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Stochastic Dominance

Scott MatthewsCourses: 12-706 / 19-702

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Admin Issues

HW 4 back todayNo Friday class this week – will do

tutorial in class

HW 4 Results

Average: 47; Median: 52Max: 90 Standard deviation: 25 (!!)

Gave easy 5 pts for Q19 alsoShow sanitized XLS

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Stochastic Dominance “Defined”

A is better than B if:Pr(Profit > $z |A) ≥ Pr(Profit > $z |B),

for all possible values of $z.Or (complementarity..)Pr(Profit ≤ $z |A) ≤ Pr(Profit ≤ $z |B),

for all possible values of $z.A FOSD B iff FA(z) ≤ FB(z) for all z

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Stochastic Dominance:Example #1

CRP below for 2 strategies shows “Accept $2 Billion” is dominated by the other.

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Stochastic Dominance (again)

Chapter 4 (Risk Profiles) introduced deterministic and stochastic dominance We looked at discrete, but similar for continuous How do we compare payoff distributions? Two concepts: A is better than B because A provides unambiguously higher

returns than B A is better than B because A is unambiguously less risky than B If an option Stochastically dominates another, it must have a

higher expected value

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First-Order Stochastic Dominance (FOSD)

Case 1: A is better than B because A provides unambiguously higher returns than B Every expected utility maximizer prefers A to B (prefers more to less) For every x, the probability of getting at least x is higher

under A than under B. Say A “first order stochastic dominates B” if:

Notation: FA(x) is cdf of A, FB(x) is cdf of B. FB(x) ≥ FA(x) for all x, with one strict inequality or .. for any non-decr. U(x), ∫U(x)dFA(x) ≥ ∫U(x)dFB(x) Expected value of A is higher than B

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FOSD

Source: http://www.nes.ru/~agoriaev/IT05notes.pdf

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FOSD Example

Option A Option B

Profit ($M) Prob.

0 ≤ x < 5 0.2

5 ≤ x < 10 0.3

10 ≤ x < 15

0.4

15 ≤ x < 20

0.1

Profit ($M) Prob.

0 ≤ x < 5 0

5 ≤ x < 10 0.1

10 ≤ x < 15

0.5

15 ≤ x < 20

0.3

20 ≤ x < 25

0.1

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First-Order Stochastic Dominance

00.20.40.60.8

1

0 5 10 15 20 25

Profit ($millions)

Cumulative Probability

AB

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Second-Order Stochastic Dominance (SOSD)

How to compare 2 lotteries based on risk Given lotteries/distributions w/ same mean

So we’re looking for a rule by which we can say “B is riskier than A because every risk averse person prefers A to B”

A ‘SOSD’ B if For every non-decreasing (concave) U(x)..

€

U(x)dFA (x)0

x

∫ ≥ U(x)dFB (x)0

x

∫

€

[FB (x) − FA (x)]dx0

x

∫ ≥ 0,∀x

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SOSD Example

Option A Option B

Profit ($M) Prob.

0 ≤ x < 5 0.1

5 ≤ x < 10 0.3

10 ≤ x < 15

0.4

15 ≤ x < 20

0.2

Profit ($M) Prob.

0 ≤ x < 5 0.3

5 ≤ x < 10 0.3

10 ≤ x < 15

0.2

15 ≤ x < 20

0.1

20 ≤ x < 25

0.1

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Second-Order Stochastic Dominance

00.20.40.60.8

1

0 5 10 15 20 25

Profit ($millions)

Cumulative Probability

AB

Area 2

Area 1

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SOSD

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SD and MCDM

As long as criteria are independent (e.g., fun and salary) then Then if one alternative SD another on

each individual attribute, then it will SD the other when weights/attribute scores combined

(e.g., marginal and joint prob distributions)

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Subjective Probabilities

Main Idea: We all have to make personal judgments (and decisions) in the face of uncertainty (Granger Morgan’s career) These personal judgments are subjective Subjective judgments of uncertainty can be

made in terms of probabilityExamples:

“My house will not be destroyed by a hurricane.”

“The Pirates will have a winning record (ever).” “Driving after I have 2 drinks is safe”.

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Outcomes and Events

Event: something about which we are uncertain Outcome: result of uncertain event

Subjectively: once event (e.g., coin flip) has occurred, what is our judgment on outcome? Represents degree of belief of outcome Long-run frequencies, etc. irrelevant - need one Example: Steelers* play AFC championship

game at home. I Tivo it instead of watching live. I assume before watching that they will lose.

*Insert Cubs, etc. as needed (Sox removed 2005)

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Next Steps

Goal is capturing the uncertainty/ biases/ etc. in these judgments Might need to quantify verbal expressions

(e.g., remote, likely, non-negligible..)What to do if question not answerable

directly? Example: if I say there is a “negligible”

chance of anyone failing this class, what probability do you assume?

What if I say “non-negligible chance that someone will fail”?

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Merging of Theories

Science has known that “objective” and “subjective” factors existed for a long time

Only more recently did we realize we could represent subjective as probabilities

But inherently all of these subjective decisions can be ordered by decision tree Where we have a gamble or bet between what

we know and what we think we know Clemen uses the basketball game gamble

example We would keep adjusting payoffs until optimal

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Continuous Distributions

Similar to above, but we need to do it a few times. E.g., try to get 5%, 50%, 95% points on

distribution Each point done with a “cdf-like” lottery

comparison

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Danger: Heuristics and Biases

Heuristics are “rules of thumb” Which do we use in life? Biased? How?

Representativeness (fit in a category)Availability (seen it before, fits memory)Anchoring/Adjusting (common base

point)Motivational Bias (perverse incentives)Idea is to consider these in advance and

make people aware of them

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Asking Experts

In the end, often we do studies like this, but use experts for elicitation Idea is we should “trust” their

predictions more, and can better deal with biases

Lots of training and reinforcement steps But in the end, get nice prob functions