A Perfect Cocktail Recipe: Mixing Decision Theories and Models of Stochastic Choice

1. A Perfect Cocktail Recipe: Mixing Decision Theories and Models of Stochastic Choice Ganna Pogrebna June 29, 2007 Blavatskyy, Pavlo and Ganna Pogrebna (2007) “Models of Stochastic Choice and Decision Theories: Why Both are Important for Analyzing Decisions” IEW Working Paper 319

2. Talk Outline • Introduction • Television shows • Affari Tuoi • Deal or No Deal UK • Data • Estimated Models of Stochastic Choice and Decision Theories • Results

3. Introduction • Non-expected utility theories • Response to violations of EUT • Tested in laboratory experiments • Natural experiment in TV shows • More representative subject pool • Significantly higher incentives • Deal or No Deal • risky lottery vs. amount for certain • high stakes • dynamic problem

4. Affari Tuoi • Italian prototype of Deal or No Deal • Aired six days a week on RAI Uno • All contestants self-select into the show • 20 contestants participate in each episode • Contestants are randomly assigned sealed boxes, numbered from 1 to 20 • Each box contains one of twenty monetary prizes ranging from 0.01 to 500,000 EUR • Independent notary company allocates prizes across boxes and seals the boxes

5. Game • Contestants receive one multiple-choice general knowledge question • Contestant, who is the first to answer this question correctly, plays the game: • Contestant keeps her own box and opens the remaining boxes one by one • Once a box is opened, the prize sealed inside is publicly revealed and deleted from the list of possible prizes

6. List of Possible Prizes (Affari Tuoi ) *Prize 5,000 Euro was replaced with prize 30,000 Euro starting from January 30, 2006

7. Timing of the game Open 6 boxes “Bank” offers a price or an exchange (8 boxes remain unopened) Exchange own box for any of 13 remaining unopened boxes? Accept price Open 3 boxes Open 3 boxes “Bank” offers a price or an exchange (5 boxes remain unopened) “Bank” offers a price for contestant’s box (11 boxes remain unopened) Accept price Open 3 boxes Accept price Open 3 boxes “Bank” offers a price or an exchange (2 boxes remain unopened) Accept price Open 2 boxes

8. List of Possible Prizes (DOND UK )

9. Timing of the game Open 5 boxes “Bank” offers a price (8 unopened boxes left) “Bank” offers a price (17 unopened boxes left) Accept price Open 3 boxes Accept price Open 3 boxes “Bank” offers a price (14 unopened boxes left) “Bank” offers a price (5 unopened boxes left) Accept price Open 3 boxes Accept price Open 3 boxes “Bank” offers a price (11 unopened boxes left) “Bank” offers a price (2 unopened boxes left) Accept price Open 3 boxes Accept price Open 2 boxes

10. Data • 114 Affari Tuoi episodes • September 20, 2005 to March 4, 2006 • 234 Deal or No Deal UK episodes • October 31, 2005 to July 22, 2006 • Distribution of possible prizes, “bank” offers, prize in own box • Gender, age, marital status, region

11. “Bank” Offers, remarks • “Bank” monetary offers are fairly predictable across episodes • In early stages of the game, they are smaller than EV of possible prizes • As the game progresses, the gap between EV and the monetary offer decreases and often disappears when there are two unopened boxes left. • Offers do not depend on prize in contestants box

12. OLS Regression Results for Affari Tuoi and DOND UK

13. Models of Stochastic Choice • Trembles (Harless and Camerer, 1994) • Fechner Model of Homoscedastic Random Errors (Hey and Orme, 1994) • Fechner Model of Heteroscedastic Random Errors (Hey, 1995 and Buschena and Zilberman, 2000) • Fechner Model of Heteroscedastic and Truncated Random Errors (Blavatskyy, 2007) • Random Utility Model (Loomes and Sugden, 1995)

14. Trembles (Harless and Camerer, 1994) • Individuals generally choose among lotteries according to a deterministic decision theory • But there is a constant probability that this deterministic choice pattern reverses (as a result of pure tremble). • - vector of parameters that characterize the parametric form of a decision theory • - utility of a lottery L according to this theory.

