Attention and Revealed Preference in a Portfolio Choice Problem

1. Attention and Revealed Preference in a Portfolio Choice Problem Peter J. Hammond, Warwick Stefan Traub, Bremen Workshop on noise and imprecision in individual and interactive decision-making Warwick, April 16th-18th 2012

2. Outline • Motivation • Theory • The Experiment • Attention • Results • Conclusions

3. 1. Motivation • Several recent economic & psychological theories addressing the allocation of attention in choice tasks in order to explaining errors, noise, and choice anomalies • Sims (2003) Theory of Rational Inattention, tested experimentally by Cheremukhin et al. (2011) • Gabaix, Laibson [et al.] (2000, 2006) Directed Cognition Model • Townsend, Busemeyer (1989), Busemeyer, Diederich (2002) Decision Field Theory • DePalma et al. (1994); Hensher et al. (2005); Cameron, DeShazo (2011); ... 3

4. 1. Motivation • Our research • Question: Is attention related to choice “quality” • Method: Choice experiment with recorded attention • Setup: Portfolio selection problem, 16 rounds with up to 3 stages • Measurement of Attention: Record the time that subjects “activate” a certain portfolio in each task • Measurement of Choice Quality: Check for first-order stochastic dominance and GARP consistency 4

5. 2. Theory • First-order stochastic dominance • Two Arrow securities (A, B), budget constraint e=pAxA+ pBxB • Safe portfolio (xA=xB) • Segment of budget line with stochastically dominated portfolios Fig. Safe portfolio and first-order stochastic dominance. Portfolios on the red part of the budget line are stochastically dominated 5

6. 2. Theory • Varian’s (1982, 2006) supporting set • Sequence of n choices • Supporting set predicts choice n+1 (if rational in terms of GARP) Fig. Example of the supporting set of a three-stage choice task. 6

7. 3. The Experiment • University of Warwick, 41 non-economics undergraduates • portfolio investment problem as in Choi et al. (2007) • 100 tokens to be allocated between 2 Arrow securities • 1 token = £ 0.2 • 2 probabilities: identical (1/2:1/2) or asymmetric (2/3:1/3) • 8 different p’s (exchange rates between tokens and securities) (1, 1.5), (2, 1), (1, 2.5), (3, 1), (1.5, 2), (2.5, 1.5), (3, 1.5), (2, 3) • 16 decision tasks with up to 3 stages • 1. stage: 16 possible combinations of price vectors and probabilities • 2. and 3. stage: individualized choices constructed from 1st stage choices (if possible)

8. 3. The Experiment • time constraint (30 seconds per choice) • payoff: £ 5 show up fee + one randomly selected choice (account) • altogether ~ £ 500 • written instructions + individual training sessions

9. 3. The Experiment Fig. First-stage choice problem. 9

10. 3. The Experiment Fig. Second-stage choice problem. 10

11. 3. The Experiment Fig. Third-stage choice problem. 11

12. 3. The Experiment Supporting set Fig. Supporting set. 12

13. 4. Attention Fig. Example of attention data for an arbitrary subject.

14. 4. Attention Fig. Example of attention data for an arbitrary subject. Details for Round 3, Choices 1-3. Time units ~50ms 1. stage 2. stage 3. stage Legend: Blue circles represent portfolios increasingly ordered by the number of tokens allocated to account B. The height of each peak specifies the number of time units (of 50ms) the respective portfolio was activated. Green markers represent actual choices. Red markers represent the left and right limit portfolios of the GARP supporting set.

15. 4. Attention Time units ~50ms Allocation of tokens to B account Fig. Construction of a histogram of the distribution of attention. Used for the computation of mean, coefficient of variation, skewness and kurtosis of the distribution of attention.

16. 5. Results Fig. Boxplot Meantime by Dominance. • Choice-based analysis • 1588 valid choices • 303 (19%) dominated • Meantime (average attention per portfolio in terms of tu’s including zeros • 3.84 (undominated) vs. 4.79 (dominated) , p<.001 • if dominated portfolio chosen, subject spent about 0.1 sec longer on each available portfolio • includes “zeros” (portfolios that did not receive any attention) • excluding zeros does not change results!

17. 5. Results Fig. Boxplots for Coefficient of Variation, Skewness, and Kurtosis by Dominance. All T-tests significant (p<.001).

18. 5. Results • Choice-based analysis • Does the shape of the density function of attention predict dominance violations? Logistic regression Number of obs = 1588 LR chi2(4) = 28.58 Prob > chi2 = 0.0000 Log likelihood = -759.68494 Pseudo R2 = 0.0185 ------------------------------------------------------------------------------ domi | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- meantime | .0484309 .0182372 2.66 0.008 .0126865 .0841752 cvartime | .1164184 .1340328 0.87 0.385 -.1462812 .3791179 skewtime | .3711099 .258087 1.44 0.150 -.1347314 .8769512 kurttime | -.067524 .0261719 -2.58 0.010 -.11882 -.016228 _cons | -2.167672 .4683899 -4.63 0.000 -3.085699 -1.249645 ------------------------------------------------------------------------------ Tab. Logistic regression. Pooled Sample. Endogenous Variable: Chosen portfolio is dominated (domi=1).

19. 5. Results • Choice-based analysis • Does the shape of the density function of attention predict dominance violations? • Yes, but with many reservations: • Variance, Skewness, Kurtosis highly correlated (rho>0.9) • Very little variance explained (<2%) • Strong gender effect (explains about 5% of the variance) • Gender, however, is not correlated with the moments (mean, kurtosis) • Including subjects & time fixed effects in order to take account panel structure: Mean insignificant, Higher moments still significant 19

20. 5. Results • Subject-based analysis • Distribution function of attention aggregated over up to 48 choices • Endogenous variable GARP consistency (based on a nonparametric test by Hammond, Traub (2012)) • Logistic regression: insignificant (low n!) Fig. Boxplots for Coefficient of Variation, Skewness, and Kurtosis by Rational (GARP consistency). All T-tests significant (p<.001). 20

21. 6. Conclusions • (Preliminary) • As it seems, subjects who performed worse • required more time per choice alternative (mean) • exhibited lower variance, skewness, kurtosis (-> less “extreme” time allocation) • Further questions: • Better measures of the distribution of attention than central moments (censored, zeros, ...) • ... 21

22. Conference on Noisy Models of Behaviour, University of Warwick A Stochastic Process Model of Attention, then Choice, Inspired by Decision Field Theory Peter J. Hammond, with Stefan Traub April 18, 2012

23. Conference on Noisy Models of Behaviour, University of Warwick Stochastic Choice Model

24. Conference on Noisy Models of Behaviour, University of Warwick Choice and Attention Data

25. Conference on Noisy Models of Behaviour, University of Warwick Stochastic Process of Attention, then Choice

26. Conference on Noisy Models of Behaviour, University of Warwick Markov Transition and Switching Probabilities

27. Conference on Noisy Models of Behaviour, University of Warwick New Experiments The task we face now is to try to devise a new experiment, and revise the software that records the attention process, in order to begin investigating empirically this version of a DFT process. More data on expectations?