grandchildren.

Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication. The analysis of signaling games and other communication games has provided insight into the evolution of communication among animals. For example, the mobbing behavior of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization. Ants have also been shown to exhibit feed-forward behavior akin to fashion.

Biologists have used the game of chicken to analyze fighting behavior and territoriality.

According to Maynard Smith, in the preface to Evolution and the Theory of Games, "paradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behaviour for which it was originally designed". Evolutionary game theory has been used to explain many seemingly incongruous phenomena in nature.

One such phenomenon is known as biological altruism. This is a situation in which an organism appears to act in a way that benefits other organisms and is detrimental to itself. This is distinct from traditional notions of altruism because such actions are not conscious, but appear to be evolutionary adaptations to increase overall fitness. Examples can be found in species ranging from vampire bats that regurgitate blood they have obtained from a night's hunting and give it to group members who have failed to feed, to worker bees that care for the queen bee for their entire lives and never mate, to vervet monkeys that warn group members of a predator's approach, even when it endangers that individual's chance of survival. All of these actions increase the overall fitness of a group, but occur at a cost to the individual.

Evolutionary game theory explains this altruism with the idea of kin selection. Altruists discriminate between the individuals they help and favor relatives. Hamilton's rule explains the evolutionary rationale behind this selection with the equation c < b × r, where the cost c to the altruist must be less than the benefit b to the recipient multiplied by the coefficient of relatedness r. The more closely related two organisms are causes the incidences of altruism to increase because they share many of the same alleles. This means that the altruistic individual, by ensuring that the alleles of its close relative are passed on through survival of its offspring, can forgo the option of having offspring itself because the same number of alleles are passed on. For example, helping a sibling (in diploid animals) has a coefficient of ½, because (on average) an individual shares ½ of the alleles in its sibling's offspring. Ensuring that enough of a sibling's offspring survive to adulthood precludes the necessity of the altruistic individual producing offspring. The coefficient values depend heavily on the scope of the playing field; for example if the choice of whom to favor includes all genetic living things, not just all relatives, we assume the discrepancy between all humans only accounts for approximately 1% of the diversity in the playing field, a co-efficient that was ½ in the smaller field becomes 0.995. Similarly if it is considered that information other than that of a genetic nature (e.g. epigenetics, religion, science, etc.) persisted through time the playing field becomes larger still, and the discrepancies smaller.

Computer science and logic

Game theory has come to play an increasingly important role in logic and in computer science. Several logical theories have a basis in game semantics. In addition, computer scientists have used games to model interactive computations. Also, game theory provides a theoretical basis to the field of multi-agent systems.

Separately, game theory has played a role in online algorithms; in particular, the k-server problem, which has in the past been referred to as games with moving costs and request-answer games. Yao's principle is a game-theoretic technique for proving lower bounds on the computational complexity of randomized algorithms, especially online algorithms.

The emergence of the internet has motivated the development of algorithms for finding equilibria in games, markets, computational auctions, peer-to-peer systems, and security and information markets. Algorithmic game theory and within it algorithmic mechanism design combine computational algorithm design and analysis of complex systems with economic theory.

Philosophy

	Stag	Hare
Stag	3, 3	0, 2
Hare	2, 0	2, 2
Stag hunt

Game theory has been put to several uses in philosophy. Responding to two papers by W.V.O. Quine (1960, 1967), Lewis (1969) used game theory to develop a philosophical account of convention. In so doing, he provided the first analysis of common knowledge and employed it in analyzing play in coordination games. In addition, he first suggested that one can understand meaning in terms of signaling games. This later suggestion has been pursued by several philosophers since Lewis. Following Lewis (1969) game-theoretic account of conventions, Edna Ullmann-Margalit (1977) and Bicchieri (2006) have developed theories of social norms that define them as Nash equilibria that result from transforming a mixed-motive game into a coordination game.

Game theory has also challenged philosophers to think in terms of interactive epistemology: what it means for a collective to have common beliefs or knowledge, and what are the consequences of this knowledge for the social outcomes resulting from the interactions of agents. Philosophers who have worked in this area include Bicchieri (1989, 1993), Skyrms (1990), and Stalnaker (1999).

