Lecture 5. Choice under Uncertainty II - Ecares

Lecture 5. Choice under Uncertainty II
1. Stochastic Dominance
Comparison of payo¤ distributions
First order stochastic dominance - two equivalent de…nitions:
i) F (.) …rst-order stochastically dominates F 0 (.) if for every nondecreasing
function u : R ! R it holds:
Z
Z
u(x)dF (x) u(x)dF 0 (x)
”Every expected utility maximizer who appreciates money prefers F (.)
over F 0 (.)."
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ii) F (.) …rst-order stochastically dominates F 0 (.) if
F (x) F 0 (x) for all x.
”The probability, that the realized payo¤ is above a certain threshold x, is
larger for F (.) than for F 0 (.) for any such threshold.”
Second order stochastic dominance:
For any two distributions F (.) and F 0 (.) with the same mean, F (.)
second-order stochastically dominates F 0 (.) if for every nondecreasing
concave function u : R ! R it holds:
Z
Z
u(x)dF (x) u(x)dF 0 (x)
"Provided that both distributions give the same expected monetary payo¤,
every risk averse agent prefers F (.) over F 0 (.)."
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2. State dependent utility
Agent cares not only about consequences, but also about reasons for
consequences.
S : set of states of nature, …nite; actual state unknown
π s > 0 : probability that s occurs; objective probabilities
Function g : S ! R + maps states of nature into monetary outcomes.
Every g(.) induces a lottery F (.) with F (x) =
∑ π s
fs:g (s)x g
Each g also represented by (x 1 ...x S ). Set of (nonnegative) g is R S +
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: rational preferences de…ned on R S +
De…nition: has an extended expected utility representation i¤ for every
s 2 S there is a function u s : R + ! R such that: (x 1 ...x S ) (x1 0...x S 0 ) if
and only if ∑ π s u s (x s ) ∑ π s u s (xs 0 ).
s2S
s2S
u s: state dependent utility function (before: state-independent or state
uniform utility functions)
Furthermore, we allow that within each state the monetary payo¤ is not a
certain amount, but a lottery with distribution F s (.).
Hence, alternative L = (F 1 , F 2 ....F S ).
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Extended independence: satis…es the extended independence axiom if
for all L, L 0 , L" and α 2 (0, 1), we have:
L L 0 i¤ αL + (1 α)L" αL 0 + (1 α)L"
Proposition: Suppose is rational, continuous and satis…es the extended
independence axiom. Then we can assign utility functions u s (.) for money
in every state s such that for any L = (F 1 , F 2 ...F S ) and L 0 = (F 0 1 , F 0 2 ...F 0 S )
we have:
Z
∑ π s (
s2S
L L 0 i¤
Z
u s (x s )dF s (x s )) ∑ π s (
s2S
u s (x s )dF 0
s (x s )).
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3. Subjective Probability Theory
Till now: objective probabilities
Now: no objective probabilities - uncertainty
Goal: Derivation of subjective probabilities from preferences
”The agent has preferences over the uncertain alternatives as if he
maximizes an expected utility function with a probability distribution that
looks like... ”
S: set of states without objective probabilities
g again represented by (x 1 ...x S ). Set of (nonnegative) g is R S +
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Again we allow that within each state the monetary payo¤ is not a certain
amount, but a lottery with distribution F s (.).
Hence, again alternative L = (F 1 , F 2 ....F S )
state dependent preferences = ( 1 , 2 , ... S )
Preferences are assumed to be rational, continuous, and satisfy the
extended independence axiom. Then we have u s (.) functions that are
Bernoulli functions for every state.
De…nition: The state preferences ( 1 , 2 , ... S ) are state uniform i¤
s = s 0for all s, s 0 2 S.
"The risk attitude towards money is the same in all states."
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Proposition: Suppose is rational, continuous, satis…es the extended
independence axiom, and is state uniform. Then there are probabilities
(π 1 , π 2 ...π S ) >> 0 and a Bernoulli function u(.) on amounts of money
such that for any (x 1 ...x S ) and(x1 0...x S 0 ) we have:
∑
s
(x 1 ...x s ) (x1...x 0 s 0 ) if and only if
π s u(x s ) ∑ π s u(xs 0 ).
s
Moreover, the probabilities are uniquely determined, and the utility
function is unique up to a scalar transformation.
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