1. Small Decision-Making under Uncertainty and Risk Takemi Fujikawa University of Western Sydney, Australia Agenda: Introduction Experimental Design Experiment 1 Experiment 2 Conclusion

2. Introduction This presentation attempts to:  examine behavioural tendency in “Small Decision-Making (SDM)” problems  present results of two experiments on SDM problems  introduce the search-assessment model  introduce EU model for SDM problems

3. What are SDM problems? SDM problems involve repeated tasks: The decision makers (DMs) face repeated-play choice problems. Each single choice is trivial: It has very similar but fairly small EV. Little time and effort is typically invested in SDM problems The DMs have to rely on the feedback obtained in the past decisions. Introduction

4. Search treatment (Experiment 1)  Experiment 1 was conducted without giving subjects prior information on payoff structure.  To construct the search-assessment model Choice treatment (Experiment 2)  Experiment 2 was conducted with giving subjects prior information on payoff structure.  To construct EU model Experimental Design

5. Experiment 1 and Experiment 2 were conducted in order at Kyoto Sangyo University Experimental Economics Laboratory (KEEL). Forty-two undergraduates at KSU served as paid subjects and participated in both experiments. Subjects received cash contingent upon performance (i.e., points they earned). Exchange rate: 1 point = 0.3 Yen (0.25 US cent). Experimental Design

6. Choice Problems Each experiment consists of Problem 1, 2 and 3. Each problem consists of 400 rounds. Subjects are asked to choose either H or L 400 times. In each round t (t=1, 2, …, 400), subjects are asked to choose either H or L. Experimental Design Problem 1 H:4 (0.8);0 (0.2) L:3 (1) Problem 3 H:32(0.1);0 (0.9) L:3(1) Problem 2 H:4 (0.2);0(0.8) L:3 (0.25);0(0.75)

7. Experiment 1: Search in SDM problems Subjects in Experiment 1 are NOT informed of payoff structure.

8. Experimental screen Basic task in each problem was a binary choice between two buttons for 400 times without giving subjects prior information on payoff structure. Experiment 1 Problem 1 H:4 (0.8);0 (0.2) L:3 (1) Problem 3 H:32(0.1);0 (0.9) L:3(1) Problem 2 H:4 (0.2);0(0.8) L:3 (0.25);0(0.75)

9. Results of Experiment 1 choiceH: “The mean proportions of H choices”. For example, if she has chosen H 100 out of 400 times, then choiceH is 0.25. posteriorH: “The posterior average points of H”. For example, if she chose H 10 times in Problem 1 and unluckily has got 24 pts, then posteriorH is 2.4 (=24/10). Note that posteriorH may or may not be the same as EV(H). Problem 1 H:4 (0.8);0 (0.2) L:3 (1) Experiment 1

10. Problem 1 ( choiceH =0.48) H:4 (0.8);0 (0.2) L:3 (1) Results: choiceH Problem 2 ( choiceH =0.55) H:4 (0.2);0(0.8) L:3 (0.25);0(0.75) Problem 3 ( choiceH =0.22) H:32(0.1);0 (0.9) L:3(1) Experiment 1

11. The tendency to select best reply to past outcomes Problem 3 ( choiceH =0.22) H:32(0.1);0 (0.9) L:3(1) Experiment 1 After the first 100 trials, posteriorH has become around 1.6. Then, subjects may have judged subjectively that EV(H)  1.6 and EV(H)<EV(L)

12. Analysis Subjects are undisclosed payoff structure in Experiment 1. In Experiment 1, the information available to subjects is limited to feedback about outcomes of their previous decisions. Subjects are required to discover payoff structure by trying both alternatives as they are undisclosed payoff distribution. Experiment 1

13. The search-assessment model Recall that only one alternative includes uncertain prospect Problem 1 and 3. To investigate Problem 1 and 3, the following Problem A is examined. Suppose each DM in Problem A is asked to choose either H or L at each round t (t=1,2, …, 400). Problem A H: x ( p );0(1- p ) L:1(1) where 0 1. Problem 1 ( choiceH =0.48) H:4 (0.8);0 (0.2) L:3 (1) Problem 3 ( choiceH =0.22) H:32(0.1);0 (0.9) L:3(1) Experiment 1

14. If she chooses H m times and gets an outcome of “x” k times, then her posteriorH is greater than or equal to 1, which is EV(L), with the probability P(H m ): This allows us to analyse the number of H choices required for judging that EV(H)>EV(L). Experiment 1 Problem A H: x ( p );0(1- p ) L:1(1) where 0 1. Problem 1 ( choiceH =0.48) H:4 (0.8);0 (0.2) L:3 (1) Problem 3 ( choiceH =0.22) H:32(0.1);0 (0.9) L:3(1)

