1 Stochastic Dominance Scott Matthews Courses: /

and Admin Issues  HW 4 back today  No Friday class this week – will do tutorial in class

HW 4 Results  Average: 47; Median: 52  Max: 90  Standard deviation: 25 (!!)  Gave easy 5 pts for Q19 also  Show sanitized XLS and

4 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 F A (z) ≤ F B (z) for all z

and Stochastic Dominance: Example #1  CRP below for 2 strategies shows “Accept $2 Billion” is dominated by the other.

and 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

and 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: F A (x) is cdf of A, F B (x) is cdf of B.  F B (x) ≥ F A (x) for all x, with one strict inequality  or.. for any non-decr. U(x), ∫U(x)dF A (x) ≥ ∫U(x)dF B (x)  Expected value of A is higher than B

and FOSD Source:

and FOSD Example  Option A  Option B Profit ($M)Prob. 0 ≤ x < ≤ x < ≤ x < ≤ x < Profit ($M)Prob. 0 ≤ x < 50 5 ≤ x < ≤ x < ≤ x < ≤ x < 250.1

and

and 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)..

and SOSD Example  Option A  Option B Profit ($M)Prob. 0 ≤ x < ≤ x < ≤ x < ≤ x < Profit ($M)Prob. 0 ≤ x < ≤ x < ≤ x < ≤ x < ≤ x < 250.1

and Area 2 Area 1

and SOSD

and 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)

and 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 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”.

and 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)

and 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”?

and 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

and 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

and 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

and 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