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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 Axioms • Completeness: induces an ordering over all pairs of the outcome space (allowing ties) • Transitivity: if o1 o2 and o2 o3, then o1 o3 • substitutability : if o1 ~ o2 then [p:o1, p3:o3 … pn:o4] ~ [p:o2, p3:o3 … pn:o4] • monotonicity: if o1 o2 and p > q then [p:o1, (1-p):o2] > [q:o1, (1-q):o2] Axioms • continuity: if o1 >o2 and o2 >o3 then there exists p such that o2 ~ [p:o1, (1-p):o3] • decomposibility: if (for all oi ) the probability of lottery 1 selecting oi is the same as the probability of lottery 2 selecting oi, then the lotteries are equivalent. 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 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. 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 buy a nicer car (utility = 3) Risk averse The first you get is valued the most. buy a car (utility = 2) buy a bike (utility = 1) $200 $1500 How would you describe this curve? $5000 money Maximizing expected utility utility buy a nicer car (utility = 3) buy a car (utility = 2) buy a bike (utility = 1) $200 $1500 $5000 money • 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 Maximizing expected utility utility buy a nicer car (utility = 3) buy a car (utility = 2) buy a bike (utility = 1) $200 $1500 $5000 money • 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? 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 Grey’s marginal utility is sometimes increasing, sometimes decreasing → neither risk-averse (everywhere) nor risk-seeking (everywhere) What is utility, anyway? • Function u: O → (O is the set of “outcomes” that lotteries randomize over) • What are its units? – It doesn’t 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 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 p*L+(1-p)L’ p=50% p*L+(1-p)L’ 1-p=50% 50% o1 = L’ L 25% o2 25% 75% o3 o2 25% o4 25% 50% o1 o2 12.5% o3 12.5% o4 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) 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(o1) U(o2) iff o1 o2 [utility is an accurate measure of goodness] – U[p1:o1…pn:on] = p1U(o1) [ the total utility is based on utility of individual outcomes]