15. P(H m ) for Problem 3 Problem A H:x(p);0(1-p) L:1(1) where 0 1. Problem 3 (choiceH=0.22) H:32(0.1);0 (0.9) L:3 (1) P(H m ) is calibrated by setting p=0.1 and x=32/3. Calibration implies the probability that posteriorH>3 does not exceed 0.98 until H is chosen 10,000 times in Problem 3. Experiment 1

16. Experiment 2: Choice in SDM problems Subjects in Experiment 2 are clearly disclosed payoff structure.

17. Experimental screen Basic task in each problem was a binary choice between two buttons for 400 times with prior information on payoff structure. Problem 1 H:4 (0.8);0 (0.2) L:3 (1) Problem 3 H:32(0.1);0 (0.9) L:3(1) Problem 2 H:4 (0.2);0(0.8) L:3 (0.25);0(0.75) Experiment 2

18. Results: choiceH Problem 1 ( choiceH =0.63) H:4(0.8);0(0.2) L:3(1) Problem 2 ( choiceH =0.69) H:4(0.2);0(0.8) L:3(0.25);0(0.75) Problem 3 ( choiceH =0.4) H:32(0.1);0(0.9) L:3(1) Experiment 2

19. Analysis Is it a optimal decision for risk-averse DM to choose both H and L within 400 trials? Results of Experiment 2 can be analysed within the framework of EUT since subjects are disclosed the payoff structure. Making objective probabilities available to subjects allows direct evaluation of EUT. In analysing the results, this paper presumes that subjects are asked how many times of 400 rounds they are willing to choose H once for all. Experiment 2

20. The utility function, u(x), is considered: To investigate an optimal behaviour in Problem 1 and 3, we employ the risk-averse utility function with Problem 1 ( choiceH =0.63) H:4(0.8);0(0.2) L:3(1) Problem 3 ( choiceH =0.4) H:32(0.1);0(0.9) L:3(1) Experiment 2

21. The EU model for Problem 1 Let V 1 (m) be EU she acquires when choosing H m (  400) times in Problem 1: where k is the number for the realised highest payoff of H in Problem 1 (i.e., 4 points). How many times out of 400 times should DM choose H to maximise V 1 (m)? Problem 1 ( choiceH =0.63) H:4(0.8);0(0.2) L:3(1) Experiment 2

22. V 1 (m) has its maximum at m=252. An theoretically-optimal number of H choices is 252 out of 400 times. DM can maximise EU by choosing H 252 out of 400 times. This coincides exactly results of Experiment 2 that H was chosen 252 times. Problem 1 ( choiceH =0.63) H:4(0.8);0(0.2) L:3(1) Analysis of Problem 1 Experiment 2

23. Conclusion: Experiment 1 (search in SDM problem) Experiment 1 includes simple binary choice problems without giving subjects any information on payoff structure. I have presented the search-assessment model, which  shows that the probability that subjects misestimate the payoff structure is large with only 400 times, even in simple and SDM problems.  implies that subjects are likely to misunderstand in such a way that EV(H)<EV(L).

24. Conclusion: Experiment 2 (choice in SDM problem) Experiment 2 is conducted with giving subjects prior information on payoff structure. Hence, the results can be analysed within the framework of EUT. In Experiment 2, subjects choose both H and L in each choice problem. This paper presents the EU models, which reveal that it is theoretically-optimal to choose H often but not all the time within given trials, to maximise EU.

25. References Allais, M. (1953). Le Comportement de l'Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l'Ecole Americaine. Econometrica, 21(4), 503-46. Barron, G., & Erev, I. (2003). Small Feedback-Based Decisions and Their Limited Correspondence to Description-Based Decisions. Journal of Behavioral Decision Making, 16(3), 215-33. Erev, I., & Barron, G. (2005). On Adaptation, Maximization, and Reinforcement Learning Among Cognitive Strategies. Psychological Review, 112(4), 912-31. Fujikawa, T. (2005). An Experimental Study of Petty Corrupt Behaviour in Small Decision Making Problems. American Journal of Applied Sciences, Special issue, 14-18. Fujikawa, T., & Oda, S. H. (2005). A Laboratory Study of Bayesian Updating in Small Feedback-Based Decision Problems. American Journal of Applied Sciences, 2(7), 1129-33. Kahneman, D., & Tversky, A. (1979). Prospect Theory: An Analysis of Decision under Risk. Econometrica, 47(2), 23-53.