In ethics, some (most notably David Gauthier, Gregory Kavka, and Jean Hampton) authors have attempted to pursue Thomas Hobbes' project of deriving morality from self-interest. Since games like the prisoner's dilemma present an apparent conflict between morality and self-interest, explaining why cooperation is required by self-interest is an important component of this project. This general strategy is a component of the general social contract view in political philosophy.

Other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and corresponding animal behaviors. These authors look at several games including the prisoner's dilemma, stag hunt, and the Nash bargaining game as providing an explanation for the emergence of attitudes about morality (see, e.g., Skyrms (1996, 2004) and Sober and Wilson (1999)).

In popular culture

• Based on the 1998 book by Sylvia Nasar, the life story of game theorist and mathematician John Nash was turned into the 2001 biopic A Beautiful Mind, starring Russell Crowe as Nash.
• The 1959 military science fiction novel Starship Troopers by Robert A. Heinlein mentioned "games theory" and "theory of games". In the 1997 film of the same name, the character Carl Jenkins referred to his military intelligence assignment as being assigned to "games and theory".
• The 1964 film Dr. Strangelove satirizes game theoretic ideas about deterrence theory. For example, nuclear deterrence depends on the threat to retaliate catastrophically if a nuclear attack is detected. A game theorist might argue that such threats can fail to be credible, in the sense that they can lead to subgame imperfect equilibria. The movie takes this idea one step further, with the Russians irrevocably committing to a catastrophic nuclear response without making the threat public.
• The 1980s power pop band Game Theory was founded by singer/songwriter Scott Miller, who described the band's name as alluding to "the study of calculating the most appropriate action given an adversary... to give yourself the minimum amount of failure."
• Liar Game, a 2005 Japanese manga and 2007 television series, presents the main characters in each episode with a game or problem that is typically drawn from game theory, as demonstrated by the strategies applied by the characters.

Signalling theory

From Wikipedia, the free encyclopedia

By stotting (also called pronking), a springbok (Antidorcas marsupialis) signals honestly that it is young, fit, and not worth chasing to predators such as cheetahs.

Within evolutionary biology, signalling theory is a body of theoretical work examining communication between individuals, both within species and across species. The central question is when organisms with conflicting interests, such as in sexual selection, should be expected to provide honest signals (no presumption being made of conscious intention) rather than cheating. Mathematical models describe how signalling can contribute to an evolutionarily stable strategy.

Signals are given in contexts such as mate selection by females, which subjects the advertising males' signals to selective pressure. Signals thus evolve because they modify the behaviour of the receiver to benefit the signaller. Signals may be honest, conveying information which usefully increases the fitness of the receiver, or dishonest. An individual can cheat by giving a dishonest signal, which might briefly benefit that signaller, at the risk of undermining the signalling system for the whole population.

The question of whether selection of signals works at the level of the individual organism or gene, or at the level of the group, has been debated by biologists such as Richard Dawkins, arguing that individuals evolve to signal and to receive signals better, including resisting manipulation. Amotz Zahavi suggested that cheating could be controlled by the handicap principle, where the best horse in a handicap race is the one carrying the largest handicap weight. According to Zahavi's theory, signallers such as male peacocks have 'tails' that are genuinely handicaps, being costly to produce. The system is evolutionarily stable as the large showy tails are honest signals. Biologists have attempted to verify the handicap principle, but with inconsistent results. The mathematical biologist Ronald Fisher analysed the contribution that having two copies of each gene (diploidy) would make to honest signalling, demonstrating that a runaway effect could occur in sexual selection. The evolutionary equilibrium depends sensitively on the balance of costs and benefits.

The same mechanisms can be expected in humans, where researchers have studied behaviours including risk taking by young men, hunting of large game animals, and costly religious rituals, finding that these appear to qualify as costly honest signals.

