TheInfoList

OR:

Game theory is the study of
mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, b ...
s of strategic interactions among
rational agent A rational agent or rational being is a person or entity that always aims to perform optimal actions based on given premises and information. A rational agent can be anything that makes decisions, typically a person, firm, machine, or software. ...
s. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs
1
vii–xi
It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an
umbrella term In linguistics, semantics, general semantics, and ontologies, hyponymy () is a semantic relation between a hyponym denoting a subtype and a hypernym or hyperonym (sometimes called umbrella term or blanket term) denoting a supertype. In other ...
for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a conv ...
s, which became a standard method in game theory and
mathematical economics Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference a ...
. His paper was followed by the 1944 book ''
Theory of Games and Economic Behavior ''Theory of Games and Economic Behavior'', published in 1944 by Princeton University Press, is a book by mathematician John von Neumann and economist Oskar Morgenstern which is considered the groundbreaking text that created the interdisciplinar ...
'', co-written with Oskar Morgenstern, which considered
cooperative game Cooperative game may refer to: * Cooperative board game, board games in which players work together to achieve a common goal * Cooperative game theory, in game theory, a game with competition between groups of players and the possibility of cooperat ...
s 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 explicitly applied to evolution 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. , with the
Nobel Memorial Prize in Economic Sciences The Nobel Memorial Prize in Economic Sciences, officially the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel ( sv, Sveriges riksbanks pris i ekonomisk vetenskap till Alfred Nobels minne), is an economics award administered ...
going to game theorists
Paul Milgrom Paul Robert Milgrom (born April 20, 1948) is an American economist. He is the Shirley and Leonard Ely Professor of Humanities and Sciences at the Stanford University School of Humanities and Sciences, a position he has held since 1987. He is a ...
and
Robert B. Wilson } Robert Butler Wilson, Jr. (born May 16, 1937) is an American economist and the Adams Distinguished Professor of Management, Emeritus at Stanford University. He was jointly awarded the 2020 Nobel Memorial Prize in Economic Sciences, together w ...
, fifteen game theorists have won the economics Nobel Prize. John Maynard Smith was awarded the
Crafoord Prize The Crafoord Prize is an annual science prize established in 1980 by Holger Crafoord, a Swedish industrialist, and his wife Anna-Greta Crafoord. The Prize is awarded in partnership between the Royal Swedish Academy of Sciences and the Crafoord Fou ...
for his application of evolutionary game theory.

# History

## Precursors

Discussions on the mathematics of games began long before the rise of modern mathematical game theory. Cardano's work on games of chance in ''Liber de ludo aleae'' (''Book on Games of Chance''), which was written around 1564 but published posthumously in 1663, formulated some of the field's basic ideas. In the 1650s,
Pascal Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, Frenc ...
and Huygens developed the concept of expectation on reasoning about the structure of games of chance, and Huygens published his gambling calculus in ''De ratiociniis in ludo aleæ'' (''On Reasoning in Games of Chance'') in 1657. In 1713, a letter attributed to Charles Waldegrave analyzed a game called "le Her". He was an active Jacobite and uncle to James Waldegrave, a British diplomat. In this letter, Waldegrave provided a
minimax Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When ...
mixed strategy solution to a two-person version of the card game le Her, and the problem is now known as Waldegrave problem. 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 A duopoly (from Greek δύο, ''duo'' "two" and πωλεῖν, ''polein'' "to sell") is a type of oligopoly where two firms have dominant or exclusive control over a market. It is the most commonly studied form of oligopoly due to its simplicity ...
and presented a solution that is the Nash equilibrium of the game. In 1913,
Ernst Zermelo Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel axiomatic ...
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''), which 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 is symmetric and provided a solution to a non-trivial infinite game (known in English as Blotto game). Borel conjectured the non-existence of mixed-strategy equilibria in finite two-person zero-sum games, a conjecture that was proved false by von Neumann.

## Birth and early developments

Game theory did not 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 set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a conv ...
s, which became a standard method in game theory and
mathematical economics Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference a ...
. His paper was followed by his 1944 book ''
Theory of Games and Economic Behavior ''Theory of Games and Economic Behavior'', published in 1944 by Princeton University Press, is a book by mathematician John von Neumann and economist Oskar Morgenstern which is considered the groundbreaking text that created the interdisciplinar ...
'' 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 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. Subsequent work focused primarily on
cooperative game Cooperative game may refer to: * Cooperative board game, board games in which players work together to achieve a common goal * Cooperative game theory, in game theory, a game with competition between groups of players and the possibility of cooperat ...
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 The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...
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 Nuclear strategy involves the development of doctrines and strategies for the production and use of nuclear weapons. As a sub-branch of military strategy, nuclear strategy attempts to match nuclear weapons as means to political ends. In addi ...
. Around this same time, John Nash developed a criterion for mutual consistency of players' strategies known as the Nash equilibrium, applicable to a wider variety of games than the criterion proposed by von Neumann and Morgenstern. Nash proved that every finite n-player, non-zero-sum (not just two-player zero-sum)
non-cooperative game In game theory, a non-cooperative game is a game with competition between individual players, as opposed to cooperative games, and in which alliances can only operate if self-enforcing (e.g. through credible threats). However, 'cooperative' an ...
has what is now known as a Nash equilibrium in mixed strategies. Game theory experienced a flurry of activity in the 1950s, during which the concepts of the core, the
extensive form game An extensive-form game is a specification of a game in game theory, allowing (as the name suggests) for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, t ...
,
fictitious play In game theory, fictitious play is a learning rule first introduced by George W. Brown. In it, each player presumes that the opponents are playing stationary (possibly mixed) strategies. At each round, each player thus best responds to the empiri ...
, repeated games, and the Shapley value were developed. The 1950s also saw the first applications of game theory to philosophy and political science.

