Game theory is "the study of mathematical models of conflict and
cooperation between intelligent rational decision-makers". Game theory
is mainly used in economics, political science, and psychology, as
well as in logic and computer science. Originally, it addressed
zero-sum games, in which one person's gains result in losses for the
other participants. Today, game theory applies to a wide range of
behavioral relations, and is now an umbrella term for the science of
logical decision making in humans, animals, and computers.
Modern game theory began with the idea regarding the existence of
mixed-strategy equilibria in two-person zero-sum games and its proof
by John von Neumann. Von Neumann's original proof used the Brouwer
fixed-point theorem on continuous mappings into compact convex sets,
which became a standard method in game theory and mathematical
economics. His paper was followed by the 1944 book Theory of Games and
Economic Behavior, co-written with Oskar Morgenstern, which considered
cooperative games of several players. The second edition of this book
provided an axiomatic theory of expected utility, which allowed
mathematical statisticians and economists to treat decision-making
This theory was developed extensively in the 1950s by many scholars.
Game theory was later explicitly applied to biology in the 1970s,
although similar developments go back at least as far as the 1930s.
Game theory has been widely recognized as an important tool in many
fields. With the
Nobel Memorial Prize in Economic Sciences
Nobel Memorial Prize in Economic Sciences going to
Jean Tirole in 2014, eleven game-theorists have now won
the economics Nobel Prize.
John Maynard Smith
John Maynard Smith was awarded the Crafoord
Prize for his application of game theory to biology.
1.1 Prize-winning achievements
2 Game types
Cooperative / Non-cooperative
2.2 Symmetric / Asymmetric
2.3 Zero-sum / Non-zero-sum
2.4 Simultaneous / Sequential
Perfect information and imperfect information
2.6 Combinatorial games
2.7 Infinitely long games
2.8 Discrete and continuous games
2.9 Differential games
2.10 Many-player and population games
2.11 Stochastic outcomes (and relation to other fields)
2.13 Pooling games
2.14 Mean field game theory
3 Representation of games
3.1 Extensive form
3.2 Normal form
3.3 Characteristic function form
4 General and applied uses
4.1 Description and modeling
4.2 Prescriptive or normative analysis
Economics and business
4.4 Political science
Computer science and logic
5 In popular culture
6 See also
8 References and further reading
8.1 Textbooks and general references
8.2 Historically important texts
8.3 Other print references
9 External links
John von Neumann
Early discussions of examples of two-person games occurred long before
the rise of modern, mathematical game theory. The first known
discussion of game theory occurred in a letter written by Charles
Waldegrave, an active Jacobite, and uncle to James Waldegrave, a
British diplomat, in 1713. In this letter, Waldegrave provides a
minimax mixed strategy solution to a two-person version of the card
game le Her, and the problem is now known as Waldegrave problem. James
Madison made what we now recognize as a game-theoretic analysis of the
ways states can be expected to behave under different systems of
taxation. In his 1838 Recherches sur les principes
mathématiques de la théorie des richesses (Researches into the
Mathematical Principles of the Theory of Wealth), Antoine Augustin
Cournot considered a duopoly and presents a solution that is a
restricted version of the Nash equilibrium.
Ernst Zermelo published Über eine Anwendung der Mengenlehre
auf die Theorie des Schachspiels (On an Application of Set Theory to
the Theory of the Game of Chess). It proved that the optimal chess
strategy is strictly determined. This paved the way for more general
In 1938, the Danish mathematical economist
Frederik Zeuthen proved
that the mathematical model had a winning strategy by using Brouwer's
fixed point theorem. In his 1938 book Applications aux Jeux de
Hasard and earlier notes,
Émile Borel proved a minimax theorem for
two-person zero-sum matrix games only when the pay-off matrix was
symmetric. Borel conjectured that non-existence of mixed-strategy
equilibria in two-person zero-sum games would occur, a conjecture that
was proved false.
Game theory did not really exist as a unique field until John von
Neumann published a paper in 1928. Von Neumann's original proof
Brouwer's fixed-point theorem
Brouwer's fixed-point theorem on continuous mappings into compact
convex sets, which became a standard method in game theory and
mathematical economics. His paper was followed by his 1944 book Theory
of Games and Economic Behavior co-authored with Oskar Morgenstern.
The second edition of this book provided an axiomatic theory of
utility, which reincarnated Daniel Bernoulli's old theory of utility
(of the money) as an independent discipline. Von Neumann's work in
game theory culminated in this 1944 book. This foundational work
contains the method for finding mutually consistent solutions for
two-person zero-sum games. During the following time period, work on
game theory was primarily focused on cooperative game theory, which
analyzes optimal strategies for groups of individuals, presuming that
they can enforce agreements between them about proper strategies.
In 1950, the first mathematical discussion of the prisoner's dilemma
appeared, and an experiment was undertaken by notable mathematicians
Merrill M. Flood and Melvin Dresher, as part of the RAND Corporation's
investigations into game theory. RAND pursued the studies because of
possible applications to global nuclear strategy. Around this same
time, John Nash developed a criterion for mutual consistency of
players' strategies, known as Nash equilibrium, applicable to a wider
variety of games than the criterion proposed by von Neumann and
Morgenstern. Nash proved that every n-player, non-zero-sum (not just
2-player zero-sum) non-cooperative game has what is now known as a
Game theory experienced a flurry of activity in the 1950s, during
which time the concepts of the core, the extensive form game,
fictitious play, repeated games, and the
Shapley value were developed.
