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Stochastic Dominance

Stochastic Dominance. Scott Matthews Courses: 12-706 / 19-702. 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.

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Stochastic Dominance

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  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 12-706 and 73-359

  3. HW 4 Results • Average: 47; Median: 52 • Max: 90 • Standard deviation: 25 (!!) • Gave easy 5 pts for Q19 also • Show sanitized XLS 12-706 and 73-359

  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 FA(z) ≤ FB(z) for all z 12-706 and 73-359

  5. Stochastic Dominance:Example #1 • CRP below for 2 strategies shows “Accept $2 Billion” is dominated by the other. 12-706 and 73-359

  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 12-706 and 73-359

  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 12-706 and 73-359

  8. FOSD 12-706 and 73-359 Source: http://www.nes.ru/~agoriaev/IT05notes.pdf

  9. Option A Option B FOSD Example 12-706 and 73-359

  10. 12-706 and 73-359

  11. 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).. 12-706 and 73-359

  12. Option A Option B SOSD Example 12-706 and 73-359

  13. Area 2 Area 1 12-706 and 73-359

  14. SOSD 12-706 and 73-359

  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) 12-706 and 73-359

  16. 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”. 12-706 and 73-359

  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) 12-706 and 73-359

  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”? 12-706 and 73-359

  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 12-706 and 73-359

  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 12-706 and 73-359

  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 12-706 and 73-359

  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 12-706 and 73-359

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