Sexual selection

When animals choose mates, traits such as signalling are subject to evolutionary pressure. For example, the male gray tree frog, Hyla versicolor, produces a call to attract females. Once a female chooses a mate, this selects for a specific style of male calling, thus propagating a specific signalling ability. The signal can be the call itself, the intensity of a call, its variation style, its repetition rate, and so on. Various hypotheses seek to explain why females would select for one call over the other. The sensory exploitation hypothesis proposes that pre-existing preferences in female receivers can drive the evolution of signal innovation in male senders, in a similar way to the hidden preference hypothesis which proposes that successful calls are better able to match some 'hidden preference' in the female. Signallers have sometimes evolved multiple sexual ornaments, and receivers have sometimes evolved multiple trait preferences.

Honest signals

Eurasian jay, Garrulus glandarius, gives honest signals—loud alarm calls—from its tree perch when it sees a predator.

In biology, signals are traits, including structures and behaviours, that have evolved specifically because they change the behaviour of receivers in ways that benefit the signaller. Traits or actions that benefit the receiver exclusively are called cues. When an alert bird deliberately gives a warning call to a stalking predator and the predator gives up the hunt, the sound is a signal. When a foraging bird inadvertently makes a rustling sound in the leaves that attracts predators and increases the risk of predation, the sound is a 'cue'.

Signalling systems are shaped by mutual interests between signallers and receivers. An alert bird such as a Eurasian jay warning off a stalking predator is communicating something useful to the predator: that it has been detected by the prey; it might as well quit wasting its time stalking this alerted prey, which it is unlikely to catch. When the predator gives up, the signaller can get back to other tasks such as feeding. Once the stalking predator is detected, the signalling prey and receiving predator thus have a mutual interest in terminating the hunt.

Within species, mutual interests increase with kinship. Kinship is central to models of signalling between relatives, for instance when broods of nestling birds beg and compete for food from their parents.

The yellow-banded poison dart frog gives an honest signal of its toxicity to warn off predators and reduce the frog's risk of injury.

The term honesty in animal communication is controversial because in non-technical usage it implies intent, to discriminate deception from honesty in human interactions. However, biologists use the phrase "honest signals" in a direct, statistical sense. Biological signals, like warning calls or resplendent tail feathers, are honest if they truly convey useful information to the receiver. That is, the signal trait conveys to the receiver the presence of an otherwise unobservable factor. Honest biological signals do not need to be perfectly informative, reducing uncertainty to zero; all they need to be useful is to be correct "on average", so that certain behavioural responses to the signal are advantageous, statistically, compared to the behaviour that would occur in absence of the signal. Ultimately the value of the signalled information depends on the extent to which it allows the receiver to increase its fitness. Hence, "honest" signals are evolutionarily stable.

One class of honest signal is the aposematic warning signal, generally visual, given by poisonous or dangerous animals such as wasps, poison dart frogs, and pufferfish. Warning signals are honest indications of noxious prey, because conspicuousness evolves in tandem with noxiousness. Thus, the brighter and more conspicuous the organism, the more toxic it usually is. The most common and effective colours are red, yellow, black and white.

Dishonest signals

Male fiddler crab signals with its enlarged fighting claw, but weak regrown claws may be dishonest signals.

Because there are both mutual and conflicting interests in most animal signalling systems, a central problem in signalling theory is dishonesty or cheating. For example, if foraging birds are safer when they give a warning call, cheats could give false alarms at random, just in case a predator is nearby. But too much cheating could cause the signalling system to collapse. Every dishonest signal weakens the integrity of the signalling system, and so reduces the fitness of the group. An example of dishonest signalling comes from Fiddler crabs such as Uca lactea mjoebergi, which have been shown to bluff (no conscious intention being implied) about their fighting ability. When a claw is lost, a crab occasionally regrows a weaker claw that nevertheless intimidates crabs with smaller but stronger claws. The proportion of dishonest signals is low enough for it not to be worthwhile for crabs to test the honesty of every signal through combat.

Richard Dawkins and John Krebs in 1978 considered whether individuals of the same species would act as if attempting to deceive each other. They applied a "selfish gene" view of evolution to animals' threat displays to see if it would be in their genes' interests to give dishonest signals. They criticised previous ethologists, such as Nikolaas Tinbergen and Desmond Morris for suggesting that such displays were "for the good of the species". They argued that such communication ought to be viewed as an evolutionary arms race in which signallers evolve to become better at manipulating receivers, while receivers evolve to become more resistant to manipulation. The game theoretical model of the war of attrition similarly suggests that threat displays ought not to convey any reliable information about intentions.

