- This article discusses the mathematical modelling of incentive structures. For other games (and their theories) see
Game (disambiguation).
Game theory is a branch of applied mathematics that uses models to study interactions with formalised incentive structures
("games"). It has applications in a variety of fields, including economics,
international relations, evolutionary biology, political science, and military strategy.
Game theorists study the predicted and actual behaviour of individuals in games, as well as optimal strategies. Seemingly
different types of interactions can exhibit similar incentive structures, thus all exemplifying one particular game.
John von Neumann and Oskar Morgenstern first formalised the subject in 1944 in their book Theory of Games and Economic
Behavior. Game theory has important applications in fields like operations research, economics, collective action, political science, psychology, and biology. It has close links with economics in
that it seeks to find rational strategies in situations where the outcome depends not only on one's own strategy and "market
conditions", but upon the strategies chosen by other players with possibly different or overlapping goals. Applications in
military strategy drove some of the early development of game
theory.
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. And computer scientists have used games to model interactive computations. Computability logic attempts to develop a comprehensive formal theory (logic) of interactive computational tasks and resources, formalising these entities as games between a
computing agent and its environment.
Game theoretic analysis can apply to simple games of entertainment or to more significant aspects of life and society. The
prisoner's dilemma, as popularized by mathematician Albert W. Tucker, furnishes an example of the application of game theory to
real life; it has many implications for the nature of human co-operation, and has even been used as the basis of a game show called Friend or Foe?.
Biologists have used game theory to understand and predict certain outcomes of
evolution, such as the concept of evolutionarily stable strategy introduced by John Maynard Smith and George R. Price in a
1973 paper in Nature (See also Maynard Smith 1982). See
also evolutionary game theory and behavioral ecology.
Analysts of games commonly use other branches of mathematics, in particular probability, statistics and linear programming, in conjunction with game theory.
Mathematical definitions
There are a few alternative definitions of the notion of a 'game'. We shall hereby give a short introduction and say a few
words about the relations between them.
Normal form game
A game in normal or strategic form combines the set of possible strategies for each player and records the payoffs for each
outcome. Let N be a set of players. For each player  there is given a set of strategies  . The game is then a function:
So that, if one knows the tuple of strategies that were chosen by the players, one is given the allocation
payments, a real number assignment. A further generalization can be achieved by splitting the game into two
functions: the normal form game', describing the way in which strategies define outcomes, and a second
function depicting player's preferences on the set of outcomes. Hence:
Where  is the outcome set of the
game. And for each player there is a
preference function
 .
A reduced normal form exists as well. The reduced normal form combines strategies for which are associated with the same
payoffs.
Extensive form game
(Main article: Extensive form game)
The normal form gives the mathematician an easy notation for the study of equilibria problems, because it
bypasses the question of how strategies are calculated, i.e. how the game is actually played. The convenient notation for
dealing with these questions, more relevant to combinatorial game theory, is the extensive form of the game. This is given by a tree, where at each vertex of the tree a different player has the choice of choosing an edge.
Simple game
The normal form and the extensive form capture the essense of non-cooperative games. But in some games
the formation of coalitions and the way cooperation is developed are more important. For dealing with questions of
cooperation, the notion of a simple game was developed.
Types of games and examples
Game theory classifies games into many categories that determine which particular methods one can apply to solving them (and
indeed how one defines "solved" for a particular category). Common categories include:
Zero-sum and non-zero-sum games
In zero-sum games the total benefit to all players in the game, for every
combination of strategies, always adds to zero (or more informally put, a player benefits only at the expense of others).
Go, chess and poker exemplify zero-sum games, because one wins exactly the amount one's opponents lose. Most
real-world examples in business and politics, as well as the famous prisoner's
dilemma are non-zero-sum games, because some outcomes have net results greater or less than zero. Informally, a gain
by one player does not necessarily correspond with a loss by another. For example, a business contract ideally involves a
positive-sum outcome, where each side ends up better off than if they did not make the deal.
Note that one can more easily analyse a zero-sum game; and it turns out that one can transform any game into a zero-sum game
by adding an additional dummy player (often called "the board"), whose losses compensate the players' net winnings.
A game's payoff matrix represents convenient way of representation.
Consider for example the two-player zero-sum game with the following matrix:
Player 2
Action A Action B Action C
Action 1 30 -10 20
Player 1
Action 2 10 20 -20
This game proceeds as follows: the first player chooses one of the two actions 1 or 2; and the second player, unaware of the
first player's choice, chooses one of the three actions A, B or C. Once the players have made their choices, the payoff gets
allocated according to the table; for instance, if the first player chose action 2 and the second player chose action B, then the
first player gains 20 points and the second player loses 20 points. Both players know the payoff matrix and attempt to maximize
the number of their points. What should they do? (Clearly, Player 2 will not pick action A.)
Player 1 could reason as follows: "with action 2, I could lose up to 20 points and can win only 20, while with action 1 I can
lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, player 2 would choose action C
(negative numbers in the table represent payoff for him). If both players take these actions, the first player will win 20
points. But what happens if player 2 anticipates the first player's reasoning and choice of action 1, and deviously goes for
action B, so as to win 10 points? Or if the first player in turn anticipates this devious trick and goes for action 2, so as to
win 20 points after all?
John von Neumann had the fundamental and surprising insight that
probability provides a way out of this conundrum. Instead of deciding on a
definite action to take, the two players assign probabilities to their respective actions, and then use a random device which,
according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimise the
maximum expected point-loss independent of the opponent's strategy;
this leads to a linear programming problem with a unique
solution for each player. This minimax method can compute provably optimal strategies
for all two-player zero-sum games.
