Everything about Game Theory totally explained
Game theory is a branch of
applied mathematics that's used in the
social sciences (most notably
economics),
biology,
political science,
computer science, and
philosophy. Game theory attempts to mathematically capture behavior in
strategic situations, in which an individual's success in making choices depends on the choices of others. While initially developed to analyze competitions in which one individual does better at another's expense (
zero sum games), it has been expanded to treat a wide class of interactions, which are classified according to several
criteria.
Traditional applications of game theory attempt to find
equilibria in these games—sets of strategies in which individuals are unlikely to change their behavior. Many
equilibrium concepts have been developed (most famously the
Nash equilibrium) in an attempt to capture this idea. These equilibrium concepts are motivated differently depending on the field of application, although they often overlap or coincide. This methodology isn't without criticism, and debates continue over the appropriateness of particular equilibrium concepts, the appropriateness of equilibria altogether, and the usefulness of mathematical models more generally.
Although some developments occurred before it, the field of game theory came into being with the 1944 book
Theory of Games and Economic Behavior by
John von Neumann and
Oskar Morgenstern. 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
Nobel prizes in economics, and
John Maynard Smith was awarded the
Crafoord Prize for his application of game theory to biology.
Representation of games
The games studied by 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.
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 (for example, the players don't know at which point they are), or a closed line is drawn around them.
Normal form
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's 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
In
cooperative games with
transferable utility no individual payoffs are given. Instead, the characteristic function determines the payoff of each coalition. The standard assumption is that the empty coalition obtains a payoff of 0.
The origin of this form is to be found in the seminal book of
von Neumann and
Morgenstern who, when studying coalitional
normal form games, assumed that when a coalition
forms, it plays against the complementary coalition (
) as if they were playing a 2-player game. The equilibrium payoff of
is
characteristic. Now there are different models to derive coalitional values from normal form games, but not all games in characteristic function form can be derived from normal form games.
Formally, a characteristic function form game (also known as a TU-game) is given as a pair
, where
denotes a set of players and
is a characteristic function.
The characteristic function form has been generalised to games without the assumption of
transferable utility.
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 .
Application and challenges
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, sociology, 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 predict and explain behavior, game theory has also been used to attempt to develop theories of ethical or normative 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.
Political science
The application of game theory to
political science is focused in the overlapping areas of
fair division,
political economy,
public choice,
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, 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, he applies a hotelling firm location model to the political process. In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. The theorist shows how the political candidates will converge to the ideology preferred by the median voter. For more recent examples, see the books by
Steven Brams,
George Tsebelis,
Gene M. Grossman and
Elhanan Helpman, or
David Austen-Smith and
Jeffrey S. Banks.
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's 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 nondemocracy .
Game theory provides a theoretical description for a variety of observable consequences of changes in governmental policies. For example, in a static world where producers were not themselves decision makers attempting to optimize their own expenditure of resources while assuming risks, response to an increase in tax rates would imply an increase in revenues and vice versa. Game Theory inclusively weights the decision making of all participants and thus explains the contrary results illustrated by the
Laffer curve.
Economics and business
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. 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've 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. 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's an abstraction of some 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.
Descriptive
The first known use is to inform us about how actual human populations behave. Some scholars believe that by finding the equilibria of games they can predict how actual human populations will behave when confronted with situations analogous to the game being studied. This particular view of game theory has come under recent criticism. First, it's 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, humans behaviour is often contrary to 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 don't always hold, they can treat game theory as a reasonable scientific
ideal akin to the models used by
physicists. However, additional criticism of this use of game theory has been levied because some experiments have demonstrated that individuals don't play equilibrium strategies. For instance, in the
centipede game,
guess 2/3 of the average game, and the
dictator game, people regularly don't play Nash equilibria. There is an ongoing debate regarding the importance of these experiments.
Alternatively, some authors claim that Nash equilibria don't 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 game theorists have turned to
evolutionary game theory in order to resolve these worries. These models presume either no rationality or
bounded rationality on the part of players. Despite the name, evolutionary game theory doesn't necessarily presume
natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning (for example,
fictitious play dynamics).
