001 halpern 1 DT under certainty

# 001 halpern 1 DT under certainty - Decision Theory Decision...

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Decision Theory Decision theory is about making choices It has a normative aspect what “rational” people should do . . . and a descriptive aspect what people do do Not surprisingly, it’s been studied by economists, psychol- ogist, and philosophers. More recently, computer scientists have looked at it too: How should we design robots that make reasonable decisions What about software agents acting on our behalf agents bidding for you on eBay managed health care Algorithmic issues in decision making This course will focus on normative aspects, informed by a computer science perspective. 1 Axiomatic Decision Theory Standard (mathematical) approach to decision theory: Give axioms characterizing reasonable decisions Ones that any “rational” person should accept Then show show that these axioms characterize a par- ticular approach to decision making For example, we will discuss Savage’s axioms that characterize maximizing expected utility. An issue that arises frequently: How to represent a decision problem 2 Choice Under Certainty Assumption: you’re given a set X of objects. You have to state which one you want most More generally: you give a preference order among objects in X Notation: x follows y means x is strictly preferred to y There’s no uncertainty: you get what you choose. Goal: to understand reasonable properties of a pref- erence order Example : X = { a, b, c } , b follows a , a follows c , and c follows a . Are such cyclic preferences reasonable? Would a rational person have such preferences? 3 Axioms for Choice Under Certainty Asymmetry: If x follows y then y negationslash follows x . Negative Transitivity: If x negationslashfollows y and y negationslashfollows z then x negationslash follows z Transitivity: If x follows y and y follows z , then x follows z . Proposition: Asymmetry + NT imply Transitivity. Proof: Suppose that x follows y , y follows z , and x negationslashfollows z . By asymmetry, we have z negationslashfollows y . By NT, we have x negationslashfollows y contradiction. Is NT a good normative or descriptive property? Proposition. The binary relation follows is negatively tran- sitive iff x follows z implies that, for all y X , x follows y or y follows z . (Proof in Kreps) NT comes pretty close to saying that follows is a total order For all x , y , either (1) x follows y

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• Fall '06
• BLUME/HALPERN
• SEN, preference relation

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