1 The basic theory of choice
We consider a set X of alternatives. Alternatives are mutually exclusive in the sense
that one cannot choose two distinct alternatives at the same time. We also take the set
of feasible alternatives exhaustive so that a player’s choices will always be defined. Note
that this is a matter of modeling. For instance, if we have options Coffee and Tea, we
define alternatives as C = Coffee but no Tea, T = Tea but no Coffee, CT = Coffee and
Tea, and NT = no Coffee and no Tea.
Take a relation º on X. Note that a relation on X is a subset of X ×X. A relation
º is said to be complete if and only if, given any x, y
X, either x º y or y º x. A
relation º is said to be transitive if and only if, given any x, y, z
[x º y and y º z]
x º z.
A relation is a preference relation if and only if it is complete and transitive. Given any
preference relation º, we can define strict preference Â by
x Â y
[x º y and y 6º x],
and the indifference
[x º y and y º x].
A preference relation can be represented by a utility function u : X → R in the
x º y
u(x) ≥ u(y)
The following theorem states further that a relation needs to be a preference relation in
order to be represented by a utility function.
Theorem 1 Let X be finite. A relation can be presented by a utility function if and only
if it is complete and transitive. Moreover, if u : X → R represents º, and if f : R → R
is a strictly increasing function, then f ◦ u also represents º.
By the last statement, we call such utility functions ordinal.
In order to use this ordinal theory of choice, we should know the agent’s preferences on
the alternatives. As we have seen in the previous lecture, in game theory, a player chooses
between his strategies, and his preferences on his strategies depend on the strategies
played by the other players. Typically, a player does not know which strategies the
other players play. Therefore, we need a theory of decision-making under uncertainty.
2 Decision-making under uncertainty
We consider a finite set Z of prizes, and the set P of all probability distributions p : Z →
[0, 1] on Z, where
p(z) = 1. We call these probability distributions lotteries. A
lottery can be depicted by a tree. For example, in Figure 1, Lottery 1 depicts a situation
in which if head the player gets $10, and if tail, he gets $0.