In addition, the first applications of game theory to philosophy and
political science occurred during this time.
Reinhard Selten introduced his solution concept of subgame
perfect equilibria, which further refined the
Nash equilibrium (later
he would introduce trembling hand perfection as well). In 1994 Nash,
Selten and Harsanyi became
Economics Nobel Laureates for their
contributions to economic game theory.
In the 1970s, game theory was extensively applied in biology, largely
as a result of the work of
John Maynard Smith
John Maynard Smith and his evolutionarily
stable strategy. In addition, the concepts of correlated equilibrium,
trembling hand perfection, and common knowledge were introduced
In 2005, game theorists
Thomas Schelling and
Robert Aumann followed
Nash, Selten and Harsanyi as Nobel Laureates. Schelling worked on
dynamic models, early examples of evolutionary game theory. Aumann
contributed more to the equilibrium school, introducing an equilibrium
coarsening, correlated equilibrium, and developing an extensive formal
analysis of the assumption of common knowledge and of its
In 2007, Leonid Hurwicz, together with
Eric Maskin and Roger Myerson,
was awarded the Nobel Prize in
Economics "for having laid the
foundations of mechanism design theory". Myerson's contributions
include the notion of proper equilibrium, and an important graduate
text: Game Theory, Analysis of Conflict. Hurwicz introduced and
formalized the concept of incentive compatibility.
Alvin E. Roth
Alvin E. Roth and Lloyd S. Shapley were awarded the Nobel
Economics "for the theory of stable allocations and the
practice of market design" and, in 2014, the Nobel went to game
theorist Jean Tirole.
Cooperative / Non-cooperative
Cooperative game and Non-cooperative game
A game is cooperative if the players are able to form binding
commitments externally enforced (e.g. through contract law). A game is
non-cooperative if players cannot form alliances or if all agreements
need to be self-enforcing (e.g. through credible threats).
Cooperative games are often analysed through the framework of
cooperative game theory, which focuses on predicting which coalitions
will form, the joint actions that groups take and the resulting
collective payoffs. It is opposed to the traditional non-cooperative
game theory which focuses on predicting individual players' actions
and payoffs and analyzing Nash equilibria. 
Cooperative game theory provides a high-level approach as it only
describes the structure, strategies and payoffs of coalitions, whereas
non-cooperative game theory also looks at how bargaining procedures
will affect the distribution of payoffs within each coalition. As
non-cooperative game theory is more general, cooperative games can be
analyzed through the approach of non-cooperative game theory (the
converse does not hold) provided that sufficient assumptions are made
to encompass all the possible strategies available to players due to
the possibility of external enforcement of cooperation. While it would
thus be optimal to have all games expressed under a non-cooperative
framework, in many instances insufficient information is available to
accurately model the formal procedures available to the players during
the strategic bargaining process, or the resulting model would be of
too high complexity to offer a practical tool in the real world. In
such cases, cooperative game theory provides a simplified approach
that allows analysis of the game at large without having to make any
assumption about bargaining powers.
Symmetric / Asymmetric
An asymmetric game
Main article: Symmetric game
A symmetric game is a game where the payoffs for playing a particular
strategy depend only on the other strategies employed, not on who is
playing them. If the identities of the players can be changed without
changing the payoff to the strategies, then a game is symmetric. Many
of the commonly studied 2×2 games are symmetric. The standard
representations of chicken, the prisoner's dilemma, and the stag hunt
are all symmetric games. Some[who?] scholars would consider certain
asymmetric games as examples of these games as well. However, the most
common payoffs for each of these games are symmetric.
Most commonly studied asymmetric games are games where there are not
identical strategy sets for both players. For instance, the ultimatum
game and similarly the dictator game have different strategies for
each player. It is possible, however, for a game to have identical
strategies for both players, yet be asymmetric. For example, the game
pictured to the right is asymmetric despite having identical strategy
sets for both players.
Zero-sum / Non-zero-sum
A zero-sum game
Main article: Zero-sum game
Zero-sum games are a special case of constant-sum games, in which
choices by players can neither increase nor decrease the available
resources. In zero-sum games the total benefit to all players in the
game, for every combination of strategies, always adds to zero (more
informally, a player benefits only at the equal expense of
Poker exemplifies a zero-sum game (ignoring the
possibility of the house's cut), because one wins exactly the amount
one's opponents lose. Other zero-sum games include matching pennies
and most classical board games including Go and chess.
Many games studied by game theorists (including the famed prisoner's
dilemma) are non-zero-sum games, because the outcome has net results
greater or less than zero. Informally, in non-zero-sum games, a gain
by one player does not necessarily correspond with a loss by another.
Constant-sum games correspond to activities like theft and gambling,
but not to the fundamental economic situation in which there are
potential gains from trade. It is possible to transform any game into
a (possibly asymmetric) zero-sum game by adding a dummy player (often
called "the board") whose losses compensate the players' net winnings.
