Stochastic Dominance

1
Stochastic Dominance

• Scott Matthews
• Courses 12-706 / 19-702

2
Admin Issues

• HW 4 back today
• No Friday class this week will do tutorial in
  class

3
HW 4 Results

• Average 47 Median 52
• Max 90
• Standard deviation 25 (!!)
• Gave easy 5 pts for Q19 also
• Show sanitized XLS

4
Stochastic Dominance Defined

• A is better than B if
• Pr(Profit gt z A) Pr(Profit gt 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

5
Stochastic DominanceExample 1

• CRP below for 2 strategies shows Accept 2
  Billion is dominated by the other.

6
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

7
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

8
FOSD
Source http//www.nes.ru/agoriaev/IT05notes.pdf
9
FOSD Example

• Option A

• Option B

10
(No Transcript)
11
Second-Order Stochastic Dominance (SOSD)

• How to compare 2 lotteries based on risk
• Given lotteries/distributions w/ same mean
• So were 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)..

12
SOSD Example

• Option A

• Option B

13
Area 2
Area 1
14
SOSD
15
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)

16
Subjective Probabilities

• Main Idea We all have to make personal
  judgments (and decisions) in the face of
  uncertainty (Granger Morgans career)
• These personal judgments are subjective
• Subjective judgments of uncertainty can be made
  in terms of probability
• Examples
• 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.

17
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)

18
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?

19
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

20
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

21
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

22
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