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1 Stochastic Dominance Scott Matthews Courses: 12-706 / 19-702
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12-706 and 73-3592 Admin Issues HW 4 back today No Friday class this week – will do tutorial in class
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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 73-3593
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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
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12-706 and 73-3595 Stochastic Dominance: Example #1 CRP below for 2 strategies shows “Accept $2 Billion” is dominated by the other.
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12-706 and 73-3596 Stochastic Dominance (again) Chapter 4 (Risk Profiles) introduced deterministic and stochastic dominance We looked at discrete, but similar for continuous How do we compare payoff distributions? Two concepts: A is better than B because A provides unambiguously higher returns than B A is better than B because A is unambiguously less risky than B If an option Stochastically dominates another, it must have a higher expected value
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12-706 and 73-3597 First-Order Stochastic Dominance (FOSD) Case 1: A is better than B because A provides unambiguously higher returns than B Every expected utility maximizer prefers A to B (prefers more to less) For every x, the probability of getting at least x is higher under A than under B. Say A “first order stochastic dominates B” if: Notation: F A (x) is cdf of A, F B (x) is cdf of B. F B (x) ≥ F A (x) for all x, with one strict inequality or.. for any non-decr. U(x), ∫U(x)dF A (x) ≥ ∫U(x)dF B (x) Expected value of A is higher than B
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12-706 and 73-3598 FOSD Source: http://www.nes.ru/~agoriaev/IT05notes.pdf
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12-706 and 73-3599 FOSD Example Option A Option B Profit ($M)Prob. 0 ≤ x < 50.2 5 ≤ x < 100.3 10 ≤ x < 150.4 15 ≤ x < 200.1 Profit ($M)Prob. 0 ≤ x < 50 5 ≤ x < 100.1 10 ≤ x < 150.5 15 ≤ x < 200.3 20 ≤ x < 250.1
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12-706 and 73-35910
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12-706 and 73-35911 Second-Order Stochastic Dominance (SOSD) How to compare 2 lotteries based on risk Given lotteries/distributions w/ same mean So we’re looking for a rule by which we can say “B is riskier than A because every risk averse person prefers A to B” A ‘SOSD’ B if For every non-decreasing (concave) U(x)..
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12-706 and 73-35912 SOSD Example Option A Option B Profit ($M)Prob. 0 ≤ x < 50.1 5 ≤ x < 100.3 10 ≤ x < 150.4 15 ≤ x < 200.2 Profit ($M)Prob. 0 ≤ x < 50.3 5 ≤ x < 100.3 10 ≤ x < 150.2 15 ≤ x < 200.1 20 ≤ x < 250.1
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12-706 and 73-35913 Area 2 Area 1
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12-706 and 73-35914 SOSD
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12-706 and 73-35915 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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12-706 and 73-35916 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”.
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12-706 and 73-35917 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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12-706 and 73-35918 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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12-706 and 73-35919 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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12-706 and 73-35920 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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12-706 and 73-35921 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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12-706 and 73-35922 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
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