Simultaneous / Sequential
Simultaneous game and Sequential game
Simultaneous games are games where both players move simultaneously,
or if they do not move simultaneously, the later players are unaware
of the earlier players' actions (making them effectively
simultaneous). Sequential games (or dynamic games) are games where
later players have some knowledge about earlier actions. This need not
be perfect information about every action of earlier players; it might
be very little knowledge. For instance, a player may know that an
earlier player did not perform one particular action, while he does
not know which of the other available actions the first player
The difference between simultaneous and sequential games is captured
in the different representations discussed above. Often, normal form
is used to represent simultaneous games, while extensive form is used
to represent sequential ones. The transformation of extensive to
normal form is one way, meaning that multiple extensive form games
correspond to the same normal form. Consequently, notions of
equilibrium for simultaneous games are insufficient for reasoning
about sequential games; see subgame perfection.
In short, the differences between sequential and simultaneous games
are as follows:
Normally denoted by
of opponent's move?
Also known as
Perfect information and imperfect information
Main article: Perfect information
A game of imperfect information (the dotted line represents ignorance
on the part of player 2, formally called an information set)
An important subset of sequential games consists of games of perfect
information. A game is one of perfect information if all players know
the moves previously made by all other players. Most games studied in
game theory are imperfect-information games. Examples
of perfect-information games include tic-tac-toe, checkers, infinite
chess, and Go.
Many card games are games of imperfect information, such as poker and
Perfect information is often confused with complete
information, which is a similar concept. Complete
information requires that every player know the strategies and payoffs
available to the other players but not necessarily the actions taken.
Games of incomplete information can be reduced, however, to games of
imperfect information by introducing "moves by nature".
Games in which the difficulty of finding an optimal strategy stems
from the multiplicity of possible moves are called combinatorial
games. Examples include chess and go. Games that involve imperfect
information may also have a strong combinatorial character, for
instance backgammon. There is no unified theory addressing
combinatorial elements in games. There are, however, mathematical
tools that can solve particular problems and answer general
Games of perfect information have been studied in combinatorial game
theory, which has developed novel representations, e.g. surreal
numbers, as well as combinatorial and algebraic (and sometimes
non-constructive) proof methods to solve games of certain types,
including "loopy" games that may result in infinitely long sequences
of moves. These methods address games with higher combinatorial
complexity than those usually considered in traditional (or
"economic") game theory. A typical game that has been solved
this way is hex. A related field of study, drawing from computational
complexity theory, is game complexity, which is concerned with
estimating the computational difficulty of finding optimal
Research in artificial intelligence has addressed both perfect and
imperfect information games that have very complex combinatorial
structures (like chess, go, or backgammon) for which no provable
optimal strategies have been found. The practical solutions involve
computational heuristics, like alpha-beta pruning or use of artificial
neural networks trained by reinforcement learning, which make games
more tractable in computing practice.
Infinitely long games
Main article: Determinacy
Games, as studied by economists and real-world game players, are
generally finished in finitely many moves. Pure mathematicians are not
so constrained, and set theorists in particular study games that last
for infinitely many moves, with the winner (or other payoff) not known
until after all those moves are completed.
The focus of attention is usually not so much on the best way to play
such a game, but whether one player has a winning strategy. (It can be
proven, using the axiom of choice, that there are games – even
with perfect information and where the only outcomes are "win" or
"lose" – for which neither player has a winning strategy.) The
existence of such strategies, for cleverly designed games, has
important consequences in descriptive set theory.
Discrete and continuous games
Much of game theory is concerned with finite, discrete games, that
have a finite number of players, moves, events, outcomes, etc. Many
concepts can be extended, however. Continuous games allow players to
choose a strategy from a continuous strategy set. For instance,
Cournot competition is typically modeled with players' strategies
being any non-negative quantities, including fractional quantities.
Differential games such as the continuous pursuit and evasion game are
continuous games where the evolution of the players' state variables
is governed by differential equations. The problem of finding an
optimal strategy in a differential game is closely related to the
optimal control theory. In particular, there are two types of
strategies: the open-loop strategies are found using the Pontryagin
maximum principle while the closed-loop strategies are found using
Bellman's Dynamic Programming method.
A particular case of differential games are the games with a random
time horizon. In such games, the terminal time is a random
variable with a given probability distribution function. Therefore,
the players maximize the mathematical expectation of the cost
function. It was shown that the modified optimization problem can be
reformulated as a discounted differential game over an infinite time
Many-player and population games
Games with an arbitrary, but finite, number of players are often
called n-person games.
Evolutionary game theory
Evolutionary game theory considers games
involving a population of decision makers, where the frequency with
which a particular decision is made can change over time in response
to the decisions made by all individuals in the population. In
biology, this is intended to model (biological) evolution, where
genetically programmed organisms pass along some of their strategy
programming to their offspring. In economics, the same theory is
intended to capture population changes because people play the game
many times within their lifetime, and consciously (and perhaps
rationally) switch strategies.
Stochastic outcomes (and relation to other fields)
Individual decision problems with stochastic outcomes are sometimes
considered "one-player games". These situations are not considered
game theoretical by some authors.[by whom?] They may be modeled using
similar tools within the related disciplines of decision theory,
operations research, and areas of artificial intelligence,
AI planning (with uncertainty) and multi-agent system.
Although these fields may have different motivators, the mathematics
involved are substantially the same, e.g. using Markov decision
processes (MDP).
Stochastic outcomes can also be modeled in terms of game theory by
adding a randomly acting player who makes "chance moves" ("moves by
nature"). This player is not typically considered a third player
in what is otherwise a two-player game, but merely serves to provide a
roll of the dice where required by the game.
