Lecture 2 - 1 The basic theory of choice We consider a set...

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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, [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  by x  y ⇐⇒ [x º y and y 6º x], and the indifference by x y ⇐⇒ [x º y and y º x]. A preference relation can be represented by a utility function u : X → R in the following sense: x º y ⇐⇒ u(x) ≥ u(y) x, y X. 1 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 Z 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. Lottery 1 1/2 1/2 10 0
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Figure 1: Unlike the situation we just described, in game theory and more broadly when agents make their decision under uncertainty, we do not have the lotteries as in casinos where the probabilities are generated by some machines or given. Fortunately, it has been shown by Savage (1954) under certain conditions that a player’s beliefs can be represented by 2 a (unique) probability distribution. Using these probabilities, we can represent our acts by lotteries. We would like to have a theory that constructs a player’s preferences on the lotteries
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Lecture 2 - 1 The basic theory of choice We consider a set...

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