14.123 Decision Making Uncertainty Lecture Summary

14.123 Decision Making Uncertainty Lecture Summary -...

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Chapter 3 Decision Making under Uncertainty 21
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22 CHAPTER 3. DECISION MAKING UNDER UNCERTAINTY the conditions such consistent beliefs impose on the preferences, the elicitation of the beliefs from the preferences, and the representation of the beliefs by a probability distribution. 3.1 Acts, States, Consequences, and Expected Util- ity Representation Consider a fi nite set C of consequences. Let S be the set of all states of the world. Take a set F of acts f : S C as the set of alternatives (i.e., set X = F ). Each state s S describes all the relevant aspects of the world, hence the states are mutually exclusive. Moreover, the consequence f ( s ) of act f depends on the true state of the world. Hence, the decision maker may be uncertain about the consequences of his acts. Recall that the decision maker cares only about the consequences, but he needs to choose an act. Example 1 (Game as a Decision Problem) Consider a complete information game with set N = { 1 , . . . , n } of players in which each player i N has a strategy space S i . The decision problem of a player i can be described as follows. Since he cares about the strategy pro fi les, the set of consequences is C = S 1 × · · · × S n . Since he does not know Q what the other players play, the set of states is S = S i j = 6 i S j . Since he chooses among his strategies, the set of acts is F = S i , where each strategy s i is represented as a function s i 7 ( s i , s i ) . (Here, ( s i , s i ) is the strategy pro fi le in which i plays s i and the others play s i .) Traditionally, a complete-information game is de fi ned by also including the VNM utility function u i : S 1 × · · · × S n R for each player. Fixing such a utility function is equivalent to fi xing the preferences on all lotteries on S 1 × · · · × S n . We would like to represent the decision maker’s preference relation º on F by some U : F R such that U ( f ) E [ u f ] (in the sense of (OR)) where u : C R is a “utility function” on C and E is an expectation operator on S . That is, we want f º g ⇐⇒ U ( f ) E [ u f ] E [ u g ] U ( g ) . (EUR)
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3.2. ANSCOMBE-AUMANN MODEL 23 In the formulation of Von Neumann and Morgenstern, the probability distribution (and hence the expectation operator E ) is objectively given. In fact, acts are formulated as lotteries, i.e., probability distributions on C . In such a world, as we have seen in the last lecture, º is representable in the sense of (EUR) if and only if it is a continuous preference relation and satis fi es the Independence Axiom. For the cases of our concern in this lecture, there is no objectively given probability distribution on S . We therefore need to determine the decision maker’s (subjective) probability assessment on S . This is done in two important formulations. First, Savage carefully elicits the beliefs and represents them by a probability distribution in a world with no objective probability is given. Second, Anscombe and Aumann simply uses indi ff erence between some lotteries and acts to elicit preferences. I will
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