For some problems, different approaches to modeling stochastic
outcomes may lead to different solutions. For example, the difference
in approach between MDPs and the minimax solution is that the latter
considers the worst-case over a set of adversarial moves, rather than
reasoning in expectation about these moves given a fixed probability
distribution. The minimax approach may be advantageous where
stochastic models of uncertainty are not available, but may also be
overestimating extremely unlikely (but costly) events, dramatically
swaying the strategy in such scenarios if it is assumed that an
adversary can force such an event to happen. (See Black swan
theory for more discussion on this kind of modeling issue,
particularly as it relates to predicting and limiting losses in
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.
These are games the play of which is the development of the rules for
another game, the target or subject game. Metagames seek to maximize
the utility value of the rule set developed. The theory of metagames
is related to mechanism design theory.
The term metagame analysis is also used to refer to a practical
approach developed by Nigel Howard. whereby a situation is framed
as a strategic game in which stakeholders try to realise their
objectives by means of the options available to them. Subsequent
developments have led to the formulation of confrontation analysis.
These are games prevailing over all forms of society. Pooling games
are repeated plays with changing payoff table in general over an
experienced path and their equilibrium strategies usually take a form
of evolutionary social convention and economic convention. Pooling
game theory emerges to formally recognize the interaction between
optimal choice in one play and the emergence of forthcoming payoff
table update path, identify the invariance existence and robustness,
and predict variance over time. The theory is based upon topological
transformation classification of payoff table update over time to
predict variance and invariance, and is also within the jurisdiction
of the computational law of reachable optimality for ordered
Mean field game theory
Mean field game theory is the study of strategic decision making in
very large populations of small interacting agents. This class of
problems was considered in the economics literature by Boyan Jovanovic
and Robert W. Rosenthal, in the engineering literature by Peter E.
Caines and by mathematician
Pierre-Louis Lions and Jean-Michel Lasry.
Representation of games
See also: List of games in game theory
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
Main article: Extensive form game
An extensive form game
The extensive form can be used to formalize games with a time
sequencing of moves. Games here are played on trees (as pictured
here). Here each vertex (or node) represents a point of choice for a
player. The player is specified by a number listed by the vertex. The
lines out of the vertex represent a possible action for that player.
The payoffs are specified at the bottom of the tree. The extensive
form can be viewed as a multi-player generalization of a decision
tree. To solve any extensive form game, backward induction must be
used. It involves working backwards up the game tree to determine what
a rational player would do at the last vertex of the tree, what the
player with the previous move would do given that the player with the
last move is rational, and so on until the first vertex of the tree is
The game pictured consists of two players. The way this particular
game is structured (i.e., with sequential decision making and perfect
information), Player 1 "moves" first by choosing either F or U (Fair
or Unfair). Next in the sequence, Player 2, who has now seen Player
1's move, chooses to play either A or R. Once Player 2 has made his/
her choice, the game is considered finished and each player gets their
respective payoff. Suppose that Player 1 chooses U and then Player 2
chooses A: Player 1 then gets a payoff of "eight" (which in real-world
terms can be interpreted in many ways, the simplest of which is in
terms of money but could mean things such as eight days of vacation or
eight countries conquered or even eight more opportunities to play the
same game against other players) and Player 2 gets a payoff of "two".
The extensive form can also capture simultaneous-move games and games
with imperfect information. To represent it, either a dotted line
connects different vertices to represent them as being part of the
same information set (i.e. the players do not know at which point they
are), or a closed line is drawn around them. (See example in the
imperfect information section.)
Normal form or payoff matrix of a 2-player, 2-strategy game
Main article: Normal-form game
The normal (or strategic form) game is usually represented by a matrix
which shows the players, strategies, and payoffs (see the example to
the right). More generally it can be represented by any function that
associates a payoff for each player with every possible combination of
actions. In the accompanying example there are two players; one
chooses the row and the other chooses the column. Each player has two
strategies, which are specified by the number of rows and the number
of columns. The payoffs are provided in the interior. The first number
is the payoff received by the row player (Player 1 in our example);
the second is the payoff for the column player (Player 2 in our
example). Suppose that Player 1 plays Up and that Player 2 plays Left.
Then Player 1 gets a payoff of 4, and Player 2 gets 3.
When a game is presented in normal form, it is presumed that each
player acts simultaneously or, at least, without knowing the actions
of the other. If players have some information about the choices of
other players, the game is usually presented in extensive form.
Every extensive-form game has an equivalent normal-form game, however
the transformation to normal form may result in an exponential blowup
in the size of the representation, making it computationally
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
John von Neumann and Oskar
Morgenstern's book; when looking at these instances, they guessed that
when a union
displaystyle mathbf C
appears, it works against the fraction
displaystyle left( frac mathbf N mathbf C right)
as if two individuals were playing a normal game. The balanced payoff
of C is a basic function. Although there are differing examples that
help determine coalitional amounts from normal games, not all appear
that in their function form can be derived from such.
Formally, a characteristic function is seen as: (N,v), where N
represents the group of people and
displaystyle v:2^ N to mathbf R
is a normal utility.
Such characteristic functions have expanded to describe games where
there is no removable utility.