## 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 Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
, 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 Common knowledge is knowledge that is publicly known by everyone or nearly everyone, usually with reference to the community in which the knowledge is referenced. Common knowledge can be about a broad range of subjects, such as science, litera ...
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 equilibrium coarsening and correlated equilibria, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences. In 2007,
Leonid Hurwicz Leonid Hurwicz (; August 21, 1917 – June 24, 2008) was a Polish-American economist and mathematician, known for his work in game theory and mechanism design. He originated the concept of incentive compatibility, and showed how desired outcome ...
, Eric Maskin, and Roger Myerson were 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 A mechanism is called incentive-compatible (IC) if every participant can achieve the best outcome to themselves just by acting according to their true preferences. There are several different degrees of incentive-compatibility: * The stronger ...
. In 2012,
Alvin E. Roth Alvin Eliot Roth (born December 18, 1951) is an American academic. He is the Craig and Susan McCaw professor of economics at Stanford University and the Gund professor of economics and business administration emeritus at Harvard University.
and Lloyd S. Shapley were awarded the Nobel Prize in Economics "for the theory of stable allocations and the practice of market design". In 2014, the Nobel went to game theorist
Jean Tirole Jean Tirole (born 9 August 1953) is a French professor of economics at Toulouse 1 Capitole University. He focuses on industrial organization, game theory, banking and finance, and economics and psychology. In 2014 he was awarded the Nobel Mem ...
.

# 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 contract is a legally enforceable agreement between two or more Party (law), parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, Service (economics), ser ...
). 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 analyzed 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 In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equil ...
. The focus on individual payoff can result in a phenomenon known as Tragedy of the Commons, where resources are used to a collectively inefficient level. The lack of formal
negotiation Negotiation is a dialogue between two or more people or parties to reach the desired outcome regarding one or more issues of conflict. It is an interaction between entities who aspire to agree on matters of mutual interest. The agreement ...
leads to the deterioration of public goods through over-use and under provision that stems from private incentives. Cooperative game theory provides a high-level approach as it describes only 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 using a single theory may be desirable, in many instances insufficient information is available to accurately model the formal procedures available during the strategic bargaining process, or the resulting model would be too complex 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

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. That is, 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 chicken (''Gallus gallus domesticus'') is a domesticated junglefowl species, with attributes of wild species such as the grey and the Ceylon junglefowl that are originally from Southeastern Asia. Rooster or cock is a term for an adult m ...
, the
prisoner's dilemma The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...
, 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. The 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 in this section's graphic is asymmetric despite having identical strategy sets for both players.

## Zero-sum / non-zero-sum

Zero-sum games (more generally, constant-sum games) are games in which choices by players can neither increase nor decrease the available resources. In zero-sum games, the total benefit goes to all players in a game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the equal expense of others).
Poker Poker is a family of comparing card games in which players wager over which hand is best according to that specific game's rules. It is played worldwide, however in some places the rules may vary. While the earliest known form of the game wa ...
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 Matching pennies is the name for a simple game used in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously ...
and most classical board games including Go and chess. Many games studied by game theorists (including the famed
prisoner's dilemma The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...
) 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 constant-sum 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 instead the later players are unaware of the earlier players' actions (making them ''effectively'' simultaneous).
Sequential game In game theory, a sequential game is a game where one player chooses their action before the others choose theirs. The other players must have information on the first player's choice so that the difference in time has no strategic effect. Seq ...
s (or dynamic games) are games where later players have some knowledge about earlier actions. This need not be
perfect information In economics, perfect information (sometimes referred to as "no hidden information") is a feature of perfect competition. With perfect information in a market, all consumers and producers have complete and instantaneous knowledge of all market pr ...
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 they do 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 An extensive-form game is a specification of a game in game theory, allowing (as the name suggests) for the explicit representation of a number of key aspects, like the sequencing of players' possible moves, their choices at every decision point, th ...
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:

### Cournot Competition

The Cournot competition model involves players choosing quantity of a homogenous product to produce independently and simultaneously, where
marginal cost In economics, the marginal cost is the change in the total cost that arises when the quantity produced is incremented, the cost of producing additional quantity. In some contexts, it refers to an increment of one unit of output, and in others i ...
can be different for each firm and the firm's payoff is profit. The production costs are public information and the firm aims to find their profit-maximising quantity based on what they believe the other firm will produce and behave like monopolies. In this game firms want to produce at the monopoly quantity but there is a high incentive to deviate and produce more, which decreases the market-clearing price. For example, firms may be tempted to deviate from the monopoly quantity if there is a low monopoly quantity and high price, with the aim of increasing production to maximise profit. However this option does not provide the highest payoff, as a firm's ability to maximise profits depends on its market share and the elasticity of the market demand. The Cournot equilibrium is reached when each firm operates on their reaction function with no incentive to deviate, as they have the best response based on the other firms output. Within the game, firms reach the Nash equilibrium when the Cournot equilibrium is achieved.

### Bertrand Competition

The Bertrand competition, assumes homogenous products and a constant marginal cost and players choose the prices. The equilibrium of price competition is where the price is equal to marginal costs, assuming complete information about the competitors' costs. Therefore, the firms have an incentive to deviate from the equilibrium because a homogenous product with a lower price will gain all of the market share, known as a cost advantage.

## Perfect information and imperfect information

An important subset of sequential games consists of games of
perfect information In economics, perfect information (sometimes referred to as "no hidden information") is a feature of perfect competition. With perfect information in a market, all consumers and producers have complete and instantaneous knowledge of all market pr ...
. A game is one of perfect information if all players, at every move in the game, know the moves previously made by all other players. In reality, this can be applied to firms and consumers having information about price and quality of all the available goods in a market. An imperfect information game is played when the players do not know all moves already made by the opponent such as a simultaneous move game. Most games studied in game theory are imperfect-information games. Examples of perfect-information games include
tic-tac-toe Tic-tac-toe (American English), noughts and crosses ( Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid with ''X'' or ''O''. ...
,
checkers Checkers (American English), also known as draughts (; British English), is a group of strategy board games for two players which involve diagonal moves of uniform game pieces and mandatory captures by jumping over opponent pieces. Checkers ...
, chess, and Go. Many card games are games of imperfect information, such as
poker Poker is a family of comparing card games in which players wager over which hand is best according to that specific game's rules. It is played worldwide, however in some places the rules may vary. While the earliest known form of the game wa ...
and
bridge A bridge is a structure built to span a physical obstacle (such as a body of water, valley, road, or rail) without blocking the way underneath. It is constructed for the purpose of providing passage over the obstacle, which is usually someth ...
. 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, whereas perfect information is knowledge of all aspects of the game and players. Games of incomplete information can be reduced, however, to games of imperfect information by introducing " moves by nature".

### Bayesian game

One of the assumptions of the Nash equilibrium is that every player has correct beliefs about the actions of the other players. However, there are many situations in game theory where participants do not fully understand the characteristics of their opponents. Negotiators may be unaware of their opponent's valuation of the object of negotiation, companies may be unaware of their opponent's cost functions, combatants may be unaware of their opponent's strengths, and jurors may be unaware of their colleague's interpretation of the evidence at trial. In some cases, participants may know the character of their opponent well, but may not know how well their opponent knows his or her own character. Bayesian game means a strategic game with incomplete information. For a strategic game, decision makers are players, and every player has a group of actions. A core part of the imperfect information specification is the set of states. Every state completely describes a collection of characteristics relevant to the player such as their preferences and details about them. There must be a state for every set of features that some player believes may exist. For example, where Player 1 is unsure whether Player 2 would rather date her or get away from her, while Player 2 understands Player 1's preferences as before. To be specific, supposing that Player 1 believes that Player 2 wants to date her under a probability of 1/2 and get away from her under a probability of 1/2 (this evaluation comes from Player 1's experience probably: she faces players who want to date her half of the time in such a case and players who want to avoid her half of the time). Due to the probability involved, the analysis of this situation requires to understand the player's preference for the draw, even though people are only interested in pure strategic equilibrium.

## 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 Backgammon is a two-player board game played with counters and dice on tables boards. It is the most widespread Western member of the large family of tables games, whose ancestors date back nearly 5,000 years to the regions of Mesopotamia and Pe ...
. There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve some particular problems and answer some general questions. Games of
perfect information In economics, perfect information (sometimes referred to as "no hidden information") is a feature of perfect competition. With perfect information in a market, all consumers and producers have complete and instantaneous knowledge of all market pr ...
have been studied in
combinatorial game theory Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a ''position'' that the playe ...
, which has developed novel representations, e.g.
surreal numbers In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals ...
, as well as
combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...
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 Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech rec ...
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 Alpha–beta pruning is a search algorithm that seeks to decrease the number of nodes that are evaluated by the minimax algorithm in its search tree. It is an adversarial search algorithm used commonly for machine playing of two-player games ( ...
or use of
artificial neural network Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains. An ANN is based on a collection of connected units ...
s 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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
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 Determinacy is a subfield of set theory, a branch of mathematics, that examines the conditions under which one or the other player of a game has a winning strategy, and the consequences of the existence of such strategies. Alternatively and simi ...
. (It can be proven, using the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, that there are gameseven 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 In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has applications to ...
.

