576wk3a_x4 - The rest of the course Subtleties involved...

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Unformatted text preview: The rest of the course Subtleties involved with maximizing expected utility: • Finding the right state space: ◦ The wrong state space leads to intuitively incorrect answers when conditioning • Taking causality into account ◦ If you don’t, again you have problems • Computational issues: ◦ Computing probabilities efficiently using graphical representations ◦ Computing utilities efficiently ◦ Eliciting utilities efficiently • Problems with maximizing expected utility ◦ Effects of framing ◦ Ellsburg paradox, Allais paradox ◦ Dealing with large state/outcome spaces • (Possibly) some discussion on alternatives to proba- bility for representing belief (and how this impacts decision making) • Current research by Blume, Easley, Halpern 1 Savage: Summary Prof. Blume showed that if your preferences satisfy pos- tulates A1-A5, then that determines a qualitative proba- bility triangleright on events: • A triangleright B if A is more likely than B (where A and B are sets of states. With two more (arguably less plausible) axioms, A6 and A7, Savage is able to prove his major theorem: Theorem: If follows is a preference order on acts satisfying A1–A7, then there exists a probability Pr on states and a utility u on acts such that f follows g iff EU Pr ( f ) > EU Pr ( g ) . Moreover, Pr is unique and u is unique up to affine trans- formations. 2 Savage: Interpretation • Savage’s theorem says that, if a decision maker obeys Savage’s postulates, she is acting as if she has a prob- ability on states and a utility on outcomes, and is maximizing expected utility. • Note surprisingly Pr( A ) = 0 iff A is null • Pr extends the qualitative probability determined by A1–A5. • For each set B , follows B also satisfies Savage’s postulates; Pr( · | B ) is the probability determined by this pref- erence order. Today’s topic: Savage assumes you’re handed a state space. • How do we now we have the right state space? • The wrong state space leads to funny results. • The Blume-Easley-Halpern approach avoids states al- together. 3 Three-Prisoners Puzzle Computing the value of information involves condition- ing. Conditioning can be subtle ... Consider the three-prisoner’s puzzle: • Two of three prisoners a , b , and c are chosen at ran- dom to be executed, • a ’s prior that he will be executed is 2/3. • a asks the jailer whether b or c will be executed • The jailer says b . It seems that the jailer gives a no useful information about his own chances of being executed. • a already knew that one of b or c was going to be executed But conditioning seems to indicate that a ’s posterior prob- ability of being executed should be 1 / 2....
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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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576wk3a_x4 - The rest of the course Subtleties involved...

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