# Lecture 15. Ranking Payoff Distributions: Stochastic Dominance. First-Order Stochastic Dominance: higher distribution

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1 Lecture 15 Ranking Payoff Distributions: Stochastic Dominance First-Order Stochastic Dominance: higher distribution Definition 6.D.1: The distribution F( ) first-order stochastically dominates G( ) if for every nondecreasing function u : we have u(x)df(x) u(x)dg(x) Proposition 6.D.1: The distribution of monetary payoffs F( ) first order stochastically dominates the distribution G( ) if and only if F(x) G(x) for every x. Proof: First show that stochastic dominance implies that F(x) G(x) for every x. Use proof by contradiction. Assume that F( ) stochastically dominates G( ) but that for some value of x denoted Define the nondecreasing function u(x), where u(x) = 1 for all and u(x) = 0 otherwise. We know and But if contradiction. then u(x)dg(x) > u(x)df(x), a

2 And the other direction: Assume F(x) G(x) for all x, and show that stochastic dominance follows. u(x)df(x) = = u(x)dg(x) + u(x)(df(x) - dg(x)) = = u(x)dg(x) + u(x)d(f(x) - G(x)) Let H(x) = F(x) - G(x), so we need to know if u(x)dh(x) 0 for all nondecreasing functions u(x). To do this, we use integration by parts: but H(o) = 0 and limx H(x) = 0 so that The second term is negative if H(x) 0 everywhere, which is true under the maintained assumption.

3 Second Order Stochastic Dominance: riskier distribution Definition 6.D.1: For any two distributions F( ) and G( ) with the same mean, F( ) second-order stochastically dominates (or is less risky than) G( ) if for every nondecreasing concave function + u : we have u(x)df(x) u(x)dg(x) Other definition: the variable y is a mean-preserving spread of x, if y = x + z where zdh(z) = 0. Proposition 6.D.2: Consider two distributions F( ) and G( ) with the same mean. Then the following statements are equivalent: (1) F( ) second-order stochastically dominates G( ) (2) G( ) is a mean-preserving spread of F( ) (3) for all x

4 Demonstration that (2) implies (1). If G( ) is a mean preserving spread of F( ), then u(x)dg(x) = u(x + z)dh(z)df(x) but since zdh(z) = 0 (and xdh(z) = x) u(x)df(x) = u( (x + z)dh(z))df(x) by Jensen s inequality the concavity of u( ) implies that u(x)df(x) > u(x)dg(x)

5 Rabin Critique: (taken from Risk Aversion by M. Rabin and R. Thaler, J. Econ. Perspectives, Winter 2001) Suppose we know that Johnny is a risk-averse expected utility maximizer, and that he will always turn down the gamble of losing \$10 or gaining \$11. What else can we say about Johnny? Specifically, can we say anything about bets Johnny will be willing to accept in which there is a 50 percent chance of losing \$100 and a 50 percent chance of winning some amount \$Y? Answer: Johnny will reject the bet no matter what Y is. The logic behind this result is that within the expected utility framework, turning down a moderate stakes gamble means that the marginal utility of money must diminish very quickly. Suppose that you have initial wealth of W, and you reject a lose \$10/gain \$11 gamble because of diminishing marginal utility of wealth. Then it must be that U(W + 11) - U(W) U(W) - U(W - 10). Hence, on average you value each of the dollars between W and W + 11 by at most 10/11 as much as you, on average, value each of the dollars between W - 10 and W. By concavity, this implies that you value the dollar W + 11 at most 10/11 as much as you value the dollar W Iterating this observation, if you have the same aversion to the lose \$10/gain \$11 bet at wealth level W + 21, then you value dollar W = W + 32 by at most 10/11 as you value dollar W = W + 11, which means you value dollar W + 32 by at most 10/11 x 10/11 5/6 as much as dollar W You will th value the W dollar by at most 40 percent as much as th dollar W - 10, and the W dollar by at most 2 percent as

6 much as dollar W In words, rejecting the lose \$10/gain \$11 gamble implies a 10 percent decline in marginal utility for each \$21 in additional lifetime wealth, meaning that the marginal utility plummets for substantial changes in lifetime wealth. You care less than 2 percent as much about an additional dollar when you are \$900 wealthier than you are now. This rate of deterioration for the value of money is absurdly high, and hence leads to absurd risk aversion. Rubinstein Response: (from Rubinstein Lecture Notes in Microeconomic Theory) Nevertheless, in the economic literature it is usually assumed that a decision maker s preferences over wealth changes are induced from his preferences with regard to final wealth levels. Formally, when starting with wealth w, denote by the decision maker s preferences over lotteries in which the prizes are interpreted as changes in wealth. By the doctrine of

7 consequentialism all relations are derived from the same preference relation,, defined over the final wealth levels by p q iff w + p w + q (where w + p is the lottery that awards a prize w + x with probability p(x)). If is represented by a vnm utility function u, this doctrine implies that for all w, the function v w(x) = u(w + x) is a vnm utility function representing the preferences.

8 Propsect Theory to the Rescue: Theory motivated by experimental evidence that people evaluate wealth relative to a reference level. Two Key Features: Loss Aversion: the displeasure from a monetary loss is greater than the pleasure from a same-sized gain (losses resonate more than gains). Diminishing sensitivity: The marginal change in perceived well-being is greater for changes that are close to one s reference level than for changes that are far away. Under loss aversion the value function abruptly changes slope at the reference level. People are significantly risk-averse for even small amounts of money. Example: the Rabin example above: people dislike losing \$10 much more than they like gaining \$11, and hence prefer their status quo to a 50/50 bet of losing \$10 or gaining \$11. There is a kink in the utility function at the reference level. Diminishing sensitivity implies that a person s utility function becomes less steep as her wealth gets further away from her reference level. For losses relative to the reference level, we have a striking implication: while people are risk averse over gains, they are often risk loving over losses.

9 Kahneman and Tversky (again): Consider the following two distributions: F : \$0 with prob. 3/4 and \$6000 with prob. 1/4 G : \$0 with prob. 2/4, \$4000 with prob. 1/4, and \$2000 with prob. 1/4. K&T found that 70% of subjects report that they would prefer F to G. This is consistent with diminishing sensitivity. But F is a mean-preserving spread of G, so 70% of responses are inconsistent with the assumption that utility functions are concave.

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