## 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 equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
s. The problem of finding an optimal strategy in a differential game is closely related to the
optimal control Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering an ...
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 In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
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 A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
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 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". They may be modeled using similar tools within the related disciplines of decision theory,
operations research Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve dec ...
, and areas of
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech rec ...
, particularly AI planning (with uncertainty) and
multi-agent system A multi-agent system (MAS or "self-organized system") is a computerized system composed of multiple interacting intelligent agents.Hu, J.; Bhowmick, P.; Jang, I.; Arvin, F.; Lanzon, A.,A Decentralized Cluster Formation Containment Framework fo ...
. 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. (See
Black swan theory The black swan theory or theory of black swan events is a metaphor that describes an event that comes as a surprise, has a major effect, and is often inappropriately rationalized after the fact with the benefit of hindsight. The term is based o ...
for more discussion on this kind of modeling issue, particularly as it relates to predicting and limiting losses in investment banking.) 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 realize 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 A convention is a set of agreed, stipulated, or generally accepted standards, norms, social norms, or criteria, often taking the form of a custom. In a social context, a convention may retain the character of an "unwritten law" of custom (for e ...
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 Mean-field game theory is the study of strategic decision making by small interacting agents in very large populations. It lies at the intersection of game theory with stochastic analysis and control theory. The use of the term "mean field" is insp ...
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 mathematicians
Pierre-Louis Lions Pierre-Louis Lions (; born 11 August 1956) is a French mathematician. He is known for a number of contributions to the fields of partial differential equations and the calculus of variations. He was a recipient of the 1994 Fields Medal and the 19 ...
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

The extensive form can be used to formalize games with a time sequencing of moves. Games here are played on
trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...
(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 A decision tree is a decision support tool that uses a tree-like model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm that only contains condi ...
. To solve any extensive form game,
backward induction Backward induction is the process of reasoning backwards in time, from the end of a problem or situation, to determine a sequence of optimal actions. It proceeds by examining the last point at which a decision is to be made and then identifying wha ...
must be used. It involves working backward 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 or (fair or unfair). Next in the sequence, ''Player 2'', who has now seen ''Player 1''s move, chooses to play either or . Once ''Player 2'' has made their choice, the game is considered finished and each player gets their respective payoff. Suppose that ''Player 1'' chooses and then ''Player 2'' chooses : ''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

The normal (or strategic form) game is usually represented by a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
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 $\mathbf$ appears, it works against the fraction $\left\left(\frac\right\right)$ 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 $v:2^N \to \mathbf$ is a normal utility. Such characteristic functions have expanded to describe games where there is no removable utility.

## Alternative game representations

Alternative game representation forms are used for some subclasses of games or adjusted to the needs of interdisciplinary research. In addition to classical game representations, some of the alternative representations also encode time related aspects.

# 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 Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analy ...
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 Cournot competition is an economic model used to describe an industry structure in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time. It is named after Antoine A ...
. 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 1982 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 Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
. An alternative version of game theory, called chemical game theory, represents the player's choices as metaphorical chemical reactant molecules called "knowlecules".  Chemical game theory then calculates the outcomes as equilibrium solutions to a system of chemical reactions.

## Description and modeling

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
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate ca ...
s. However, empirical work has shown that in some classic games, such as the
centipede game In game theory, the centipede game, first introduced by Robert Rosenthal in 1981, is an extensive form game in which two players take turns choosing either to take a slightly larger share of an increasing pot, or to pass the pot to the other playe ...
,
guess 2/3 of the average In game theory, "guess of the average" is a game that explores how a player’s strategic reasoning process takes into account the mental process of others in the game. In this game, players simultaneously select a real number between 0 and 100, ...
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 George Robert Price (October 6, 1922 – January 6, 1975) was an American population geneticist. Price is often noted for his formulation of the Price equation in 1967. Originally a physical chemist and later a science journalist, he moved ...
, 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 In game theory, fictitious play is a learning rule first introduced by George W. Brown. In it, each player presumes that the opponents are playing stationary (possibly mixed) strategies. At each round, each player thus best responds to the empiri ...
dynamics).

## Prescriptive or normative analysis

Some scholars 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.

