Rubinstein2005-page105 - that Z is a Fnite set A lottery is...

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October 21, 2005 12:18 master Sheet number 103 Page number 87 LECTURE 8 Expected Utility Lotteries When thinking about decision making, we often distinguish be- tween actions and consequences. An action is chosen and leads to a consequence. The rational man has preferences over these conse- quences and is meant to choose a feasible action that leads to the most desired consequence. In our discussion of the rational man, we have so far not distinguished between actions and consequences since it was unnecessary for modeling situations where each action deterministically leads to a particular consequence. In this lecture we will discuss a decision maker in an environment in which the correspondence between actions and consequences is not deterministic but stochastic . The choice of an action is viewed as choosing a “lottery ticket” where the prizes are the consequences. We will be interested in preferences and choices over the set of lotteries. Let Z be a set of consequences (prizes). In this lecture we assume
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Unformatted text preview: that Z is a Fnite set. A lottery is a probability measure on Z , i.e., a lottery p is a function that assigns a nonnegative number p ( z ) to each prize z , where 6 z ∈ Z p ( z ) = 1. The number p ( z ) is taken to be the objective probability of obtaining the prize z given the lottery p . Denote by [ z ] the degenerate lottery for which [ z ] ( z ) = 1. We will use the notation α x ⊕ ( 1 − α) y to denote the lottery in which the prize x is realized with probability α and the prize y with probability 1 − α . Denote by L ( Z ) the (inFnite) space containing all lotteries with prizes in Z . Given the set of consequences Z , the space of lotter-ies L ( Z ) can be identiFed with a simplex in Euclidean space: { x ∈ < Z + | 6 x z = 1 } . Recall that < Z + is the set of functions from Z into < + . The extreme points of the simplex correspond to the degenerate lotteries....
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This note was uploaded on 12/29/2011 for the course ECO 443 taught by Professor Aswa during the Fall '10 term at SUNY Stony Brook.

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