Sports handicapping metaphor

The best horses in a handicap race carry the largest weights, so the size of the handicap is a measure of the animal's quality.

In 1975, Amotz Zahavi proposed a verbal model for how signal costs could constrain cheating and stabilize an "honest" correlation between observed signals and unobservable qualities, based on an analogy to sports handicapping systems. He called this idea the handicap principle. The purpose of a sports handicapping system is to reduce disparities in performance, making the contest more competitive. In a handicap race, intrinsically faster horses are given heavier weights to carry under their saddles. Similarly, in amateur golf, better golfers have fewer strokes subtracted from their raw scores. This creates correlations between the handicap and unhandicapped performance, and if the handicaps work as they are supposed to, between the handicap and handicapped performance. If you knew nothing about two race horses or two amateur golfers except their handicaps, you could infer which is most likely to win: the horse with the bigger weight handicap, and the golfer with the smaller stroke handicap. By analogy, if peacock 'tails' (large tail covert feathers) act as a handicapping system, and a peahen knew nothing about two peacocks but the sizes of their tails, she could "infer" that the peacock with the bigger tail has greater unobservable intrinsic quality. Display costs can include extrinsic social costs, in the form of testing and punishment by rivals, as well as intrinsic production costs. Another example given in textbooks is the extinct Irish elk, Megaloceros giganteus. The male Irish elk's enormous antlers could perhaps have evolved as displays of ability to overcome handicap, though biologists point out that if the handicap is inherited, its genes ought to be selected against.

Peacock signals reproductive fitness with its large colourful tail, possibly because it is a handicap.

The essential idea here is intuitive and probably qualifies as folk wisdom. It was articulated by Kurt Vonnegut in his 1961 short story Harrison Bergeron. In Vonnegut’s futuristic dystopia, the Handicapper General uses a variety of handicapping mechanisms to reduce inequalities in performance. A spectator at a ballet comments: "it was easy to see that she was the strongest and most graceful of all dancers, for her handicap bags were as big as those worn by two hundred pound men." Zahavi interpreted this analogy to mean that higher quality peacocks with bigger tails are signalling their ability to "waste" more of some resource by trading it off for a bigger tail. This resonates with Thorstein Veblen's idea that conspicuous consumption and extravagant status symbols can signal wealth.

The enormous antlers of the extinct Irish elk, Megaloceros giganteus may have evolved as displays of ability to overcome handicap.

Zahavi’s conclusions rest on his verbal interpretation of a metaphor, and initially the handicap principle was not well received by evolutionary biologists. However, in 1984, Nur and Hasson used life history theory to show how differences in signalling costs, in the form of survival-reproduction tradeoffs, could stabilize a signalling system roughly as Zahavi imagined. Genetic models also suggested this was possible. In 1990 Alan Grafen showed that a handicap-like signalling system was evolutionarily stable if higher quality signallers paid lower marginal survival costs for their signals.

In 1982, W.D. Hamilton proposed a specific but widely applicable handicap mechanism, parasite-mediated sexual selection. He argued that in the never-ending co-evolutionary race between hosts and their parasites, sexually selected signals indicate health. This idea was tested in 1994 in barn swallows, a species where males have long tail streamers. Møller found that the males with longer tails, and their offspring, did have fewer bloodsucking mites, whereas fostered young did not. The effect was therefore genetic, confirming Hamilton's theory.

Another example is Lozano's hypothesis that carotenoids have dual but mutually incompatible roles in immune function and signalling. Given that animals cannot synthesize carotenoids de novo, these must be obtained from food. The hypothesis states that animals with carotenoid-depended sexual signals are demonstrating their ability to "waste" carotenoids on sexual signals at the expense of their immune system.