Prescriptive or normative analysis
On the other hand, some scholars see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave. Since a
Nash equilibrium of a game constitutes one's
best response to the actions of the other players, playing a strategy that's part of a Nash equilibrium seems appropriate. However, this use for game theory has also come under criticism. First, in some cases it's 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.
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
Evolutionarily stable strategy or (ESS), and was first introduced by
John Maynard Smith (described in his 1982 book). Although its initial motivation didn't 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.
Biologists have used the
hawk-dove game (also known as chicken) to analyze fighting behavior and territoriality.
Maynard Smith, in the preface to Evolution of the Theory of Games writes, "[p]aradoxically, it has turned out that game theory is more readily applied to biology than to the field of economic behaviour for which it was originally designed." Evolutionary game theory has been used to explain many seemingly incongruous phenomena in nature.
One such phenomena 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've obtained from a night’s hunting and give it to group members who have failed to feed, to worker bees that care for the queen bee for their entire lives and never mate, to Vervet monkeys that warn group members of a predator’s approach, even when it endangers that individual’s chance of survival. All of these actions increase the overall fitness of a group, but occur at a cost to the individual.
Evolutionary game theory explains this altruism with the idea of kin selection. Altruists discriminate between the individuals they help and favor relatives. Hamilton’s rule explains the evolutionary reasoning behind this selection with the equation c
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, and especially of online algorithms.
Philosophy
Game theory has been put to several uses in philosophy. Responding to two papers by, used game theory to develop a philosophical account of convention. In so doing, he provided the first analysis of common knowledge 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 ().
In ethics, some 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 and .
Other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and corresponding animal behaviors. These authors look at several games including the Prisoner's dilemma, Stag hunt, and the Nash bargaining game as providing an explanation for the emergence of attitudes about morality (see, for example, and ).
Some assumptions used in some parts of game theory have been challenged in philosophy; psychological egoism states that rationality reduces to self-interest—a claim debated among philosophers. (see Psychological egoism#Criticism)
Types of games
Cooperative or noncooperative
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 isn't possible.
Often it's assumed that communication among players is allowed in cooperative games, but not in noncooperative ones. This classification on two binary criteria has been rejected .
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.
Symmetric and asymmetric
A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. 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 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 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 doesn't 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 don't 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 didn't perform one particular action, while he doesn't 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; although this isn't a strict rule in a technical sense.
Perfect information and imperfect information
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 an infinite number of 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.
Discrete and continuous games
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. A typical example of a differential game is the continuous pursuit and evasion game. Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics.
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.
History
The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713. In this letter, Waldegrave provides a minimax mixed strategy solution to a two-person version of the card game le Her. It wasn't until the publication of Antoine Augustin Cournot's Researches into the Mathematical Principles of the Theory of Wealth in 1838 that a general game theoretic analysis was pursued. In this work Cournot considers a duopoly and presents a solution that's a restricted version of the Nash equilibrium.
Although Cournot's analysis is more general than Waldegrave's, game theory didn't really exist as a unique field until John von Neumann published a series of papers in 1928. While the French mathematician Émile Borel did some earlier work on games, Von Neumann can rightfully be credited as the inventor of game theory. Von Neumann was a brilliant mathematician whose work was far-reaching from set theory to his calculations that were key to development of both the Atom and Hydrogen bombs and finally to his work developing computers. Von Neumann's work in game theory culminated in the 1944 book Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern. This profound work contains the method for finding mutually consistent solutions for two-person zero-sum games. During this 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 discussion of the prisoner's dilemma appeared, and an experiment was undertaken on this game at the RAND corporation. 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, allowing 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'd 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 were introduced and analysed.
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, Roger Myerson, together with Leonid Hurwicz and Eric Maskin, was awarded of the Nobel Prize in Economics "for having laid the foundations of mechanism design theory." Among his contributions, is also the notion of proper equilibrium, and an important graduate text: Game Theory, Analysis of Conflict, published in 1991.
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