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. 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:
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.