For the example given above, it turns out that the first player should choose action 1 with probability 57% and action 2 with
43%, while the second player should assign the probabilities 0%, 57% and 43% to the three actions A, B and C. Player one will
then win 2.85 points on average per game.
Co-operative games
A cooperative game is characterized by an enforceable contract. Theory of co-operative games gives justifications of
plausible contracts. The plausibility of a contract is closely related with stability.
Axiomatic bargaining
Two players may bargain how much share they want in a contract. The theory of axiomatic bargaining tells you how much share is
reasonable for you. For example, Nash bargaining solution demands that the share is fair and efficient (see an advanced
textbook for the complete formal description).
However, you may not be concerned with fairness and may demand more. How does Nash bargaining solution deal with this problem? Actually, there is a non-cooperative game of alternating
offers (by Rubinstein) supporting Nash bargaining solution as the unique Nash equilibrium.
Characteristic function games
Many players, instead of two players, may cooperate to get a better outcome. Again, how much share should be given to each
player of the total output is not clear. Core gives a reasonable set of possible shares. A combination of shares is in a
core if there exists no subcoalition in which its members may gain a higher total outcome than the share of concern. If the share
is not in a core, some members may be frustrated and may think of leaving the whole group with some other members and form a
smaller group.
Games of complete information
In games of complete information each player has the same game-relevant information as every other player. Chess and the prisoner's
dilemma exemplify complete-information games, while poker illustrates the opposite.
Complete information games occur only rarely in the real world, and game theorists usually use them only as approximations of the
actual game played.
Risk aversion
Main article: Risk aversion
For the above example to work, one must assume risk-neutral participants in the
game. For example, this means that they would place an equal value on a bet with a 50% chance of receiving 20 points and a bet
with a 100% chance of receiving 10 points. However, in reality people often exhibit risk averse behaviour and prefer a
more certain outcome - they will only take a risk if they expect to make money on average. Subjective expected utility theory explains how to
derive a measure of utility which will always satisfy the criterion of risk
neutrality, and hence serve as a measure for the payoff in game theory.
Game shows often provide examples of risk aversion. For example, if a person
has a 1 in 3 chance of winning $50,000, or can take a sure $10,000, many people will take the sure $10,000.
Lotteries can show the opposite behaviour of risk seeking: for example many
people will risk $1 to buy a 1 in 14,000,000 chance of winning $7,000,000. This illustrates the nature of people's preferences
over risk: they are risk-loving where losses are small and risk averse where losses are high, even if potential gains are greater
- people care less about a marginal dollar than say a marginal $1000 - most people would not risk $1000 for the same chance of
winning $7,000,000,000.
Games and numbers
John Conway developed a notation for certain complete information games
and defined several operations on those games, originally in order to study Go endgames, though much of the analysis focused on Nim. This developed
into combinatorial game theory.
In a surprising connection, he found that a certain subclass of these games can be used as numbers as described in his book
On Numbers and Games, leading to the very general
class of surreal numbers.
History
Though touched on by earlier mathematical results, modern game theory became a prominent branch of mathematics in the
1940s, especially after the 1944 publication of
The Theory of Games and Economic Behavior by John von
Neumann and Oskar Morgenstern. This profound work contained
the method -- alluded to above -- for finding optimal solutions for two-person zero-sum games.
Around 1950, John Nash
developed a definition of an "optimum" strategy for multi-player games where no such optimum was previously defined, known as
Nash equilibrium. Reinhard Selten with his ideas of trembling hand perfect and subgame perfect equilibria further refined this concept. These men won The Bank of Sweden Prize in Economic Sciences in Memory
of Alfred Nobel (also known as The Nobel Prize in Economics) in 1994 for their work on game theory, along with John Harsanyi who developed the analysis of games of incomplete information.
Research into game theory continues, and there remain games which produce counter-intuitive optimal strategies even under
advanced analytical techniques like trembling hand equilibrium. One example of this occurs in the Centipede Game, where at every decision players have the option of increasing their opponents' payoff
at some cost to their own.
Some experimental tests of games indicate that in many situations people respond instinctively by picking a 'reasonable'
solution or a 'social norm' rather than adopting the strategy indicated by a rational analytic concept.
The finding of Conway's number-game connection occurred in the early 1970s.
Specific applications
External links and references
- Paul Walker, An Outline of the History of Game Theory
(http://william-king.www.drexel.edu/top/class/histf.html).
- Oskar Morgenstern, John von Neumann: The Theory of Games and
Economic Behavior, 3rd ed., Princeton University Press 1953
- William Poundstone, Prisoner's Dilemma: John Von Neumann, Game
Theory and the Puzzle of the Bomb, ISBN: 038541580X
- Alvin Roth: Game Theory and Experimental Economics page
(http://www.economics.harvard.edu/~aroth/alroth.html) - Comprehensive list of links to game theory information on
the Web
- Mike Shor: Game Theory
.net (http://www.gametheory.net) - Lecture notes, interactive illustrations and
other information.
- Giorgi
Japaridze (http://www.csc.villanova.edu/~japaridz): Game Semantics or Linear Logic?
(http://www.csc.villanova.edu/~japaridz/CL/gsoll.html) - Discussion of games in logic, and links.
- Maynard Smith: Evolution and the Theory of Games, Cambridge University Press 1982
- Don Ross: Review Of Game Theory
(http://plato.stanford.edu/entries/game-theory/).
- Important publications in game theory
- Chris Yiu's Game Theory
Lounge (http://www.yiu.co.uk/gametheory/)
Game Theory was also the name of an independent rock band from the 1980s and early 1990s. See Game Theory (band).
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