Game theory is a study of strategic decision making. More
formally, it is "the study of mathematical models of conflict and
cooperation between intelligent rational decision-makers."[1] An
alternative term suggested "as a more descriptive name for the discipline"
is interactive decision theory.[2] Game theory is mainly used in economics,
political science, and psychology, as well as logic and biology. The subject
first addressed zero-sum games, such that one person's gains exactly equal net
losses of the other participant(s). Today, however, game theory applies to a
wide range of class relations, and has developed into an umbrella term for the
logical side of science, to include both human and non-humans, like computers.
Classic uses include a sense of balance in numerous games, where each person
has found or developed a tactic that cannot successfully better his results,
given the other approach.
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 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, with Oskar
Morgenstern, which considered cooperative games of several players. The second
edition of this book provided an axiomatic theory of expected utility, which
allowed mathematical statisticians and economists to treat decision-making
under uncertainty.
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. Eight
game-theorists have won the Nobel Memorial Prize in Economic Sciences, and John
Maynard Smith was awarded the Crafoord Prize for his application of game theory
to biology.
Contents [hide]
1 Representation of games
1.1 Extensive form
1.2 Normal form
1.3 Characteristic function form
1.4 Partition function form
2 General and applied uses
2.1 Description and modeling
2.2 Prescriptive or normative analysis
2.3 Economics and business
2.4 Political science
2.5 Biology
2.6 Computer science and logic
2.7 Philosophy
3 Types of games
3.1 Cooperative or non-cooperative
3.2 Symmetric and asymmetric
3.3 Zero-sum and non-zero-sum
3.4 Simultaneous and sequential
3.5 Perfect information and imperfect information
3.6 Combinatorial games
3.7 Infinitely long games
3.8 Discrete and continuous games
3.9 Differential games
3.10 Many-player and population games
3.11 Stochastic outcomes (and relation to other fields)
3.12 Metagames
4 History
5 Popular culture
6 See also
7 Notes
8 References and further reading
8.1 Textbooks and general references
8.2 Historically important texts
8.3 Other print references
8.4 Websites
[edit]Representation of games
See also: List of games in game theory
The games studied in game theory are well-defined
mathematical objects. A game consists of a set of players, a set of moves (or
strategies) available to those players, and a specification of payoffs for each
combination of strategies. Most cooperative games are presented in the
characteristic function form, while the extensive and the normal forms are used
to define noncooperative games.
[edit]Extensive form
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 to the
left). 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. (Fudenberg & Tirole 1991, p. 67)
In the game pictured to the left, there are two players.
Player 1 moves first and chooses either F or U. Player 2 sees Player 1's move
and then chooses A or R. Suppose that Player 1 chooses U and then Player 2
chooses A, then Player 1 gets 8 and Player 2 gets 2.
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.)
[edit]Normal form
Player 2
chooses Left Player
2
chooses Right
Player 1
chooses Up 4, 3 –1, –1
Player 1
chooses Down 0, 0 3, 4
Normal form or payoff matrix of a 2-player, 2-strategy game
Main article: Normal-form game
The normal (or strategic form) game is usually represented
by a matrix which shows the players, strategies, and pay-offs (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. (Leyton-Brown & Shoham 2008, p. 35)
[edit]Characteristic function form
Main article: Cooperative game
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 appears, it works
against the fraction 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 is
a normal utility.
Such characteristic functions have expanded to describe
games where there is no removable utility.
[edit]Partition function form
The characteristic function form ignores the possible
externalities of coalition formation. In the partition function form the payoff
of a coalition depends not only on its members, but also on the way the rest of
the players are partitioned (Thrall & Lucas 1963).
[edit]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 use of game theory in
the social sciences has expanded, and game theory has been applied to
political, sociological, and psychological behaviors as well.
Game-theoretic analysis was initially used to study animal
behavior by Ronald Fisher in the 1930s (although even Charles Darwin makes a
few informal game-theoretic statements). 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.[3] 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.[4]
[edit]Description and modeling
A three stage Centipede Game
The first known use 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 come under recent criticism. First, it is criticized
because the assumptions made by game theorists are often violated. Game
theorists may assume players always act in a way to directly maximize their
wins (the Homo economicus model), but in practice, human behavior often
deviates from this model. Explanations of this phenomenon are many;
irrationality, new models of deliberation, or even different motives (like that
of altruism). 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, in the centipede game, guess 2/3 of the average game, and
the dictator game, people regularly do not play Nash equilibria. These
experiments have demonstrated that individuals do not play equilibrium
strategies. There is an ongoing debate regarding the importance of these
experiments.[5]
Alternatively, some[who?] authors claim that Nash equilibria
do not provide predictions for human populations, but rather provide an
explanation for why populations that play Nash equilibria remain in that state.
