Transcript Weighted Almost Stochastic Dominance

Weighted Almost Stochastic
Dominance
and Expected Utility Theory
TAN Chin Hon
Department of Industrial and Systems Engineering
National University of Singapore
Motivation
O Decision making under uncertainty is generally
challenging.
O This problem quickly grows in difficulty as the number of
choices increase.
O The ill effects of having too many choices is described in
The Paradox of Choice: Why Less is More by Barry
Schwartz.
O We seek a simple rule that can be used to appropriately
reduce a large number of choices to a smaller
manageable number.
Expected Utility Theory
Four reasonable assumptions:
O
Completeness: Either 𝑋 ≻ 𝑌, 𝑌 ≻ 𝑋 or 𝑋~𝑌
O
Transitivity: If 𝑋 ≻ 𝑌 ≻ 𝑍, 𝑋 ≻ 𝑍
O
Independence: If 𝑋 ≻ 𝑌, 𝜌𝑋 + 1 − 𝜌 𝑍 ≻ 𝜌𝑌 + 1 − 𝜌 𝑍
O
Continuity: If 𝑋 ≻ 𝑌 ≻ 𝑍, there exists 𝜌 such that 𝜌𝑋 + 1 − 𝜌 𝑍~𝑌
Expected Utility Theory
O Expected utility theory: The preference of an
individual that satisfies the four axioms
above can be represented by a utility
function:
𝑋≽𝑌⇔𝐸 𝑢 𝑋
≥𝐸 𝑢 𝑌
O Practical challenge: Utility function of
individuals are unknown in practice.
Observations
O Preferences are generally robust to minor deviations in the
utility function.
O Individuals often display the same preference when
presented with mutually exclusive choices.
O For example, assuming that the economy is equally likely to
be good or poor, most people prefer X over Y in the problem
below:
Investment
Good Economy
Poor Economy
X
$2,000,000
-$210,000
Y
-$200,000
$850,000
Weighted Almost Stochastic
Dominance
Weighted almost stochastic
dominance (WASD) determines the
robustness of a preference with
respect to deviations in utility.
Weighted Almost Stochastic
Dominance
O Suppose we are comparing between two uncertain
prospects X and Y.
O 5 step procedure:
O Estimate the utility function of the decision maker.
O Compute the estimated marginal utility of the decision
maker.
O Adjust the cumulative density functions (CDFs)
O Compute the area between the two adjusted CDFs.
O Compute the parameter 𝜀 as follows:
𝜀=
Area where 𝐹𝑋 is above 𝐹𝑌
Area between 𝐹𝑋 and 𝐹𝑌
Example
O Consider the following investment problem:
O Suppose that the decision maker’s utility is estimated to be:
𝑢 𝑡 = 0.5𝑡
O The estimated marginal utility is:
𝑢′ 𝑡 = 0.5
Example
O The CDFs of X and Y are illustrated below:
1
0.5
-210K
-200K
850K
2M
Example
O Multiply by 0.5 to obtain adjusted CDFs of X and Y :
0.5
287.5K
0.25
2.5K
-210K
-200K
850K
𝜀=
2.5
= 0.00862
2.5 + 287.5
2M
Marginal Utility Deviation
O
WASD states that 𝐸 𝑢 𝑋
≥𝐸 𝑢 𝑌
as long as marginal utility deviates
from estimated marginal utility by less than a factor of
O
For this example, 𝑢′ 𝑡 = 0.5 and 𝜀 = 0.00862.
O
Therefore, 𝐸 𝑢 𝑋
≥𝐸 𝑢 𝑌
1
𝜀
− 1.
as long as:
0.5
1
−1
0.00862
≤ 𝑢′ ≤ 0.5
1
−1
0.00862
which is equivalent to 0.047 ≤ 𝑢′ ≤ 5.36.
O
In particular, the smaller the value of 𝜀, the larger the robustness.
Reference
For technical details and further information,
see:
Tan, C. 2015. Weighted Almost Stochastic
Dominance: Revealing the Preferences of Most
Decision Makers in the St. Petersburg
Paradox. Decision Analysis, Published online.