General and applied uses
As a method of applied mathematics, game theory has been used to study
a wide variety of human and animal behaviors. It was initially
developed in economics to understand a large collection of economic
behaviors, including behaviors of firms, markets, and consumers. The
first use of game-theoretic analysis was by Antoine Augustin Cournot
in 1838 with his solution of the Cournot duopoly. The use of game
theory in the social sciences has expanded, and game theory has been
applied to political, sociological, and psychological behaviors as
Although pre-twentieth century naturalists such as
Charles Darwin made
game-theoretic kinds of statements, the use of game-theoretic analysis
in biology began with Ronald Fisher's studies of animal behavior
during the 1930s. This work predates the name "game theory", but it
shares many important features with this field. The developments in
economics were later applied to biology largely by John Maynard Smith
in his book
Evolution and the Theory of Games.
In addition to being used to describe, predict, and explain behavior,
game theory has also been used to develop theories of ethical or
normative behavior and to prescribe such behavior. In economics
and philosophy, scholars have applied game theory to help in the
understanding of good or proper behavior. Game-theoretic arguments of
this type can be found as far back as Plato.
Description and modeling
A four-stage centipede game
The primary use of game theory is to describe and model how human
populations behave. Some[who?] scholars believe that by finding the
equilibria of games they can predict how actual human populations will
behave when confronted with situations analogous to the game being
studied. This particular view of game theory has been criticized. It
is argued that the assumptions made by game theorists are often
violated when applied to real world situations. Game theorists usually
assume players act rationally, but in practice, human behavior often
deviates from this model. Game theorists respond by comparing their
assumptions to those used in physics. Thus while their assumptions do
not always hold, they can treat game theory as a reasonable scientific
ideal akin to the models used by physicists. However, empirical work
has shown that in some classic games, such as the centipede game,
guess 2/3 of the average game, and the dictator game, people regularly
do not play Nash equilibria. There is an ongoing debate regarding the
importance of these experiments and whether the analysis of the
experiments fully captures all aspects of the relevant situation.
Some game theorists, following the work of
John Maynard Smith
John Maynard Smith and
George R. Price, have turned to evolutionary game theory in order to
resolve these issues. These models presume either no rationality or
bounded rationality on the part of players. Despite the name,
evolutionary game theory does not necessarily presume natural
selection in the biological sense.
Evolutionary game theory
Evolutionary game theory includes
both biological as well as cultural evolution and also models of
individual learning (for example, fictitious play dynamics).
Prescriptive or normative analysis
The Prisoner's Dilemma
Some scholars, like Leonard Savage, see game theory
not as a predictive tool for the behavior of human beings, but as a
suggestion for how people ought to behave. Since a strategy,
corresponding to a
Nash equilibrium of a game constitutes one's best
response to the actions of the other players – provided they are in
Nash equilibrium – playing a strategy that is part of a
Nash equilibrium seems appropriate. This normative use of game theory
has also come under criticism.
Economics and business
Game theory is a major method used in mathematical economics and
business for modeling competing behaviors of interacting agents.
Applications include a wide array of economic phenomena and
approaches, such as auctions, bargaining, mergers & acquisitions
pricing, fair division, duopolies, oligopolies, social network
formation, agent-based computational economics, general
equilibrium, mechanism design, and voting systems; and across
such broad areas as experimental economics, behavioral
economics, information economics, industrial organization,
and political economy.
This research usually focuses on particular sets of strategies known
as "solution concepts" or "equilibria". A common assumption is that
players act rationally. In non-cooperative games, the most famous of
these is the Nash equilibrium. A set of strategies is a Nash
equilibrium if each represents a best response to the other
strategies. If all the players are playing the strategies in a Nash
equilibrium, they have no unilateral incentive to deviate, since their
strategy is the best they can do given what others are doing.
The payoffs of the game are generally taken to represent the utility
of individual players.
A prototypical paper on game theory in economics begins by presenting
a game that is an abstraction of a particular economic situation. One
or more solution concepts are chosen, and the author demonstrates
which strategy sets in the presented game are equilibria of the
appropriate type. Naturally one might wonder to what use this
information should be put. Economists and business professors suggest
two primary uses (noted above): descriptive and prescriptive.
The application of game theory to political science is focused in the
overlapping areas of fair division, political economy, public choice,
war bargaining, positive political theory, and social choice theory.
In each of these areas, researchers have developed game-theoretic
models in which the players are often voters, states, special interest
groups, and politicians.
Early examples of game theory applied to political science are
provided by Anthony Downs. In his book An Economic Theory of
Democracy, he applies the Hotelling firm location model to the
political process. In the Downsian model, political candidates commit
to ideologies on a one-dimensional policy space. Downs first shows how
the political candidates will converge to the ideology preferred by
the median voter if voters are fully informed, but then argues that
voters choose to remain rationally ignorant which allows for candidate
divergence. Game Theory was applied in 1962 to the Cuban missile
crisis during the presidency of John F. Kennedy.
It has also been proposed that game theory explains the stability of
any form of political government. Taking the simplest case of a
monarchy, for example, the king, being only one person, does not and
cannot maintain his authority by personally exercising physical
control over all or even any significant number of his subjects.
Sovereign control is instead explained by the recognition by each
citizen that all other citizens expect each other to view the king (or
other established government) as the person whose orders will be
followed. Coordinating communication among citizens to replace the
sovereign is effectively barred, since conspiracy to replace the
sovereign is generally punishable as a crime. Thus, in a process that
can be modeled by variants of the prisoner's dilemma, during periods
of stability no citizen will find it rational to move to replace the
sovereign, even if all the citizens know they would be better off if
they were all to act collectively.
