Small Decision Making Under Uncertainty And Risk

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Small Decision-Making under Uncertainty and Risk
Takemi FujikawaUniversity of Western Sydney,
Australia
Agenda

• Introduction
• Experimental Design
• Experiment 1
• Experiment 2
• Conclusion

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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

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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.

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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

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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).

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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)
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Experiment 1 Search in SDM problems

• Subjects in Experiment 1 are NOT informed of
  payoff structure.

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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.
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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)
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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)
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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)

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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.

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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)
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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)
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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.
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Experiment 2 Choice in SDM problems

• Subjects in Experiment 2 are clearly disclosed
  payoff structure.

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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.
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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)
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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.

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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)
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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)
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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)
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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).

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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.

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