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Utility theory U: O-> R (utility maps from outcomes to a real number) represents preferences over outcomes ~ means indifference We need a way to talk about how preferences interact with uncertainty: A lottery is a random selection of one of a set of outcomes according to specified probabilities

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Axioms Completeness: induces an ordering over all pairs of the outcome space (allowing ties) Transitivity: if o 1 o 2 and o 2 o 3, then o 1 o 3 substitutability : if o 1 ~ o 2 then [p:o 1, p 3 :o 3 … p n :o 4 ] ~ [p:o 2, p 3 :o 3 … p n :o 4 ] monotonicity: if o 1 o 2 and p > q then [p:o 1, (1-p):o 2 ] > [q:o 1, (1-q):o 2 ]

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Axioms continuity: if o 1 >o 2 and o 2 >o 3 then there exists p such that o 2 ~ [p:o 1, (1-p):o 3 ] decomposibility: if (for all o i ) the probability of lottery 1 selecting o i is the same as the probability of lottery 2 selecting o i, then the lotteries are equivalent.

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Risk attitudes Which would you prefer? –A lottery ticket that pays out $10 with probability.5 and $0 otherwise, or –A lottery ticket that pays out $3 with probability 1 How about: –A lottery ticket (A) that pays out $100,000,000 with probability.5 and $0 otherwise, or –A lottery ticket (B) that pays out $30,000,000 with probability 1

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Risk attitudes Usually, people do not simply go by expected value An agent is risk-neutral if she only cares about the expected value of the lottery ticket An agent is risk-averse if she always prefers the expected value of the lottery ticket to the lottery ticket –Most people are like this An agent is risk-seeking if she always prefers the lottery ticket to the expected value of the lottery ticket We capture the attitude about risk by mapping the value to a utility.

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Decreasing marginal utility Typically, at some point, having an extra dollar does not make people much happier (decreasing marginal utility). Marginal utility: utility of x given you already have y. utility money $200$1500$5000 buy a bike (utility = 1) buy a car (utility = 2) buy a nicer car (utility = 3) How would you describe this curve? Risk averse The first you get is valued the most.

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Maximizing expected utility Lottery 1: get $1500 with probability 1 –gives expected utility 2 Lottery 2: get $5000 with probability.4, $200 otherwise –gives expected utility.4*3 +.6*1 = 1.8 –(expected amount of money =.4*$5000 +.6*$200 = $2120 > $1500) Any lottery is going to be less than the weighted known value utility money $200 $1500 $5000 buy a bike (utility = 1) buy a car (utility = 2) buy a nicer car (utility = 3)

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Maximizing expected utility Any weighting of 200 and 5000 – represents a point on the red line above Even if you got the exact same amount as a no-risk value, you would prefer the no risk value. Can you see why? Lottery 1: get $X with probability 1 Lottery 2: get $5000 with probability p, $200 otherwise Is there ever a case where (in this scenario) you would prefer the lottery? utility money $200 $1500 $5000 buy a bike (utility = 1) buy a car (utility = 2) buy a nicer car (utility = 3)

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Different possible risk attitudes under expected utility maximization utility money Green has decreasing marginal utility risk-averse Blue has constant marginal utility risk-neutral Red has increasing marginal utility risk-seeking Greys marginal utility is sometimes increasing, sometimes decreasing neither risk-averse (everywhere) nor risk-seeking (everywhere)

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What is utility, anyway? Function u: O (O is the set of outcomes that lotteries randomize over) What are its units? –It doesnt really matter –If you replace your utility function by u(o) = a + b*u(o) (an affine transformation), your behavior will be unchanged Why would you want to maximize expected utility rather than expected value? –This is a question about preferences over lotteries

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Compound lotteries – a lottery which gives out tickets to another lottery For two lottery tickets L and L, let p*L + (1-p)L be the compound lottery ticket where you get lottery ticket L with probability p, and L with probability 1-p o1o1 o2o2 o3o3 o2o2 o4o4 50% 25% 75%25% p=50%1-p=50% o1o1 o2o2 o3o3 o4o4 25% 50% 12.5% = L L p*L+(1-p)L

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Goals of axioms Researchers are trying to show that (in certain circumstances) we ONLY need to worry about a single dimension utility function (rather than something in multiple dimensions)

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Sufficient conditions for expected utility Idea: A decision maker chooses between prospects by comparing expected utility (weighted sums of utility values multiplied by probabilities) Expected utility theorem. (von Neumann and Morgenstern 1944) if an agent can choose between the lotteries, the utility of an arbitrary lottery can be calculated as a linear combination of the utility of its parts Suppose we have a relationship over outcomes which satisfies the axioms of completeness, transitivity, substitutability, decomposability, monotinicity, and continuity then there exists a function u: O so that L L if and only if L gives a higher expected value of u than L as defined by –U(o 1 ) U(o 2 ) iff o 1 o 2 [utility is an accurate measure of goodness] –U[p 1 :o 1 …p n :o n ] = p 1 U(o 1 ) [ the total utility is based on utility of individual outcomes]

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