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Business theory
Game theory is a branch of applied mathematics that is often used in the context of economics. It
studies strategic interactions between agents. In strategic games, agents choose strategies that
will maximize their return, given the strategies the other agents choose. The essential feature is
that it provides a formal modelling approach to social situations in which decision makers interact
with other agents. Game theory extends the simpler optimisation approach developed in
neoclassical economics.The field of game theory came into being with the 1944 classic Theory of
Games and Economic Behavior by John von Neumann and Oskar Morgenstern. A major center
for the development of game theory was RAND Corporation where it helped to define nuclear
strategies.
Game theory has played, and continues to play a large role in the social sciences, and is now also
used in many diverse academic fields. Beginning in the 1970s, game theory has been applied to
animal behaviour, including evolutionary theory. Many games, especially the prisoner's dilemma,
are used to illustrate ideas in political science and ethics. Game theory has recently drawn
attention from computer scientists because of its use in artificial intelligence and cybernetics.
In addition to its academic interest, game theory has received attention in popular culture. A Nobel
Prize–winning game theorist, John Nash, was the subject of the 1998 biography by Sylvia Nasar
and the 2001 film A Beautiful Mind. Game theory was also a theme in the 1983 film WarGames.
Several game shows have adopted game theoretic situations, including Friend or Foe? and to
some extent Survivor. The character Jack Bristow on the television show Alias is one of the few
fictional game theorists in popular culture.Although some game theoretic analyses appear similar
to decision theory, game theory studies decisions made in an environment in which players
interact. In other words, game theory studies choice of optimal behavior when costs and benefits
of each option depend upon the choices of other individuals.
Extensive form
The extensive form can be used to formalize games with some important order. Games here are
often presented as 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.In the game pictured here, 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 incomplete 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.
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.
Characteristic function form
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. This classification on two binary criteria has been 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 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.
A symmetric game is a game where the payoffs for playing a particular strategy depend only on
the other strategies employed, not on who is playing them. If the identities of the players can be
changed without changing the payoff to the strategies, then a game is symmetric. Many of the
commonly studied 2×2 games are symmetric. The standard representations of chicken, the
prisoner's dilemma, and the stag hunt are all symmetric games. Some scholars would consider
certain asymmetric games as examples of these games as well. However, the most common
payoffs for each of these games are symmetric.Most commonly studied asymmetric games are
games where there are not identical strategy sets for both players. For instance, the ultimatum
game and similarly the dictator game have different strategies for each player. It is possible,
however, for a game to have identical strategies for both players, yet be asymmetric. For example,
the game pictured to the right is asymmetric despite having identical strategy sets for both
players.
Zero sum and non-zero sum
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 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 famous prisoner's dilemma) are
non-zero-sum games, because some outcomes have 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.
Simultaneous and sequential
Simultaneous games are games where both players move simultaneously, or if they do not move
simultaneously, the later players are unaware of the earlier players' actions (making them
effectively simultaneous). Sequential games (or dynamic games) are games where later players
have some knowledge about earlier actions. This need not be perfect information about every
action of earlier players; it might be very little knowledge. For instance, a player may know that an
earlier player did not perform one particular action, while 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. Normal form is
used to represent simultaneous games, and extensive form is used to represent sequential ones.
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, since in simultaneous games not
every player knows the actions of the others. Most games studied in game theory are imperfect
information games, although there are some interesting examples of perfect information games,
including the ultimatum game and centipede game. Perfect information games include also chess,
go, mancala, and arimaa.Perfect information is often confused with complete information, which is
a similar concept. Complete information requires that every player know the strategies and
payoffs of the other players but not necessarily the actions.
Infinitely long games
Games, as studied by economists and real-world game players, are generally finished in a finite
number of 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.
Most of the objects treated in most branches of game theory are discrete, with a finite number of
players, moves, events, outcomes, etc. However, the concepts can be extended into the realm of
real numbers. This branch has sometimes been called differential games, because they map to a
real line, usually time, although the behaviors may be mathematically discontinuous. Much of this
is discussed under such subjects as optimization theory and extends into many fields of
engineering and physics.Economists have long used game theory to analyze a wide array of
economic phenomena, including auctions, bargaining, duopolies, fair division, oligopolies, social
network formation, and voting systems. 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. 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.
Biology
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 Evolutionary stable strategy or (ESS), and was first introduced by John
Maynard Smith (described in his 1982 book). 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. Ronald 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 (Maynard Smith & Harper, 2003). The analysis of signaling
games and other communication games has provided some 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.Finally, biologists have used the hawk-dove game (also known as chicken)
to analyze fighting behavior and territoriality.
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