A game-theoretic explanation for democratic peace is that public and
open debate in democracies send clear and reliable information
regarding their intentions to other states. In contrast, it is
difficult to know the intentions of nondemocratic leaders, what effect
concessions will have, and if promises will be kept. Thus there will
be mistrust and unwillingness to make concessions if at least one of
the parties in a dispute is a non-democracy.
On the other hand, game theory predicts that two countries may still
go to war even if their leaders are cognizant of the costs of
fighting. War may result from asymmetric information; two countries
may have incentives to mis-represent the amount of military resources
they have on hand, rendering them unable to settle disputes agreeably
without resorting to fighting. Moreover, war may arise because of
commitment problems: if two countries wish to settle a dispute via
peaceful means, but each wishes to go back on the terms of that
settlement, they may have no choice but to resort to warfare. Finally,
war may result from issue indivisibilities.
Game theory could also help predict a nation's responses when there is
a new rule or law to be applied to that nation. One example would be
Peter John Wood's (2013) research when he looked into what nations
could do to help reduce climate change. Wood thought this could be
accomplished by making treaties with other nations to reduce green
house gas emissions. However, he concluded that this idea could not
work because it would create a prisoner's dilemma to the nations.
The hawk-dove game
Main article: Evolutionary game theory
Unlike those in economics, the payoffs for games in biology are often
interpreted as corresponding to fitness. In addition, the focus has
been less on equilibria that correspond to a notion of rationality and
more on ones that would be maintained by evolutionary forces. The best
known equilibrium in biology is known as the evolutionarily stable
strategy (ESS), first introduced in (Smith & Price 1973). Although
its initial motivation did not involve any of the mental requirements
of the Nash equilibrium, every ESS is a Nash equilibrium.
In biology, game theory has been used as a model to understand many
different phenomena. It was first used to explain the evolution (and
stability) of the approximate 1:1 sex ratios. (Fisher 1930) suggested
that the 1:1 sex ratios are a result of evolutionary forces acting on
individuals who could be seen as trying to maximize their number of
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's
Biologists have used the game of chicken to analyze fighting behavior
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
Evolutionary game theory has been
used to explain many seemingly incongruous phenomena in nature.
One such phenomenon is known as biological altruism. This is a
situation in which an organism appears to act in a way that benefits
other organisms and is detrimental to itself. This is distinct from
traditional notions of altruism because such actions are not
conscious, but appear to be evolutionary adaptations to increase
overall fitness. Examples can be found in species ranging from vampire
bats that regurgitate blood they have obtained from a night's hunting
and give it to group members who have failed to feed, to worker bees
that care for the queen bee for their entire lives and never mate, to
vervet monkeys that warn group members of a predator's approach, even
when it endangers that individual's chance of survival. All of
these actions increase the overall fitness of a group, but occur at a
cost to the individual.
Evolutionary game theory
Evolutionary game theory explains this altruism with the idea of kin
selection. Altruists discriminate between the individuals they help
and favor relatives.
Hamilton's rule explains the evolutionary
rationale behind this selection with the equation c<b*r where the
cost (c) to the altruist must be less than the benefit (b) to the
recipient multiplied by the coefficient of relatedness (r). The more
closely related two organisms are causes the incidences of altruism to
increase because they share many of the same alleles. This means that
the altruistic individual, by ensuring that the alleles of its close
relative are passed on, (through survival of its offspring) can forgo
the option of having offspring itself because the same number of
alleles are passed on. Helping a sibling for example (in diploid
animals), has a coefficient of ½, because (on average) an individual
shares ½ of the alleles in its sibling's offspring. Ensuring that
enough of a sibling’s offspring survive to adulthood precludes the
necessity of the altruistic individual producing offspring. The
coefficient values depend heavily on the scope of the playing field;
for example if the choice of whom to favor includes all genetic living
things, not just all relatives, we assume the discrepancy between all
humans only accounts for approximately 1% of the diversity in the
playing field, a co-efficient that was ½ in the smaller field becomes
0.995. Similarly if it is considered that information other than that
of a genetic nature (e.g. epigenetics, religion, science, etc.)
persisted through time the playing field becomes larger still, and the
Computer science and logic
Game theory has come to play an increasingly important role in logic
and in computer science. Several logical theories have a basis in game
semantics. In addition, computer scientists have used games to model
interactive computations. Also, game theory provides a theoretical
basis to the field of multi-agent systems.
Separately, game theory has played a role in online algorithms; in
particular, the k-server problem, which has in the past been referred
to as games with moving costs and request-answer games. Yao's
principle is a game-theoretic technique for proving lower bounds on
the computational complexity of randomized algorithms, especially
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.
Game theory has been put to several uses in philosophy. Responding to
two papers by W.V.O. Quine (1960, 1967), Lewis (1969) used game
theory to develop a philosophical account of convention. In so doing,
he provided the first analysis of common knowledge and employed it in
analyzing play in coordination games. In addition, he first suggested
that one can understand meaning in terms of signaling games. This
later suggestion has been pursued by several philosophers since
Lewis. Following Lewis (1969) game-theoretic account of
conventions, Edna Ullmann-Margalit (1977) and Bicchieri (2006) have
developed theories of social norms that define them as Nash equilibria
that result from transforming a mixed-motive game into a coordination
Game 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 agents' interactions.