Game theory is a major method used in
mathematical economics Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference a ...
and business for modeling competing behaviors of interacting agents. Applications include a wide array of economic phenomena and approaches, such as
auction An auction is usually a process of buying and selling goods or services by offering them up for bids, taking bids, and then selling the item to the highest bidder or buying the item from the lowest bidder. Some exceptions to this definition e ...
s,
bargaining In the social sciences, bargaining or haggling is a type of negotiation in which the buyer and seller of a good or service debate the price or nature of a transaction. If the bargaining produces agreement on terms, the transaction takes pl ...
,
mergers and acquisitions Mergers and acquisitions (M&A) are business transactions in which the ownership of companies, other business organizations, or their operating units are transferred to or consolidated with another company or business organization. As an aspec ...
pricing,N. Agarwal and P. Zeephongsekul
Psychological Pricing in Mergers & Acquisitions using Game Theory
School of Mathematics and Geospatial Sciences, RMIT University, Melbourne
fair division, duopolies, oligopolies, social network formation,
agent-based computational economics Agent-based computational economics (ACE) is the area of computational economics that studies economic processes, including whole economies, as dynamic systems of interacting agents. As such, it falls in the paradigm of complex adaptive systems. ...
, general equilibrium, mechanism design, and voting systems; and across such broad areas as experimental economics, behavioral economics, information economics, industrial organization, and
political economy Political economy is the study of how economic systems (e.g. markets and national economies) and political systems (e.g. law, institutions, government) are linked. Widely studied phenomena within the discipline are systems such as labour ...
. 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. Economists and business professors suggest two primary uses (noted above): ''descriptive'' and '' prescriptive''. The
Chartered Institute of Procurement & Supply The Chartered Institute of Procurement & Supply (CIPS), formerly the Chartered Institute of Purchasing & Supply, is a global professional body working for the procurement and supply profession in many regions of the world. It promotes best prac ...
(CIPS) promotes knowledge and use of game theory within the context of business
procurement Procurement is the method of discovering and agreeing to terms and purchasing goods, services, or other works from an external source, often with the use of a tendering or competitive bidding process. When a government agency buys goods or s ...
. CIPS and TWS Partners have conducted a series of surveys designed to explore the understanding, awareness and application of game theory among
procurement Procurement is the method of discovering and agreeing to terms and purchasing goods, services, or other works from an external source, often with the use of a tendering or competitive bidding process. When a government agency buys goods or s ...
professionals. Some of the main findings in their third annual survey (2019) include: *application of game theory to procurement activity has increased – at the time it was at 19% across all survey respondents *65% of participants predict that use of game theory applications will grow *70% of respondents say that they have "only a basic or a below basic understanding" of game theory *20% of participants had undertaken on-the-job training in game theory *50% of respondents said that new or improved software solutions were desirable *90% of respondents said that they do not have the software they need for their work.

## Project management

Sensible decision-making is critical for the success of projects. In project management, game theory is used to model the decision-making process of players, such as investors, project managers, contractors, sub-contractors, governments and customers. Quite often, these players have competing interests, and sometimes their interests are directly detrimental to other players, making project management scenarios well-suited to be modeled by game theory. Piraveenan (2019) Material was copied from this source, which is available under
in his review provides several examples where game theory is used to model project management scenarios. For instance, an investor typically has several investment options, and each option will likely result in a different project, and thus one of the investment options has to be chosen before the project charter can be produced. Similarly, any large project involving subcontractors, for instance, a construction project, has a complex interplay between the main contractor (the project manager) and subcontractors, or among the subcontractors themselves, which typically has several decision points. For example, if there is an ambiguity in the contract between the contractor and subcontractor, each must decide how hard to push their case without jeopardizing the whole project, and thus their own stake in it. Similarly, when projects from competing organizations are launched, the marketing personnel have to decide what is the best timing and strategy to market the project, or its resultant product or service, so that it can gain maximum traction in the face of competition. In each of these scenarios, the required decisions depend on the decisions of other players who, in some way, have competing interests to the interests of the decision-maker, and thus can ideally be modeled using game theory. Piraveenan summarises that two-player games are predominantly used to model project management scenarios, and based on the identity of these players, five distinct types of games are used in project management. * Government-sector–private-sector games (games that model
public–private partnership A public–private partnership (PPP, 3P, or P3) is a long-term arrangement between a government and private sector institutions.Hodge, G. A and Greve, C. (2007), Public–Private Partnerships: An International Performance Review, Public Administ ...
s) * Contractor–contractor games * Contractor–subcontractor games * Subcontractor–subcontractor games * Games involving other players In terms of types of games, both cooperative as well as non-cooperative, normal-form as well as extensive-form, and zero-sum as well as non-zero-sum are used to model various project management scenarios.

