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Decision Theory

Decision Theory
Decision TheoryRisk 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 that pays out 100,000,000 with probability .5 and 0 otherwise, or – A lottery ticket that pays out 30,000,000 with probability 1 • Usually, people do not simply go by expected value • An agent is riskneutral if she only cares about the expected value of the lottery ticket • An agent is riskaverse if she always prefers the expected value of the lottery ticket to the lottery ticket – Most people are like this • An agent is riskseeking if she always prefers the lottery ticket to the expected value of the lottery ticketDecreasing marginal utility • Typically, at some point, having an extra dollar does not make people much happier (decreasing marginal utility) utility buy a nicer car (utility = 3) buy a car (utility = 2) buy a bike (utility = 1) money 200 1500 5000Maximizing expected utility utility buy a nicer car (utility = 3) buy a car (utility = 2) buy a bike (utility = 1) money 200 1500 5000 • Lottery 1: get 1500 with probability 1 – gives expected utility 2 • Lottery 2: get 5000 with probability .4, 200 otherwise – gives expected utility .43 + .61 = 1.8 – (expected amount of money = .45000 + .6200 = 2120 1500) • So: maximizing expected utility is consistent with risk aversionDifferent possible risk attitudes under expected utility maximization utility money • Green has decreasing marginal utility → riskaverse • Blue has constant marginal utility → riskneutral • Red has increasing marginal utility → riskseeking • Grey’s marginal utility is sometimes increasing, sometimes decreasing → neither riskaverse (everywhere) nor riskseeking (everywhere)Acting optimally over time • Finite number of periods: • Overall utility = sum of rewards in individual periods • Infinite number of periods: • … are we just going to add up the rewards over infinitely many periods – Always get infinity • (Limit of) average payoff: lim Σ r(t)/n n→∞ 1≤t≤n – Limit may not exist… t • Discounted payoff: Σ δ r(t) for some δ 1 t • Interpretations of discounting: – Interest rate r: δ= 1/(1+r) – World ends with some probability 1 δ • Discounting is mathematically convenient
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