001 halpern 1 DT under certainty

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

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Unformatted text preview: 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)...
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This homework help was uploaded on 02/19/2008 for the course ECON 4760 taught by Professor Blume/halpern during the Fall '06 term at Cornell.

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

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