The handicap principle has proven hard to test empirically, partly because of inconsistent interpretations of Zahavi’s metaphor and Grafen’s marginal fitness model, and partly because of conflicting empirical results: in some studies individuals with bigger signals seem to pay higher costs, in other studies they seem to be paying lower costs. A possible explanation for the inconsistent empirical results is given in a series of papers by Getty, who shows that Grafen’s proof of the handicap principle is based on the critical simplifying assumption that signallers trade off costs for benefits in an additive fashion, the way humans invest money to increase income in the same currency. But the assumption that costs and benefits trade off in an additive fashion is true only on a logarithmic scale; for the survival cost – reproduction benefit tradeoff is assumed to mediate the evolution of sexually selected signals. Fitness depends on producing offspring, which is a multiplicative function of reproductive success given an individual is still alive times the probability of still being alive, given investment in signals.

Costly signalling and Fisherian diploid dynamics

The effort to discover how costs can constrain an "honest" correlation between observable signals and unobservable qualities within signallers is built on strategic models of signalling games, with many simplifying assumptions. These models are most often applied to sexually selected signalling in diploid animals, but they rarely incorporate a fact about diploid sexual reproduction noted by the mathematical biologist Ronald Fisher in the early 20th century: if there are "preference genes" correlated with choosiness in females as well as "signal genes" correlated with display traits in males, choosier females should tend to mate with showier males. Over generations, showier sons should also carry genes associated with choosier daughters, and choosier daughters should also carry genes associated with showier sons. This can cause the evolutionary dynamic known as Fisherian runaway, in which males become ever showier. Russell Lande explored this with a quantitative genetic model, showing that Fisherian diploid dynamics are sensitive to signalling and search costs. Other models incorporate both costly signalling and Fisherian runaway. These models show that if fitness depends on both survival and reproduction, having sexy sons and choosy daughters (in the stereotypical model) can be adaptive, increasing fitness just as much as having healthy sons and daughters.

Examples

One theory is that autumnal colours are a signal from trees to aphids of powerful chemical defences.

Sam Brown and W. D. Hamilton and Marco Archetti proposed that autumn leaf colour is a signal from trees to aphids and other pest species that migrate in autumn to the trees. In their theory, bright autumn coloration with pinks and yellows is costly to trees because pigments require energy to synthesize, but the investment may help them to reduce their parasite load.

Stotting, for example in Thomson's Gazelle, is cited as an example of signalling: the gazelles jump close to a predator instead of escaping, in what could be a signal of strength.

Human honest signals

Human behaviour may also provide examples of costly signals. In general, these signals provide information about a person’s phenotypic quality or cooperative tendencies. Evidence for costly signalling has been found in many areas of human interaction including risk taking, hunting, and religion.

Costly signalling in hunting

A male hunter and a female gatherer of the Kali'na people of Guyana, drawn by Pierre Barrère in 1743. Generous sharing by male hunters may serve as a "costly signal", helping them to acquire mates.

Large game hunting has been studied extensively as a signal of men’s willingness to take physical risks, as well as showcase strength and coordination. Costly signalling theory is a useful tool for understanding food sharing among hunter gatherers because it can be applied to situations in which delayed reciprocity is not a viable explanation. Instances that are particularly inconsistent with the delayed reciprocity hypothesis are those in which a hunter shares his kill indiscriminately with all members of a large group. In these situations, the individuals sharing meat have no control over whether or not their generosity will be reciprocated, and free riding becomes an attractive strategy for those receiving meat. Free riders are people who reap the benefits of group-living without contributing to its maintenance. Fortunately, costly signalling theory can fill some of the gaps left by the delayed reciprocity hypothesis. Hawkes has suggested that men target large game and publicly share meat to draw social attention or to show off. Such display and the resulting favorable attention can improve a hunter’s reputation by providing information about his phenotypic quality. High quality signallers are more successful in acquiring mates and allies. Thus, costly signalling theory can explain apparently wasteful and altruistic behaviour.

In order to be effective, costly signals must fulfill specific criteria. Firstly, signallers must incur different levels of cost and benefit for signalling behaviour. Secondly, costs and benefits must reflect the signallers’ phenotypic quality. Thirdly, the information provided by a signal should be directed at and accessible to an audience. A receiver can be anyone who stands to benefit from information the signaller is sending, such as potential mates, allies, or competitors. Honesty is guaranteed when only individuals of high quality can pay the (high) costs of signalling. Hence, costly signals make it impossible for low-quality individuals to fake a signal and fool a receiver.

