Lecture 3 Axioms & elicitation

Contents Axioms Elicitation Quick recap ”But we just studied these…” -> these are vital to understanding! Elicitation How can we elicit someone’s utility function? What does a utility function look like?

Axioms Translation We can compare any two alternatives in utility If 10 > 5 and 5 > 2, then 10 > 2 Lotteries of consequences make sense Reduction of compound events Sure-thing principle No ”joy in gambling” Utility is bounded 15.11.2018

If the axioms are satisfied… We can model alternatives with utilities The optimal or rational choice is maximizing expected utility Utility is an interval scale Size of unit is arbitrary Zero point is arbitrary You can question whether people truly behave like the axioms describe (lecture 4 onwards) 15.11.2018

Violation of solvability Continuity It has been argued that with extreme outcomes (such as death), one alternative is strictly preferred to the other, unless p = 1. Counterexample: 15.11.2018

Violation of monotonicity The axiom implies that, for two uncertain events with identical payoffs, an individual’s preference will depend only on the probability of winning. Suppose that: C represents a flip of a coin today with a payoff in 1 year, and D represents a flip of the same coin 1 year from now with immediate payoff; Would you still be indifferent? On the other hand, are the outcomes really the same? 15.11.2018

Violation of substitution P=0.10 P=0.01 P=0.89 A 1M B 5M P=0.10 P=0.01 P=0.89 A’ 1M B’ 5M 15.11.2018

Substitution - continued A frequent choice pattern is A and B’! We have: A = 0.11 1M + 0.89 1M B = 0.10 5M + 0.89 1M A’ = 0.11 1M B’ = 0.10 5M This problem is called Allais’ paradox 15.11.2018

Problem of payoffs What are actually the outcomes or payoffs? Are these the same[3]: [3] Sen, A. (1997). Maximization and the Act of Choice. Econometrica, vol. 65(4) pp. 745-779 15.11.2018

Elicitation 15.11.2018

Interlude: what is probability? Frequency The number of events in a sequence, taken to the limit Naturally, no infinite sequences exist So typically estimated from data Example Of all the courses you have taken, how many times did you get a grade 5? Subjective Your subjective opinion on how likely something is Can be influenced by data/experience, doesn’t have to be Hard to show someone is wrong! Example How likely is it that you will get a grade 5 on this course?

The components of a decision What we need to elicit: Probabilities How likely is it that you receive consequence A…B…C…N? Utilities How desirable is consequence A…B…C…N? 15.11.2018

Components example Suppose you have a few extra euros. Should you Play the lottery Buy a coffee Save the money p U EU Play lottery 0.01 100 1 Buy coffee 0.99 10 9.9 Save 3 15.11.2018

Encoding methods Self-elicitation versus interviewer elicitation 15.11.2018 Encoding methods Self-elicitation versus interviewer elicitation Direct questioning versus inferring from choices The process (interviewer recommended!) Motivate the subject and investigate the subject’s biases Structure the uncertain quantity by having it clearly defined Make the subject think fundamentally about the problem and avoid biases (if possible) Encode the probability judgments Verify the responses by checking for consistency (multiple estimations)

Chesley’s taxonomy Elicitation approach Who elicits? Direct Indirect (or hybrid) Self Interviewer 15.11.2018

Direct asking techniques 15.11.2018 Direct asking techniques Magnitude or direct estimation What is p(x<5)? What is k such that p(x < k) =0.8? Odds method Estimate the ratio of two probabilities: what are the odds that boxer A will win boxer B? (e.g. 2-to-1 odds = 66% vs 33% probability) Quartile or fractile assessment Seeks to find equally likely points Quartile assessments can be made by three splits of the range of the distribution

Direct asking techniques 15.11.2018 Direct asking techniques Graphical techniques Histograms or smooth curves; either ask the individual to provide a graphical distribution or solicit his or her reaction to a given graph Direct specification of the distribution parameters Make assumptions about the distribution (for example normal), assess its parameters (mean and variance). Note: difficulty of assessing variances!

Indirect methods Subject makes a set of bets Difficult to do properly! Example: tumor probability [1] Difficult to do properly! We need the utility function over the prizes to do this! [1] https://dspace.library.uu.nl/bitstream/handle/1874/858/c5.pdf (p. 120) 15.11.2018

Hybrid: Lotteries, Bids Lottery Replacing p estimation with lottery Given lotteries ApB and CqD, which would you pick? Repeat with different values to find estimate for p Bid Given a gamble ApB Asking: how much would you pay to take part? 15.11.2018

Recommendations Unfortunately there is no ”one best method” Depends on what you and the DM is comfortable with Some methods (quartiles, parameters) benefit from statistics skills of the DM Using an interviewer and consistency checks is a good idea Use density functions, not cumulative distribution functions! 15.11.2018

Example: which distribution does the CDF represent? 15.11.2018

Exponential Normal Gamma 15.11.2018

Properties of a good elicitation process Two measurement issues: reliability and validity Reliability: relatively free of random error, repeatable, stable, and consistent Validity: accurately represents the person’s opinion Ease of use Not too much cognitive effort for the DM Not too long 15.11.2018

Next lecture Behavioral decision making! “On traditional economic theory: We do not play chess as if we were a grandmaster, invest as if we were Warren Buffett, or cook like an Iron Chef. It is more likely we cook like Warren Buffett, who loves to eat at Dairy Queen.” - Richard Thaler (Nobel Prize in Economics 2017) 15.11.2018