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Small Decision Making Under Uncertainty And Risk

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Title: Small Decision Making Under Uncertainty And Risk


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?
Introduction
  • 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.

4
Experimental Design
Experimental Design
  • 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

5
Experimental Design
Experimental Design
  • 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).

6
Choice Problems
Experimental Design
  • 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 (t1, 2, , 400), subjects are
    asked to choose either H or L.

Problem 1 H 4 (0.8) 0 (0.2) L 3 (1)
Problem 2 H 4 (0.2) 0 (0.8) L 3
(0.25) 0 (0.75)
Problem 3 H 32 (0.1) 0 (0.9) L 3 (1)
7
Experiment 1 Search in SDM problems
  • Subjects in Experiment 1 are NOT informed of
    payoff structure.

8
Experimental screen
Experiment 1
Problem 1 H 4 (0.8) 0 (0.2) L 3 (1)
Problem 2 H 4 (0.2) 0 (0.8) L 3
(0.25) 0 (0.75)
Problem 3 H 32 (0.1) 0 (0.9) L 3 (1)
Basic task in each problem was a binary choice
between two buttons for 400 times without giving
subjects prior information on payoff structure.
9
Results of Experiment 1
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)
10
Experiment 1
Results choiceH
Problem 1 (choiceH0.48) H 4 (0.8) 0 (0.2)
L 3 (1)
Problem 2 (choiceH0.55) H 4 (0.2) 0 (0.8)
L 3 (0.25) 0 (0.75)
Problem 3 (choiceH0.22) H 32 (0.1) 0 (0.9)
L 3 (1)
11
Experiment 1
The tendency to select best reply to past outcomes
Problem 3 (choiceH0.22) H 32 (0.1) 0 (0.9)
L 3 (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)ltEV(L)

12
Analysis
Experiment 1
  • 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.

13
The search-assessment model
Experiment 1
  • 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 (t1,2, , 400).

Problem A H x (p) 0 (1-p) L 1 (1) where
0ltplt1, pxgt1.
Problem 1 (choiceH0.48) H 4 (0.8) 0 (0.2)
L 3 (1)
Problem 3 (choiceH0.22) H 32 (0.1) 0 (0.9)
L 3 (1)
14
Experiment 1
  • 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(Hm)
  • This allows us to analyse the number of H choices
    required for judging that EV(H)gtEV(L).

Problem A H x (p) 0 (1-p) L 1 (1) where
0ltplt1, pxgt1.
Problem 1 (choiceH0.48) H 4 (0.8) 0 (0.2)
L 3 (1)
Problem 3 (choiceH0.22) H 32 (0.1) 0 (0.9)
L 3 (1)
15
P(Hm) for Problem 3
Experiment 1
  • P(Hm) is calibrated by setting p0.1 and x32/3.
  • Calibration implies the probability that
    posteriorHgt3 does not exceed 0.98 until H is
    chosen 10,000 times in Problem 3.

Problem 3 (choiceH0.22) H 32 (0.1) 0 (0.9)
L 3 (1)
Problem A H x (p) 0 (1-p) L 1 (1) where
0ltplt1, pxgt1.
16
Experiment 2 Choice in SDM problems
  • Subjects in Experiment 2 are clearly disclosed
    payoff structure.

17
Experiment 2
Experimental screen
Problem 1 H 4 (0.8) 0 (0.2) L 3 (1)
Problem 2 H 4 (0.2) 0 (0.8) L 3
(0.25) 0 (0.75)
Problem 3 H 32 (0.1) 0 (0.9) L 3 (1)
Basic task in each problem was a binary choice
between two buttons for 400 times with prior
information on payoff structure.
18
Results choiceH
Experiment 2
Problem 1 (choiceH0.63) H 4 (0.8) 0 (0.2)
L 3 (1)
Problem 2 (choiceH0.69) H 4 (0.2) 0 (0.8)
L 3 (0.25) 0 (0.75)
Problem 3 (choiceH0.4) H 32 (0.1) 0 (0.9)
L 3 (1)
19
Analysis
Experiment 2
  • 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.

20
Experiment 2
  • 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 (choiceH0.63) H 4 (0.8) 0 (0.2)
L 3 (1)
Problem 3 (choiceH0.4) H 32 (0.1) 0 (0.9)
L 3 (1)
21
The EU model for Problem 1
Experiment 2
  • Let V1(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 V1(m)?

Problem 1 (choiceH0.63) H 4 (0.8) 0 (0.2)
L 3 (1)
22
Analysis of Problem 1
Experiment 2
  • V1(m) has its maximum at m252.
  • 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 (choiceH0.63) H 4 (0.8) 0 (0.2)
L 3 (1)
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)ltEV(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.
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