Lecture 15. Ranking Payoff Distributions: Stochastic Dominance. FirstOrder Stochastic Dominance: higher distribution


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1 Lecture 15 Ranking Payoff Distributions: Stochastic Dominance FirstOrder Stochastic Dominance: higher distribution Definition 6.D.1: The distribution F( ) firstorder 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( ) secondorder 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 meanpreserving 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( ) secondorder stochastically dominates G( ) (2) G( ) is a meanpreserving 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 riskaverse 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 samesized gain (losses resonate more than gains). Diminishing sensitivity: The marginal change in perceived wellbeing 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 riskaverse 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 meanpreserving spread of G, so 70% of responses are inconsistent with the assumption that utility functions are concave.
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