lecture2 - 14.12 Game Theory Lecture Notes Theory of Choice...

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14.12 Game Theory Lecture Notes Theory of Choice Muhamet Yildiz (Lecture 2) 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 de f ned. Note that this is a matter of modeling. For instance, if we have options Co f ee and Tea, we de f ne alternatives as C = Co f ee but no Tea, T = Tea but no Co f ee, CT = f ee and Tea, and NT = no Co f ee and no Tea. Take a relation º on X .Are la t ionon X is a subset of X × X t ion º is said to be complete if and only if, given any x, y X ,e ither x º y or y º x t 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 . Are lat ionisa preference relation if and only if it is complete and transitive. Given any preference relation º ,wecande f ne strict preference  by x  y ⇐⇒ [ x º y and y x ] , and the indi f erence x y [ x º y and y º x ] . Ap re fe rencere t ioncanbe represented by a utility function u : X R in the following sense: x º y u ( x ) u ( y ) x, y X. 1
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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 f nite. 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 ,andi f 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 f nite set Z of prizes, and the set P of all probability distributions p : Z [0 , 1] on Z ,whe re 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 the player gets $10 with probability 1/2 (e.g. if a coin toss results in Head) and $0 with probability 1/2 (e.g. if the coin toss results in Tail). Lottery 1 1/2 1/2 10 0 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 2
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by Savage (1954) under certain conditions that a player’s beliefs can be represented by a (unique) probability distribution. Using these probabilities, we can represent our acts by lotteries.
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This note was uploaded on 11/28/2011 for the course ECONOMICS - taught by Professor Muhammadyildiz during the Spring '05 term at University of Massachusetts Boston.

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lecture2 - 14.12 Game Theory Lecture Notes Theory of Choice...

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