Philosophers who have worked in this area include Bicchieri (1989,
1993), Skyrms (1990), and Stalnaker (1999).
In ethics, some[who?] authors have attempted to pursue Thomas Hobbes'
project of deriving morality from self-interest. Since games like the
prisoner's dilemma present an apparent conflict between morality and
self-interest, explaining why cooperation is required by self-interest
is an important component of this project. This general strategy is a
component of the general social contract view in political philosophy
(for examples, see Gauthier (1986) and Kavka (1986)).
Other authors have attempted to use evolutionary game theory in order
to explain the emergence of human attitudes about morality and
corresponding animal behaviors. These authors look at several games
including the prisoner's dilemma, stag hunt, and the Nash bargaining
game as providing an explanation for the emergence of attitudes about
morality (see, e.g., Skyrms (1996, 2004) and Sober and
In popular culture
Based on the book by Sylvia Nasar, the life story of game theorist
and mathematician John Nash was turned into the biopic A Beautiful
Mind starring Russell Crowe.
"Games theory" and "theory of games" are mentioned in the military
science fiction novel
Starship Troopers by Robert A. Heinlein. In
the 1997 film of the same name, the character Carl Jenkins refers to
his assignment to military intelligence as to "games and theory".
Dr. Strangelove satirizes game theoretic ideas about
deterrence theory. For example, nuclear deterrence depends on the
threat to retaliate catastrophically if a nuclear attack is detected.
A game theorist might argue that such threats can fail to be credible,
in the sense that they can lead to subgame imperfect equilibria. The
movie takes this idea one step further, with the Russians irrevocably
committing to a catastrophic nuclear response without making the
threat public.
Liar Game is a popular Japanese manga, television program and movie,
where each episode presents the main characters with a Game Theory
type game. The show's supporting characters reflect and explore game
theory's predictions around self-preservation strategies used in each
challenge. The main character however, who is portrayed as an
innocent, naive and good hearted young lady Kansaki Nao, always
attempts to convince the other players to follow a mutually beneficial
strategy where everybody wins. Kansaki Nao's seemingly simple
strategies that appear to be the product of her innocent good nature
actually represent optimal equilibrium solutions which Game Theory
attempts to solve. Other players however, usually use her naivety
against her to follow strategies that serve self-preservation. The
show improvises heavily on Game Theory predictions and strategies to
provide each episode's script, the players decisions. In a sense, each
episode exhibits a Game Theory game and the strategies/ equilibria/
solutions provide the script which is coloured in by the
Combinatorial game theory
Glossary of game theory
Quantum game theory
Quantum refereed game
Reverse game theory
Tragedy of the commons
Law and economics
List of cognitive biases
List of emerging technologies
List of games in game theory
Outline of artificial intelligence
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References and further reading
Wikiquote has quotations related to: Game theory
Wikimedia Commons has media related to Game theory.
Textbooks and general references
Aumann, Robert J (1987), "game theory", The New Palgrave: A Dictionary
of Economics, 2, pp. 460–82 .
Camerer, Colin (2003), "Introduction", Behavioral Game Theory:
Experiments in Strategic Interaction, Russell Sage Foundation,
pp. 1–25, ISBN 978-0-691-09039-9 , Description.
Dutta, Prajit K. (1999), Strategies and games: theory and practice,
MIT Press, ISBN 978-0-262-04169-0 . Suitable for
undergraduate and business students.
Fernandez, L F.; Bierman, H S. (1998),
Game theory with economic
applications, Addison-Wesley, ISBN 978-0-201-84758-1 .
Suitable for upper-level undergraduates.
Gibbons, Robert D. (1992),
Game theory for applied economists,
Princeton University Press, ISBN 978-0-691-00395-5 .
Suitable for advanced undergraduates.
Published in Europe as Gibbons, Robert (2001), A Primer in Game
Theory, London: Harvester Wheatsheaf,
ISBN 978-0-7450-1159-2 .
Gintis, Herbert (2000),
Game theory evolving: a problem-centered
introduction to modeling strategic behavior, Princeton University
Press, ISBN 978-0-691-00943-8
Green, Jerry R.; Mas-Colell, Andreu; Whinston, Michael D. (1995),
Microeconomic theory, Oxford University Press,
ISBN 978-0-19-507340-9 . Presents game theory in formal way
suitable for graduate level.
Joseph E. Harrington (2008) Games, strategies, and decision making,
Worth, ISBN 0-7167-6630-2. Textbook suitable for undergraduates
in applied fields; numerous examples, fewer formalisms in concept
Howard, Nigel (1971), Paradoxes of Rationality: Games, Metagames, and
Political Behavior, Cambridge, MA: The MIT Press,
Isaacs, Rufus (1999), Differential Games: A Mathematical Theory With
Applications to Warfare and Pursuit, Control and Optimization, New
York: Dover Publications, ISBN 978-0-486-40682-4
Miller, James H. (2003),
Game theory at work: how to use game theory
to outthink and outmaneuver your competition, New York: McGraw-Hill,
ISBN 978-0-07-140020-6 . Suitable for a general audience.
Osborne, Martin J. (2004), An introduction to game theory, Oxford
University Press, ISBN 978-0-19-512895-6 . Undergraduate
Osborne, Martin J.; Rubinstein, Ariel (1994), A course in game theory,
MIT Press, ISBN 978-0-262-65040-3 . A modern introduction at
the graduate level.