However, the question of how populations reach those points remains open.
Some[who?] game theorists have turned to evolutionary game
theory in order to resolve these issues. These models presume either no
rationality or bounded rationality on the part of players. Despite the name,
evolutionary game theory does not necessarily presume natural selection in the
biological sense. Evolutionary game theory includes both biological as well as
cultural evolution and also models of individual learning (for example,
fictitious play dynamics).
[edit]Prescriptive or normative analysis
Cooperate Defect
Cooperate -1,
-1 -10, 0
Defect 0, -10 -5, -5
The Prisoner's Dilemma
On the other hand, some[who?] 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. However, the
rationality of such a decision has been proved only for special cases. This
normative use of game theory has also come under criticism. First, in some
cases it is appropriate to play a non-equilibrium strategy if one expects
others to play non-equilibrium strategies as well. For an example, see Guess
2/3 of the average.
Second, the Prisoner's dilemma presents another potential
counterexample. In the Prisoner's Dilemma, each player pursuing his own
self-interest leads both players to be worse off than had they not pursued
their own self-interests.
[edit]Economics and business
This article is incomplete. Please help to improve the
article, or discuss the issue on the talk page. (November 2010)
Game theory is a major method used in mathematical economics
and business for modeling competing behaviors of interacting agents.[6]
Applications include a wide array of economic phenomena and approaches, such as
auctions, bargaining, mergers & acquisitions pricing,[7] fair division,
duopolies, oligopolies, social network formation, agent-based computational
economics,[8] general equilibrium, mechanism design,[9] and voting systems,[10]
and across such broad areas as experimental economics,[11] behavioral
economics,[12] information economics,[13] industrial organization,[14] and
political economy.[15][16]
This research usually focuses on particular sets of
strategies known as equilibria in games. These "solution concepts"
are usually based on what is required by norms of rationality. 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. So, 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.[17][18]
The payoffs of the game are generally taken to represent the
utility of individual players. Often in modeling situations the payoffs
represent money, which presumably corresponds to an individual's utility. This
assumption, however, can be faulty.
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 should this information be put.
Economists and business professors suggest two primary uses (noted above):
descriptive and prescriptive.[3]
[edit]Political science
The application of game theory to political science is
focused in the overlapping areas of fair division, political economy, public
choice, war bargaining, positive political theory, and social choice theory. In
each of these areas, researchers have developed game-theoretic models in which
the players are often voters, states, special interest groups, and politicians.
For early examples of game theory applied to political
science, see the work of Anthony Downs. In his book An Economic Theory of
Democracy (Downs 1957), 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.
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
(Levy & Razin 2003).
[edit]Biology
Hawk Dove
Hawk 20, 20 80, 40
Dove 40, 80 60, 60
The hawk-dove game
Unlike 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, but rather on ones
that would be maintained by evolutionary forces. The best known equilibrium in
biology is known as the evolutionarily stable strategy (or ESS), and was 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 to understand many
different phenomena. It was first used to explain the evolution (and stability)
of the approximate 1:1 sex ratios. (Fisher 1930) suggested that the 1:1 sex
ratios are a result of evolutionary forces acting on individuals who could be
seen as trying to maximize their number of grandchildren.
Additionally, biologists have used evolutionary game theory
and the ESS to explain the emergence of animal communication (Harper &
Maynard Smith 2003). 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 Butterfly Economics.
Biologists have used the game of chicken to analyze fighting
behavior and territoriality.[citation needed]
Maynard Smith, in the preface to Evolution and the Theory of
Games, writes, "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.[19]
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.[20] All of these actions
increase the overall fitness of a group, but occur at a cost to the individual.
Evolutionary game theory explains this altruism with the
idea of kin selection. Altruists discriminate between the individuals they help
and favor relatives. Hamilton's rule explains the evolutionary reasoning 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.[20] The coefficient values depend heavily on
the scope of the playing field; for example if the choice of whom to favor
includes all genetic living things, not just all relatives, we assume the
discrepancy between all humans only accounts for approximately 1% of the
diversity in the playing field, a co-efficient that was ½ in the smaller field
becomes 0.995. Similarly if it is considered that information other than that
of a genetic nature (e.g. epigenetics, religion, science, etc.) persisted
through time the playing field becomes larger still, and the discrepancies
smaller.