## Political science

The application of game theory to political science is focused in the overlapping areas of fair division,
political economy Political economy is the study of how economic systems (e.g. markets and national economies) and political systems (e.g. law, institutions, government) are linked. Widely studied phenomena within the discipline are systems such as labour ...
, public choice, war bargaining,
positive political theory Positive political theory (PPT) or explanatory political theory is the study of politics using formal methods such as social choice theory, game theory, and statistical analysis. In particular, social choice theoretic methods are often used to d ...
, 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 1957 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 The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...
, 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 sends 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. However, 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 is Peter John Wood's (2013) research looking into what nations could do to help reduce climate change. Wood thought this could be accomplished by making treaties with other nations to reduce
greenhouse gas emissions Greenhouse gas emissions from human activities strengthen the greenhouse effect, contributing to climate change. Most is carbon dioxide from burning fossil fuels: coal, oil, and natural gas. The largest emitters include coal in China and ...
. However, he concluded that this idea could not work because it would create a prisoner's dilemma for the nations.

## Biology

Unlike those in economics, the payoffs for games in
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
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 . 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. 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 (see
Paul Ormerod Paul Andrew Ormerod (born 20 March 1950) is a British economist who is a partner at Volterra Partners consultancy. Additionally, he is a visiting professor at UCL Centre for Decision Making Uncertainty. Research Ormerod has researched complexity ...
's '' Butterfly Economics''). Biologists have used the
game of chicken The game of chicken, also known as the hawk–dove game or snowdrift game, is a model of conflict for two players in game theory. The principle of the game is that while the ideal outcome is for one player to yield (to avoid the worst outcome if ...
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 monkey The vervet monkey (''Chlorocebus pygerythrus''), or simply vervet, is an Old World monkey of the family Cercopithecidae native to Africa. The term "vervet" is also used to refer to all the members of the genus ''Chlorocebus''. The five distinct ...
s 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 Kin selection is the evolutionary strategy that favours the reproductive success of an organism's relatives, even when at a cost to the organism's own survival and reproduction. Kin altruism can look like altruistic behaviour whose evolution ...
explains the evolutionary rationale behind this selection with the equation , where the cost to the altruist must be less than the benefit to the recipient multiplied by the coefficient of relatedness . 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 , because (on average) an individual shares half 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 coefficient that was 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 Game semantics (german: dialogische Logik, translated as ''dialogical logic'') is an approach to formal semantics that grounds the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player ...
. In addition, computer scientists have used games to model
interactive computation In computer science, interactive computation is a mathematical model for computation that involves input/output communication with the external world ''during'' computation. Uses Among the currently studied mathematical models of computation th ...
s. Also, game theory provides a theoretical basis to the field of
multi-agent system A multi-agent system (MAS or "self-organized system") is a computerized system composed of multiple interacting intelligent agents.Hu, J.; Bhowmick, P.; Jang, I.; Arvin, F.; Lanzon, A.,A Decentralized Cluster Formation Containment Framework fo ...
s. Separately, game theory has played a role in
online algorithm In computer science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start. In contrast, an ...
s; in particular, the -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

Game theory has been put to several uses in philosophy. Responding to two papers by , used game theory to develop a philosophical account of convention. In so doing, he provided the first analysis of
common knowledge Common knowledge is knowledge that is publicly known by everyone or nearly everyone, usually with reference to the community in which the knowledge is referenced. Common knowledge can be about a broad range of subjects, such as science, litera ...
and employed it in analyzing play in
coordination game A coordination game is a type of simultaneous game found in game theory. It describes the situation where a player will earn a higher payoff when they select the same course of action as another player. The game is not one of pure conflict, which r ...
s. 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 game-theoretic account of conventions, Edna Ullmann-Margalit (1977) and Bicchieri (2006) have developed theories of
social norms Social norms are shared standards of acceptable behavior by groups. Social norms can both be informal understandings that govern the behavior of members of a society, as well as be codified into rules and laws. Social normative influences or soc ...
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 Thomas Hobbes ( ; 5/15 April 1588 – 4/14 December 1679) was an English philosopher, considered to be one of the founders of modern political philosophy. Hobbes is best known for his 1651 book '' Leviathan'', in which he expounds an infl ...
' project of deriving morality from self-interest. Since games like the
prisoner's dilemma The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...
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 In moral and political philosophy, the social contract is a theory or model that originated during the Age of Enlightenment and usually, although not always, concerns the legitimacy of the authority of the state over the individual. Social ...
view in political philosophy (for examples, see and ). 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., and ).

