Regions of rationality: Maps for bounded agents (Decision Analysis, in press)

1. Regions of rationality: Maps for bounded agents(Decision Analysis, in press) Robin M. Hogarth ICREA & Universitat Pompeu Fabra, Barcelona & Natalia Karelaia H.E.C., Université de Lausanne

2. “Regions of rationality” The starting point: • “heuristics and biases” (Kahneman, Slovic, & Tversky, 1982) • simple decision rules can rival the predictive ability of complex algorithms (e.g., regression) (e.g., TTB: Gigerenzer, Todd, & the ABC Research Group, 1999; EW: Dawes & Corrigan, 1974). Idea: • Attention as a scarce resource (Simon, 1978) -> how much information to seek & how to combine the pieces to make decisions in different “regions”: identify decision rules that are appropriate to each region • multiple-cue prediction (multi-attribute choice) • cues are probabilistically related to the criterion

3. A theoretical approach… • Effectiveness of several heuristics: the probability that the best of m alternatives (with k cues) is identified; the environmental conditions favoring various heuristics, e.g.: • differential weighting of cues • inter-correlations of cues • continuous/binary cues (c/b) • noise in the environment • interactions of these factors 2.Illustration: 20 “artificial” and 4 empirical environments

4. Models • Single Variable (SV) models • Lexicographic – SVc • Lexicographic – SVb • DEBA (binary cues) • Equal weight (EW) models 4. EWc 5. EWb • Hybrid models 6. EW/DEBA • EW/SVb • Domran (DR) models (lower benchmark) 8. DRc • DRb • Multiple regression (MR) (upper benchmark) 10. MRc 11. MRb

5. Method Single Variable, continuous cues - SVc • Choosing between A & B • Y = criterion and X = cue • Assume: Y and X are N(0,1), >0 = error, , N(0, ), • Question:

6. Prob {SVc chooses the best b/w A & B}

7. Prob {SVc chooses the best b/w A & B} Therefore, pdf = probability density function

8. Prob {SVc chooses the best from A, B, & C} - z1 and z2 are bivariate N

9. SVc: generalizing to the case of m alternatives (m>3) (m-1) between-alternative comparisons where

10. Overall probability of correct choice by SVc • Random sampling of m=3 from the underlying population of alternatives. • Either A, B, or C is chosen -> overall probability is: 3 P{((Xa>Xb) & (Xa>Xc))&((Ya>Yb)&(Ya>Yc))} integrated across : • where • , .

11. Overall probability of correct choice by SVc: generalizing to m>3 where

12. Other models: EWc & MRc Model: Error: Vd di*

13. Models with binary cues - SVb where Therefore,

14. Models with binary cues - SVb choosing 1 of 2 where

15. Models with binary cues - DEBA & Hybrids • Prob {a given alternative is chosen correctly}= the joint probability that the sequence of decisions (or eliminations) made at each stage is correct. • Three key notions: • Appropriate model for each stage • Partial correlations: and partial st. deviations: 3. Probability theory to calculate sequence of correct eliminations

16. Illustration: 20 “artificial” environments • Choosing the best from 2, 3, and 4 alternatives • n=40 k

17. Choosing the best from 3 Low inter-cue corr High inter-cue corr 3 cues 3 cues Low inter-cue corr High inter-cue corr 5 cues 5 cues

18. Some results (1) Similarity of models’ performance • agreement between models (average between all pairs, A-D)=63% (vs. 33.(3)% of random agreement), lower when lower inter-cue corr. • Model with continuous cues outperform their binary counterparts (except DR). • DRb > DRc. Choosing at random: DRb = in 51%, DRc = in 81%. • Larger inter-cue correlation reduces performance of all models (except SV).

19. Regression of model performance

20. Illustration: 4 empirical datasets 1)Golf all-around ranking, N=60 1. Birdie average (*-1) 2. Scoring average 3. Putting average  2)Golf earnings, N=60 1. Top 10 finishes 2. All-around ranking (*-1) 3. Consecutive cuts 3) PhD economics programs: ratings-1993, http://www.phds.org, N=107 1. # of PhDs for the academic year 87-88 to 91-92 2.Total # of program citations 88-92/ number program faculty 3. % Faculty with research support 4)Consumer reports:test score for digital cameras, http://sub.which.net,N=49 1. Image quality 2. Picture download time 3. Focusing

21. Illustration: empirical datasets

22. Golf earnings Golf ranking Economics PhD programs Consumer reports

23. Discussion • Our contributions • Analytical analysis • Regions of rationality: a multidimensional terrain • Further research & implications • Non-random sampling of alternatives • Hybrids with categorical & continuous variables • Different loss functions • Predicting consumer preferences • Bounded rationality and expertise: how do people build maps of their decision making terrain?