[edit]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 (Ben David, Borodin
& Karp et al. 1994). Yao's principle is a game-theoretic technique for
proving lower bounds on the computational complexity of randomized algorithms,
and especially of 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[21] and within it algorithmic mechanism design[22] combine computational
algorithm design and analysis of complex systems with economic theory.[23]
[edit]Philosophy
Stag Hare
Stag 3, 3 0, 2
Hare 2, 0 2, 2
Stag hunt
Game theory has been put to several uses in philosophy.
Responding to two papers by W.V.O. Quine (1960, 1967), Lewis (1969) used game theory
to develop a philosophical account of convention. In so doing, he provided the
first analysis of common knowledge and employed it in analyzing play in
coordination games. In addition, he first suggested that one can understand
meaning in terms of signaling games. This later suggestion has been pursued by
several philosophers since Lewis (Skyrms (1996), Grim, Kokalis, and Alai-Tafti
et al. (2004)). 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.[24][25]
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),[26] Skyrms (1990),[27]
and Stalnaker (1999).[28]
In ethics, some[who?] authors have attempted to pursue the
project, begun by Thomas Hobbes, 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).[29]
Other authors have attempted to use evolutionary game theory
in order to explain the emergence of human attitudes about morality and
corresponding animal behaviors. These authors look at several games including
the Prisoner's dilemma, Stag hunt, and the Nash bargaining game as providing an
explanation for the emergence of attitudes about morality (see, e.g., Skyrms
(1996, 2004) and Sober and Wilson (1999)).
Some assumptions used in some parts of game theory have been
challenged in philosophy; for example, psychological egoism states that
rationality reduces to self-interest—a claim debated among philosophers. (see
Psychological egoism#Criticisms)
[edit]Types of games
[edit]Cooperative or non-cooperative
Main articles: Cooperative game and Non-cooperative game
A game is cooperative if the players are able to form
binding commitments. For instance the legal system requires them to adhere to
their promises. In noncooperative games this is not possible.
Often it is assumed that communication among players is
allowed in cooperative games, but not in noncooperative ones. However, this
classification on two binary criteria has been questioned, and sometimes
rejected (Harsanyi 1974).
Of the two types of games, noncooperative games are able to
model situations to the finest details, producing accurate results. Cooperative
games focus on the game at large. Considerable efforts have been made to link
the two approaches. The so-called Nash-programme[clarification needed] has
already established many of the cooperative solutions as noncooperative
equilibria.
Hybrid games contain cooperative and non-cooperative
elements. For instance, coalitions of players are formed in a cooperative game,
but these play in a non-cooperative fashion.
[edit]Symmetric and asymmetric
E F
E 1, 2 0, 0
F 0, 0 1, 2
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.
[edit]Zero-sum and non-zero-sum
A B
A –1, 1 3, –3
B 0, 0 –2, 2
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 others). Poker exemplifies a zero-sum
game (ignoring the possibility of the house's cut), because one wins exactly
the amount one's opponents lose. Other zero-sum games include matching pennies
and most classical board games including Go and chess.
Many games studied by game theorists (including the infamous
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 an additional dummy player (often
called "the board"), whose losses compensate the players' net
winnings.
[edit]Simultaneous and sequential
Main articles: Sequential game and Simultaneous 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 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, and extensive form is used to
represent sequential ones. The transformation of extensive to normal form is
one way, meaning that multiple extensive form games correspond to the same
normal form. Consequently, notions of equilibrium for simultaneous games are
insufficient for reasoning about sequential games; see subgame perfection.
In short, the differences between sequential and
simultaneous games are as follows:
Sequential Simultaneous
Normally denoted by: Decision
Trees Payoff Matrices
Prior knowledge of opponent's move: Yes No
Time Axis: Yes No
Also known as: Extensive
Game Strategic Game
[edit]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. Thus, only sequential games can
be games of perfect information because players in simultaneous games do not
know the actions of the other players. Most games studied in game theory are
imperfect-information games. Interesting examples of perfect-information games
include the ultimatum game and centipede game. Recreational games of perfect
information games include chess, go, and mancala. Many card games are games of
imperfect information, for instance poker or contract bridge.
Perfect information is often confused with complete
information, which is a similar concept. Complete information requires that
every player know the strategies and payoffs available to the other players but
not necessarily the actions taken. Games of incomplete information can be
reduced, however, to games of imperfect information by introducing "moves
by nature" (Leyton-Brown & Shoham 2008, p. 60).