Bliege Bird et al. observed turtle hunting and spear fishing patterns in a Meriam community in the Torres Strait of Australia, publishing their findings in 2001. Here, only some Meriam men were able to accumulate high caloric gains for the amount of time spent turtle hunting or spear fishing (reaching a threshold measured in kcal/h). Since a daily catch of fish is carried home by hand and turtles are frequently served at large feasts, members of the community know which men most reliably brought them turtle meat and fish. Thus, turtle hunting qualifies as a costly signal. Furthermore, turtle hunting and spear fishing are actually less productive (in kcal/h) than foraging for shellfish, where success depends only on the amount of time dedicated to searching, so shellfish foraging is a poor signal of skill or strength. This suggests that energetic gains are not the primary reason men take part in turtle hunting and spear fishing. A follow-up study found that successful Meriam hunters do experience greater social benefits and reproductive success than less skilled hunters.

The Hadza people of Tanzania also share food, possibly to gain in reputation. Hunters cannot be sharing meat mainly to provision their families or to gain reciprocal benefits, as teenage boys often give away their meat even though they do not yet have wives or children, so costly signalling of their qualities is the likely explanation. These qualities include good eyesight, coordination, strength, knowledge, endurance, or bravery. Hadza hunters more often pair with highly fertile, hard-working wives than non-hunters. A woman benefits from mating with a man who possesses such qualities as her children will most likely inherit qualities that increase fitness and survivorship. She may also benefit from her husband’s high social status. Thus, hunting is an honest and costly signal of phenotypic quality.

Among the men of Ifaluk atoll, costly signalling theory can also explain why men torch fish. Torch fishing is a ritualized method of fishing on Ifaluk whereby men use torches made from dried coconut fronds to catch large dog-toothed tuna. Preparation for torch fishing requires significant time investments and involves a great deal of organization. Due to the time and energetic costs of preparation, torch fishing results in net caloric losses for fishers. Therefore, torch fishing is a handicap that serves to signal men’s productivity. Torch fishing is the most advertised fishing occupation on Ifaluk. Women and others usually spend time observing the canoes as they sail beyond the reef. Also, local rituals help to broadcast information about which fishers are successful and enhance fishers’ reputations during the torch fishing season. Several ritual behavioural and dietary constraints clearly distinguish torch fishers from other men. First, males are only permitted to torch fish if they participated on the first day of the fishing season. The community is well informed as to who participates on this day, and can easily identify the torch fishers. Second, torch fishers receive all of their meals at the canoe house and are prohibited from eating certain foods. People frequently discuss the qualities of torch fishermen. On Ifaluk, women claim that they are looking for hard-working mates. With the distinct sexual division of labor on Ifaluk, industriousness is a highly valued characteristic in males. Torch fishing thus provides women with reliable information on the work ethic of prospective mates, which makes it an honest costly signal.

In many human cases, a strong reputation built through costly signalling enhances a man’s social status over the statuses of men who signal less successfully. Among northern Kalahari foraging groups, traditional hunters usually capture a maximum of two or three antelopes per year. It was said of a particularly successful hunter:

"It was said of him that he never returned from a hunt without having killed at least a wildebeest, if not something larger. Hence the people connected with him ate a great deal of meat and his popularity grew."

Although this hunter was sharing meat, he was not doing so in the framework of reciprocity. The general model of costly signalling is not reciprocal; rather, individuals who share acquire more mates and allies. Costly signalling applies to situations in Kalahari foraging groups where giving often goes to recipients who have little to offer in return. A young hunter is motivated to impress community members with daughters so that he can obtain his first wife. Older hunters may wish to attract women interested in an extramarital relationship, or to be a co-wife. In these northern Kalahari groups, the killing of a large animal indicates a man who has mastered the art of hunting and can support a family. Generally, many women seek a man who is a good hunter, has an agreeable character, is generous, and has advantageous social ties. Since hunting ability is a prerequisite for marriage, men who are good hunters enter the marriage market earliest. Costly signalling theory explains seemingly wasteful foraging displays.