Shoham, Yoav; Leyton-Brown, Kevin (2009), Multiagent Systems:
Algorithmic, Game-Theoretic, and Logical Foundations, New York:
Cambridge University Press, ISBN 978-0-521-89943-7, retrieved 8
Roger McCain's Game Theory: A Nontechnical Introduction to the
Analysis of Strategy[permanent dead link] (Revised Edition)
Webb, James N. (2007), Game theory: decisions, interaction and
evolution, Undergraduate mathematics, Springer,
ISBN 1-84628-423-6 Consistent treatment of game types
usually claimed by different applied fields, e.g. Markov decision
Historically important texts
Aumann, R.J. and Shapley, L.S. (1974), Values of Non-Atomic Games,
Princeton University Press
Cournot, A. Augustin (1838), "Recherches sur les principles
mathematiques de la théorie des richesses", Libraire des sciences
politiques et sociales, Paris: M. Rivière & C.ie
Edgeworth, Francis Y. (1881), Mathematical Psychics, London: Kegan
Farquharson, Robin (1969), Theory of Voting, Blackwell (Yale U.P. in
the U.S.), ISBN 0-631-12460-8
Luce, R. Duncan; Raiffa, Howard (1957), Games and decisions:
introduction and critical survey, New York: Wiley
reprinted edition: R. Duncan Luce ;
Howard Raiffa (1989), Games
and decisions: introduction and critical survey, New York: Dover
Publications, ISBN 978-0-486-65943-5
Maynard Smith, John (1982),
Evolution and the theory of games,
Cambridge University Press, ISBN 978-0-521-28884-2
Maynard Smith, John; Price, George R. (1973), "The logic of animal
conflict", Nature, 246 (5427): 15–18, Bibcode:1973Natur.246...15S,
Nash, John (1950), "Equilibrium points in n-person games", Proceedings
of the National Academy of Sciences of the United States of America,
36 (1): 48–49, Bibcode:1950PNAS...36...48N,
doi:10.1073/pnas.36.1.48, PMC 1063129 ,
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
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Look up game theory in Wiktionary, the free dictionary.
Wikiversity has learning resources about Game Theory
Wikibooks has a book on the topic of: Introduction to Game Theory
James Miller (2015): Introductory Game Theory Videos.
Hazewinkel, Michiel, ed. (2001) , "Games, theory of",
Encyclopedia of Mathematics, Springer Science+Business Media B.V. /
Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Paul Walker: History of Game Theory Page.
David Levine: Game Theory. Papers, Lecture Notes and much more stuff.
Alvin Roth:"Game Theory and Experimental
Economics page". Archived
from the original on 15 August 2000. Retrieved 13 September
2003. — 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: Game Theory .net — Lecture notes, interactive
illustrations and other information.
Jim Ratliff's Graduate Course in Game Theory (lecture notes).
Don Ross: Review Of Game Theory in the Stanford Encyclopedia of
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: Game Theory and Politics — Syllabuses and lecture
notes for game theory and political science.
Web sites on game theory and social interactions
Kesten Green's Conflict Forecasting at the
Wayback Machine (archived
11 April 2011) — See Papers for 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 Yale videos of the
Benjamin Moritz, Bernhard Könsgen, Danny Bures, Ronni Wiersch, (2007)
Spieltheorie-Software.de: An application for Game Theory implemented
Antonin Kucera: Stochastic Two-Player Games.
Yu-Chi Ho: What is Mathematical Game Theory; What is Mathematical Game
Theory (#2); What is Mathematical Game Theory (#3); What is
Mathematical Game Theory (#4)-Many person game theory; What is
Mathematical Game Theory ?( #5) – Finale, summing up, and my
Topics in game theory
Escalation of commitment
First-player and second-player win
Hierarchy of beliefs
Simultaneous action selection
Bayesian Nash equilibrium
Perfect Bayesian equilibrium
Evolutionarily stable strategy
Quantal response equilibrium
Strong Nash equilibrium
Markov perfect equilibrium
Tit for tat
Large Poisson game
Strictly determined game
Optional prisoner's dilemma
Battle of the sexes
Public goods game
War of attrition
El Farol Bar problem
Guess 2/3 of the average
Nash bargaining game
Prisoners and hats puzzle
Princess and Monster game
Arrow's impossibility theorem
Albert W. Tucker
David K. Levine
David M. Kreps
Donald B. Gillies
Harold W. Kuhn
John Maynard Smith
Antoine Augustin Cournot
John von Neumann
Merrill M. Flood
Robert B. Wilson
Combinatorial game theory
First-move advantage in chess
Glossary of game theory
List of game theorists
List of games in game theory
Tragedy of the commons
Tyranny of small decisions
Areas of mathematics
History of mathematics
Philosophy of mathematics
Philosophy of mathematics education
Differential equations / Dynamical systems
Mathematics and art
Marital property (USA)
Estate in land
Bundle of rights
Common good (economics)
Free rider problem
Labor theory of property
Law of rent
Right to property
Tragedy of the commons
Ejido (agrarian land)
restraint on alienation
Freedom to roam
Right of way
Two Treatises of Government
John Stuart Mill
The Great Transformation
What Is Property?
Murray N. Rothbard
Ethics of Liberty
The Social Contract
The Wealth of Nations