Game theory
Source: SFB 504
Theory of rational behavior for interactive decision problems. In a game, several
agents strive to maximize their (expected) utility index by chosing
particular courses of action, and each agent's final utility payoffs depend on the profile of
courses of action chosen by all agents.
The interactive situation, specified by the set of participants, the possible courses of action
of each agent, and the set of all possible utility payoffs, is called a game; the agents
'playing' a game are called the players.
In denegerate games, the players' payoffs only depend on their own actions. For example,
in competitive markets (competitive market equilibrium), it is enough that each
player optimizes regardless of the behavior of other traders. As soon as a small number of
agents is involved in an economic transaction, however, the payoffs to each of them depend on
the other agents' actions. For example in an oligopolistic industry or in a cartel, the price or the
quantity set optimally by each firm depends crucially on the prices or quantities set by the
competing firms. Similarly, in a market with a small number of traders, the equilibrium price
depends on each trader's own actions as well as the one of his fellow traders (see auctions).
Whenever an optimizing agent expects a reaction from other agents to his own actions,
his payoff is determined by other player's actions as well, and he is playing a game.
Game theory provides general methods of dealing with interactive optimization problems; its
methods and concepts, particularly the notion of strategy and strategic
equilibrium find a vast number of applications throughout social sciences (including
biology). Although the word 'game' suggests peaceful and 'kind' behavior, most situations revelant
in politics, psychology, biology, and economics involve rather strong conflicts of interest, competition,
and cheating, apart from leaving room for cooperation or mutually benefically actions.
Based on a model of optimizing agents that plan individually optimal course of play, knowing
that her opponents will do so as well, the basic objects of interest in strategic (or 'noncooperative')
game theory are the players' strategies. A player's strategy is a complete plan of
actions to be taken when the game is actually played; it must be completely specified before the
actual play of the game starts, and it prescribe the course of play for each decision that a
player might be called upon to take, for each possible piece of information that the player may have
at each time where he might be called upon to act.
A strategy may also include random moves. It is generally assumed that the players
evaluate uncertain payoffs according to von Neumann Morgenstern
utility.
In addition to the strategic branch of game theory, there is another one that focuses on the
interactions of groups of players that jointly strive to maximize their surplus. While this
second branch represents the analysis of coalitional games, which centers around notions
of 'coalitionally stable' payoff configurations, we focus here on strategic game theory (from which
coalitional games are derived).
Given a strategic game, a profile of strategies results in a profile of (expected) utility payoffs.
A certain payoff allocation, or a profile of final moves of the players is called an outcome
of the game. An outcome is called an equilibrium outcome if no player can unilaterally
improve the outcome (in terms of his own payoff) given that the other players stick to their
equilibrium strategies.
A profile of strategies is called a (strategic) equilibrium if, given that all players
conform to the prescribed strategies, no player can gain from unilaterally switching to another
strategy. Alternatively, a profile of strategies forms an equilibrium if the strategies form best
responses to one another.
(Unfortunately, it is impossible to describe what is an equilibrium other than in such a
selfreferential way. The best way to understand this definition is then to take it literally.)
Only equilibrium outcomes are reasonable outcomes for games, because outside an equilibrium
there is at least one player that can improve by playing according to another strategy.
An implicit assumption of game theory is that the players, being rational, are able to
reproduce any equilibrium calculations of anybody else. In particular, all the equilibrium
strategies must be known to (as they are computed by) the players. Similarly, it is assumed
that the whole structure of the game, in much the same way as the players' social context, is
known by each player (and that this knowledge itself is known etc.)