15. Trembles, continued • LL of observing N decisions of contestants to reject an offer for a risky lottery , can be written as • where is an indicator function i.e. if x is true and if x is false, and is probability of a tremble.

16. Trembles, continued • LL of observing M decisions of contestants to accept offer for a risky lottery , can be written as • Parameters and p are estimated to maximize log-likelihood .

17. Fechner Model of Homoscedastic Random Errors (Hey and Orme, 1994) • H&O (1994) estimate a Fechner model of random errors • Where a random error distorts the net advantage of one lottery over another (in terms of utility) • Net advantage is calculated according to underlying deterministic decision theory • The error term is a normally distributed random variable with zero mean and constant standard deviation

18. Fechner Model of Homoscedastic Random Errors, continued • LL of observing N decisions of contestants to reject an offer for a risky lottery , can be written as • where is the cumulative distribution function (cdf) of a normal distribution with zero mean and standard deviation .

19. Fechner Model of Homoscedastic Random Errors, continued • LL of observing M decisions of contestants to accept offer for a risky lottery , can be written as • Parameters and are estimated to maximize log-likelihood

20. Fechner Model of Heteroscedastic Random Errors (Hey, 1995 and Buschena and Zilberman, 2000) • Assume that the error term is heteroscedastic • STDEV of errors is higher in certain decision problems, e.g. when lotteries have many possible outcomes • In DOND a natural assumption is that contestants, who face risky lotteries with a smaller range of possible outcomes, have a lower volatility of random errors than contestants, who face risky lotteries with a wider range of possible outcomes. • We estimate a Fechner model of random errors when the standard deviation of random errors is proportionate to the difference between the utility of the highest outcome and the utility of the lowest outcome of a risky lottery L.

21. Fechner Model of Heteroscedastic Random Errors, continued • LL of observing N decisions of contestants to reject an offer for a risky lottery , can be written as

22. Fechner Model of Heteroscedastic Random Errors, continued • LL of observing M decisions of contestants to accept offer for a risky lottery , can be written as • Parameters and are estimated to maximize log-likelihood .

23. Fechner Model of Heteroscedastic and Truncated Random Errors (Blavatskyy, 2007) • Truncate the distribution of random errors so that an individual does not commit transparent errors. • E.g. transparent error - an individual values a risky lottery > than its highest possible outcome for certain or when an individual values a risky lottery < than its lowest possible outcome for certain (known as a violation of the internality axiom). • In DOND a rational contestant would always reject an offer, which is < than the lowest possible prize remaining and accept an offer, which > the highest of the remaining prizes • But in Fechner model - a strictly positive probability that a contestant commits such transparent error. • To disregard such transparent errors, the distribution of heteroscedastic Fechner errors is truncated from above and from below.

24. Fechner Model of Heteroscedastic and Truncated Random Errors, continued • LL of observing N decisions of contestants to reject an offer for a risky lottery , can be written as

25. Fechner Model of Heteroscedastic and Truncated Random Errors, continued • LL of observing M decisions of contestants to accept offer for a risky lottery , can be written as • Parameters and are estimated to maximize log-likelihood .

26. Random Utility Model (Loomes and Sugden, 1995) • Individual preferences over lotteries are stochastic and can be represented by a random utility model. • Individual preferences over lotteries are captured by a decision theory with a parametric form that is characterized by a vector of parameters . • We will assume that one of the parameters is normally distributed with mean and standard deviation and the remaining parameters are non-stochastic.

27. Random Utility Model, continued • Let denote a value of parameter • Such that given other parameters , a contestant is exactly indifferent between accepting and rejecting an offer O for a risky lottery L • i.e. • and for all an individual prefers to accept an offer.

28. Random Utility Model, continued • LL of observing N decisions of contestants to reject an offer for a risky lottery , can be written as

29. Random Utility Model, continued • LL of observing M decisions of contestants to accept offer for a risky lottery , can be written as • Parameters , and are estimated to maximize log-likelihood .