Physical risks as a costly signal

Young men may take part in risky sports like motorcycle racing to signal their strength and skill.

Costly signalling can be applied to situations involving physical strain and risk of physical injury or death. Research on physical risk taking is important because information regarding why people, especially young men, take part in high risk activities can help in the development of prevention programs. Reckless driving is a lethal problem among young men in western societies. A male who takes a physical risk is sending the message that he has enough strength and skill to survive extremely dangerous activities. This signal is directed at peers and potential mates. When those peers are criminals or gang members, sociologists Diego Gambetta and James Densley find that risk-taking signals can help expedite acceptance into the group.

In a study of risk taking, some types of risk, such as physical or heroic risk for others' benefit, are viewed more favorably than other types of risk, such as taking drugs. Males and females valued different degrees of heroic risk for mates and same-sex friends. Males valued heroic risk taking by male friends, but preferred less of it in female mates. Females valued heroic risk taking in male mates and less of it in female friends. Females may be attracted to males inclined to physically defend them and their children. Males may prefer heroic risk taking by male friends as they could be good allies.

In western societies, voluntary blood donation is a common, yet less extreme, form of risk taking. Costs associated with these donations include pain and risk of infection. If blood donation is an opportunity to send costly signals, then donors will be perceived by others as generous and physically healthy. In a survey, both donors and non-donors expressed perceptions of the health, generosity, and ability of blood donors to operate in stressful situations.

Religion as a costly signal

Religious rituals such as snake handling may be explainable as costly signals.

Costly religious rituals such as male circumcision, food and water deprivation, and snake handling look paradoxical in evolutionary terms. Devout religious beliefs wherein such traditions are practiced therefore appear maladaptive. Religion may have arisen to increase and maintain intragroup cooperation. Cooperation leads to altruistic behaviour, and costly signalling could explain this. All religions may involve costly and elaborate rituals, performed publicly, to demonstrate loyalty to the religious group. In this way, group members increase their allegiance to the group by signalling their investment in group interests. However, as group size increases among humans, the threat of free riders grows. Costly signalling theory accounts for this by proposing that these religious rituals are costly enough to deter free riders.

Irons proposed that costly signalling theory could explain costly religious behaviour. He argued that hard-to-fake religious displays enhanced trust and solidarity in a community, producing emotional and economic benefits. He showed that display signals among the Yomut Turkmen of northern Iran helped to secure trade agreements. These "ostentatious" displays signalled commitment to Islam to strangers and group members. Sosis demonstrated that people in religious communities are four times more likely to live longer than their secular counterparts, and that these longer lifespans were positively correlated with the number of costly requirements demanded from religious community members. However, confounding variables may not have been excluded. Wood found that religion offers a subjective feeling of well-being within a community, where costly signalling protects against free riders and helps to build self-control among committed members. Iannaccone studied the effects of costly signals on religious communities. In a self-reported survey, as the strictness of a church increased, the attendance and contributions to that church increased proportionally. In effect, people were more willing to participate in a church that has more stringent demands on its members. Despite this observation, costly donations and acts conducted in a religious context does not itself establish that membership in these clubs is actually worth the entry costs imposed.

Despite the experimental support for this hypothesis, it remains controversial. A common critique is that devoutness is easy to fake, such as simply by attending a religious service. However, the hypothesis predicts that people are more likely to join and contribute to a religious group when its rituals are costly. Another critique specifically asks: why religion? There is no evolutionary advantage to evolving religion over other signals of commitment such as nationality, as Irons admits. However, the reinforcement of religious rites as well as the intrinsic reward and punishment system found in religion makes it an ideal candidate for increasing intragroup cooperation. Finally, there is insufficient evidence for increase in fitness as a result of religious cooperation. However, Sosis argues for benefits from religion itself, such as increased longevity, improved health, assistance during crises, and greater psychological well being though both the supposed benefits from religion and the costly-signaling mechanism have been contested.