## Retail and consumer product pricing

Game theory applications are often used in the pricing strategies of retail and consumer markets, particularly for the sale of inelastic goods. With retailers constantly competing against one another for consumer market share, it has become a fairly common practice for retailers to discount certain goods, intermittently, in the hopes of increasing foot-traffic in brick and mortar locations (websites visits for e-commerce retailers) or increasing sales of ancillary or complimentary products. Black Friday, a popular shopping holiday in the US, is when many retailers focus on optimal pricing strategies to capture the holiday shopping market. In the Black Friday scenario, retailers using game theory applications typically ask "what is the dominant competitor's reaction to me?" In such a scenario, the game has two players: the retailer, and the consumer. The retailer is focused on an optimal pricing strategy, while the consumer is focused on the best deal. In this closed system, there often is no dominant strategy as both players have alternative options. That is, retailers can find a different customer, and consumers can shop at a different retailer. Given the market competition that day, however, the dominant strategy for retailers lies in outperforming competitors. The open system assumes multiple retailers selling similar goods, and a finite number of consumers demanding the goods at an optimal price. A blog by a Cornell University professor provided an example of such a strategy, when
Amazon Amazon most often refers to: * Amazons, a tribe of female warriors in Greek mythology * Amazon rainforest, a rainforest covering most of the Amazon basin * Amazon River, in South America * Amazon (company), an American multinational technology c ...
priced a Samsung TV \$100 below retail value, effectively undercutting competitors. Amazon made up part of the difference by increasing the price of HDMI cables, as it has been found that consumers are less price discriminatory when it comes to the sale of secondary items. Retail markets continue to evolve strategies and applications of game theory when it comes to pricing consumer goods. The key insights found between simulations in a controlled environment and real-world retail experiences show that the applications of such strategies are more complex, as each retailer has to find an optimal balance between
pricing Pricing is the process whereby a business sets the price at which it will sell its products and services, and may be part of the business's marketing plan. In setting prices, the business will take into account the price at which it could acqui ...
, supplier relations,
brand image A brand is a name, term, design, symbol or any other feature that distinguishes one seller's good or service from those of other sellers. Brands are used in business, marketing, and advertising for recognition and, importantly, to create a ...
, and the potential to cannibalize the sale of more profitable items.

## Epidemiology

Since the decision to take a vaccine for a particular disease is often made by individuals, who may consider a range of factors and parameters in making this decision (such as the incidence and prevalence of the disease, perceived and real risks associated with contracting the disease, mortality rate, perceived and real risks associated with vaccination, and financial cost of vaccination), game theory has been used to model and predict vaccination uptake in a society.

# 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 biographical film or biopic () is a film that dramatizes the life of a non-fictional or historically-based person or people. Such films show the life of a historical person and the central character's real name is used. They differ from docu ...
'' A Beautiful Mind'', starring
Russell Crowe Russell Ira Crowe (born 7 April 1964) is an actor. He was born in New Zealand, spent ten years of his childhood in Australia, and moved there permanently at age twenty one. He came to international attention for his role as Roman General Max ...
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 Soviet Union 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 Manga ( Japanese: 漫画 ) are comics or graphic novels originating from Japan. Most manga conform to a style developed in Japan in the late 19th century, and the form has a long prehistory in earlier Japanese art. The term ''manga'' is used ...
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. * The 1974 novel '' Spy Story'' by
Len Deighton Leonard Cyril Deighton (; born 18 February 1929) is a British author. His publications have included cookery books, history and military history, but he is best known for his spy novels. After completing his national service in the Royal Air Fo ...
explores elements of Game Theory in regard to cold war army exercises. * The 2008 novel ''
The Dark Forest ''The Dark Forest'' () is a 2008 science fiction novel by the Chinese writer Liu Cixin. It is the sequel to the Hugo Award-winning novel '' The Three-Body Problem'' in the trilogy titled "''Remembrance of Earth's Past''", but Chinese readers g ...
'' by
Liu Cixin Liu Cixin (, pronounced ; born 23 June 1963) is a Chinese science fiction writer. He is a nine-time winner of China's Galaxy Award and has also received the 2015 Hugo Award for his novel '' The Three-Body Problem'' as well as the 2017 Lo ...
explores the relationship between extraterrestrial life, humanity, and game theory. * The prime antagonist Joker in the movie '' The Dark Knight'' presents game theory concepts—notably the
prisoner's dilemma The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defe ...
in a scene where he asks passengers in two different ferries to bomb the other one to save their own. * In the 2018 film ''
Crazy Rich Asians ''Crazy Rich Asians'' is a satirical 2013 romantic comedy novel by Kevin Kwan. Kwan stated that his intention in writing the novel was to "introduce a contemporary Asia to a North American audience". He claimed the novel was loosely based on his ...
'', the female lead Rachel Chu is a professor of economics and game theory at New York University. At the beginning of the film she is seen in her NYU classroom playing a game of
poker Poker is a family of comparing card games in which players wager over which hand is best according to that specific game's rules. It is played worldwide, however in some places the rules may vary. While the earliest known form of the game wa ...
with her teaching assistant and wins the game by bluffing; then in the
climax Climax may refer to: Language arts * Climax (narrative), the point of highest tension in a narrative work * Climax (rhetoric), a figure of speech that lists items in order of importance Biology * Climax community, a biological community ...
of the film, she plays a game of mahjong with her boy friend's disapproving mother Eleanor, losing the game to Eleanor on purpose but winning her approval as a result.