30. 7 Decision Theories Embedded in Models of Stochastic Choice

31. Risk Neutrality (RN) • Maximize expected value (EV) • utility of a risky lottery that delivers outcome xi with probability piis • There are no free parameters to be estimated for this decision theory, i.e. vector θ is the empty set

32. Expected Utility Theory (EUT) • utility of lottery is • u is a (Bernoulli) utility function over money • We will estimate expected utility theory with two utility functions: constant relative risk aversion (CRRA) and expo-power (EP) • CRRA utility function is • (vector θ is just r ) • EP utility function is • (vector θ is )

33. Regret Theory (RT) and Skew-Symmetric Bilinear Utility Theory (SSB) • SSB: an individual chooses a risky lottery over a sure amount O if • where ψ is a skew-symmetric function • SSB coincides with regret theory if ψ is convex (assumption of regret aversion) • We will estimate RT (SSB) with function

34. RT and SSB, continued • This function satisfies assumption of regret aversion when δ>1 • When δ=1, RT(SBB)=EUT+CRRA • When r=0, RT(SBB)=CPT with current offer as a reference point, no loss aversion and linear prob. weighting • When δ=1 and r=0 RT(SBB)=RN • Vector θ is

35. Yaari’s Dual Model (YDM) • the utility of a risky lottery is • probability weighting function • vector consists only of one element—the coefficient of the probability weighting function .

36. Rank-Dependent Expected Utility Theory • the utility of a risky lottery is • probability weighting function • and CRRA utility function • vector consists of two elements— CRRA coefficient and the coefficient of the probability weighting function:

37. Disappointment Aversion Theory (DAT) • the utility of a risky lottery is • is a number of disappointing outcomes in lottery L • is a subjective parameter that captures disappointment preferences • vector consists of two elements— CRRA coefficient and the disappointment aversion parameter

38. Results • We estimate static and dynamic decision problem • In a static problem, an individual treats all remaining prizes as equiprobable • In a dynamic problem, an individual anticipates future bank offers • In both problems: • Result 1: Estimates of parameters of decision theories differ substantially, depending on which model of stochastic choice the theories are embedded in

39. Results (goodness of fit) • We use Vuong’s likelihood ratio test (and Clarke test) for non-nested models • Result 2: For every decision theory, the best fit to the data is obtained when this theory is embedded into a Fechner model with heteroscedastic truncated errors • Result 3: For every model of stochastic choice, the worst fit to the data is obtained when this model is combined with RN.

40. Results, dynamic vs. static • Tremble model yields the same LL in static and dynamic problems • Random utility model provides a better fit to the data in a dynamic rather than a static decision problem • No statistically significant difference for Fechner model • Estimated parameters are similar in a dynamic and a static problem but differ significantly across different models of stochastic choice

41. Results, UK static • Decision theories embedded in a tremble model perform significantly worse compared to other models • Decision theories embedded in a Fechner model with homoscedastic errors yield similar goodness of fit as in a random utility model • Best fit to the data: • RDEU embedded into a Fechner model with heteroscedastic errors • RDEU and EUT+EP in a truncated Fechner model

42. Results, AT dynamic • In a tremble model, for all theories that have RN as a special case, the estimates are the same as for RN • Mass point in „bank“ offers (2.2%=EV) • In a dynamic decision problem: • standard deviation of stochastic parameters in a random utility model tends to be higher • the variance of random errors in a Fechner model tends to be lower • Best fit to the data: • EUT+EP or RT (SSB) in a truncated Fechner model • EUT+EP or RT (SSB) in a random utility model

43. Results, UK dynamic • Tremble model—estimates are the same as in a static case (except for RT(SSB) and DAT) even w/o mass point • In a dynamic decision problem: • standard deviation of stochastic parameters in a random utility model tends to be higher • the variance of random errors in a Fechner model tends to be higher too (expect for YDM) • Best fit to the data: • RDEU or EUT+EP in a truncated Fechner model • RT (SSB) or RDEU in a random utility model

44. Conclusion • Correctly selected model of stochastic choice matters just as much as a correctly selected decision theory • Estimated parameters differ a lot • Best model of stochastic choice is a truncated Fechner model (and a random utility model in dynamic problems) • In this model, EUT performs not significantly worse than non-EUT • CRRA gives (nearly always) worse fit than EP