[edit]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 or incomplete
information may also have a strong combinatorial character, for instance
backgammon. There is no unified theory addressing combinatorial elements in
games. There are, however, mathematical tools that can solve particular
problems and answer general questions.[30]
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.[31][32] 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.[33]
Research in artificial intelligence has addressed both
perfect and imperfect (or incomplete) 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.[30][34]
[edit]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 what is the
best way to play such a game, but simply on whether one or the other 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.
[edit]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.
[edit]Differential games
Differential games such as the continuous pursuit and
evasion game are continuous games where the evolution of the players' state
variables is governed by differential equations. The problem of finding an
optimal strategy in a differential game is closely related to the optimal
control theory. In particular, there are two types of strategies: the open-loop
strategies are found using the Pontryagin Maximum Principle while the
closed-loop strategies are found using Bellman's Dynamic Programming method.
A particular case of differential games are the games with
random time horizon.[35] In such games, the terminal time is a random variable
with a given probability distribution function. Therefore, the players maximize
the mathematical expectation of the cost function. It was shown that the
modified optimization problem can be reformulated as a discounted differential
game over an infinite time interval.
[edit]Many-player and population games
Games with an arbitrary, but finite, number of players are
often called n-person games (Luce & Raiffa 1957). 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 (Webb 2007).
[edit]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, particularly AI
planning (with uncertainty) and multi-agent system. Although these fields may
have different motivators, the mathematics involved are substantially the same,
e.g. using Markov decision processes (MDP).[citation needed]
Stochastic outcomes can also be modeled in terms of game
theory by adding a randomly acting player who makes "chance moves",
also known as "moves by nature" (Osborne & Rubinstein 1994). 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.[36] (See black swan theory
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.[36]
[edit]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 (Howard 1971) 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.
[edit]History
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 James Waldegrave in
1713.[37] In this letter, Waldegrave provides a minimax mixed strategy solution
to a two-person version of the card game le Her. 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.[38][39] 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.
The Danish mathematician 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.[40] Von Neumann's original proof
used Brouwer's fixed-point theorem on continuous mappings into compact convex
sets, which became a standard method in game theory and mathematical economics.
His paper was followed by his 1944 book Theory of Games and Economic Behavior.
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.[41]
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.[37] 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. This equilibrium is sufficiently
general to allow for the analysis of non-cooperative games in addition to
cooperative ones.
Game theory experienced a flurry of activity in the 1950s,
during which time the concepts of the core, the extensive form game, fictitious
play, repeated games, and the Shapley value were developed. In addition, the
first applications of Game theory to philosophy and political science occurred
during this time.
In 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 1967, John Harsanyi
developed the concepts of complete information and Bayesian games. Nash, Selten
and Harsanyi became Economics Nobel Laureates in 1994 for their contributions
to economic game theory.
In the 1970s, game theory was extensively applied in
biology, largely as a result of the work of John Maynard Smith and his
evolutionarily stable strategy. In addition, the concepts of correlated
equilibrium, trembling hand perfection, and common knowledge[42] were
introduced and analyzed.
In 2005, game theorists Thomas Schelling and Robert Aumann
followed Nash, Selten and Harsanyi as Nobel Laureates. Schelling worked on
dynamic models, early examples of evolutionary game theory. Aumann contributed
more to the equilibrium school, introducing an equilibrium coarsening,
correlated equilibrium, and developing an extensive formal analysis of the
assumption of common knowledge and of its consequences.
In 2007, Leonid Hurwicz, together with Eric Maskin and Roger
Myerson, was awarded the Nobel Prize in Economics "for having laid the
foundations of mechanism design theory." Myerson's contributions include
the notion of proper equilibrium, and an important graduate text: Game Theory,
Analysis of Conflict (Myerson 1997). Hurwicz introduced and formalized the concept
of incentive compatibility.
[edit]Popular culture
This section relies on references to primary sources. Please
add references to secondary or tertiary sources. (October 2012)
The life story of game theorist and mathematician John Nash
was turned into a biopic, A Beautiful Mind starring Russell Crowe,[43] based on
the namesake book by Sylvia Nasar.[44]
"Games-theory" and "theory of games" are
mentioned in the military science fiction novel Starship Troopers by Robert A.
Heinlein.[45] In the 1997 film of the same name the character Carl Jenkins
refers to his assignment, military intelligence, as "Games and
Theory."
One of the main gameplay decision-making mechanics of the
video game Zero Escape: Virtue's Last Reward is based on game theory. Some of
the characters even reference the Prisoner's Dilemma.