* Applied ethics * Bandwidth-sharing game *
Chainstore paradox The chainstore paradox is an apparent game theory paradox involving the chain store game, where a "deterrence strategy" appears optimal instead of the backward induction strategy of standard game theory reasoning. The chain store game A monopolist ...
*
Collective intentionality In the philosophy of mind, collective intentionality characterizes the intentionality that occurs when two or more individuals undertake a task together. Examples include two individuals carrying a heavy table up a flight of stairs or dancing a ta ...
* Glossary of game theory *
Intra-household bargaining Intra-household bargaining refers to negotiations that occur between members of a household in order to arrive at decisions regarding the household unit, like whether to spend or save, whether to study or work. Bargaining is traditionally defined i ...
*
Kingmaker scenario In game theory, a kingmaker scenario in a game of three or more players, is an endgame situation where a player who is unable to win has the capacity to determine which player among others will win. This player is referred to as the ''kingmaker' ...
*
Law and economics Law and economics, or economic analysis of law, is the application of microeconomic theory to the analysis of law, which emerged primarily from scholars of the Chicago school of economics. Economic concepts are used to explain the effects of laws ...
* Outline of artificial intelligence * Parrondo's paradox *
Precautionary principle The precautionary principle (or precautionary approach) is a broad epistemological, philosophical and legal approach to innovations with potential for causing harm when extensive scientific knowledge on the matter is lacking. It emphasizes caut ...
*
Quantum refereed game Quantum refereed game in quantum information processing is a class of games in the general theory of quantum games. It is played between two players, Alice and Bob, and arbitrated by a referee. The referee outputs the pay-off for the players after ...
* Risk management * Self-confirming equilibrium * Tragedy of the commons *
Wilson doctrine (economics) In economic theory, the Wilson doctrine (or Wilson critique) stipulates that game theory should not rely excessively on common knowledge assumptions. Most prominently, it is interpreted as a request for institutional designs to be "detail-free". Th ...
Lists * List of cognitive biases *
List of emerging technologies This is a list of emerging technologies, in-development technical innovations with significant potential in their applications. The criteria for this list is that the technology must: # Exist in some way; purely hypothetical technologies can ...
* List of games in game theory

# References

* . *
Description

## Historically important texts

* * * * * :*reprinted edition: * * * * Shapley, L.S. (1953), A Value for n-person Games, In: Contributions to the Theory of Games volume II, H. W. Kuhn and A. W. Tucker (eds.) * Shapley, L.S. (1953), Stochastic Games, Proceedings of National Academy of Science Vol. 39, pp. 1095–1100. * English translation: "On the Theory of Games of Strategy," in A. W. Tucker and R. D. Luce, ed. (1959), ''Contributions to the Theory of Games'', v. 4, p
42.
Princeton University Press. * *

## Other material

* * * *
Allan Gibbard Allan may refer to: People * Allan (name), a given name and surname, including list of people and characters with this name * Allan (footballer, born 1984) (Allan Barreto da Silva), Brazilian football striker * Allan (footballer, born 1989) (A ...
, "Manipulation of voting schemes: a general result", ''Econometrica'', Vol. 41, No. 4 (1973), pp. 587–601. * * * , (2002 edition) * . A layman's introduction. * . * * * * * * * * * * *

* James Miller (2015)
Introductory Game Theory Videos
* * Paul Walker

* David Levine
Game Theory. Papers, Lecture Notes and much more stuff.
* Alvin Roth: — Comprehensive list of links to game theory information on the Web * Adam Kalai
Game Theory and Computer Science
— Lecture notes on Game Theory and Computer Science * Mike Shor
GameTheory.net
— Lecture notes, interactive illustrations and other information. * Jim Ratliff'
(lecture notes). * Don Ross
Review Of Game Theory
in the ''Stanford Encyclopedia of Philosophy''. * Bruno Verbeek and Christopher Morris
Game Theory and Ethics
* Elmer G. Wiens
Game Theory
— Introduction, worked examples, play online two-person zero-sum games. * Marek M. Kaminski

— Syllabuses and lecture notes for game theory and political science.

* Kesten Green's — Se
Papers
fo
evidence on the accuracy of forecasts from game theory and other methods
. * McKelvey, Richard D., McLennan, Andrew M., and Turocy, Theodore L. (2007)
Gambit: Software Tools for Game Theory
'. * Benjamin Polak
Open Course on Game Theory at Yalevideos of the course
* Benjamin Moritz, Bernhard Könsgen, Danny Bures, Ronni Wiersch, (2007)
Spieltheorie-Software.de: An application for Game Theory implemented in JAVA
'. * Antonin Kucera
Stochastic Two-Player Games
* Yu-Chi Ho
What is Mathematical Game TheoryWhat is Mathematical Game Theory (#2)What is Mathematical Game Theory (#3)What is Mathematical Game Theory (#4)-Many person game theoryWhat is Mathematical Game Theory ?( #5) – Finale, summing up, and my own view
{{DEFAULTSORT:Game Theory Artificial intelligence Formal sciences Mathematical economics John von Neumann