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Stochastic Dominance

This guide explores the concept of Stochastic Dominance (SD) in decision-making, particularly in evaluating risky alternatives. It delves into First-Order Stochastic Dominance (FOSD) and Second-Order Stochastic Dominance (SOSD) principles, providing clear definitions and examples. The document illustrates how to compare payoff distributions and make informed choices under uncertainty. Additionally, it discusses subjective probabilities and their impact on decisions, equipping you with essential tools for effective risk assessment and management in variations of lotteries and investment strategies.

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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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