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An Investigation of Ambiguity Affi nity
The Devil You Don’t Know
Tristan Gray-Le CozColorado State University
RISK •Decisions under which the probability distribution for the outcomes is known or easily calculable•Has traditionally been viewed as explicit probabilities•90% chance of Winning •50 black balls and 50 red balls
Ambiguity and Risk
Ambiguity•Situations in which the probability distributions are unknown•Can be conceived of as inexact probabilities•Explicitly stated as unknown chances•Decision outcome is influenced by too many factors to keep track of
Ambiguity and Risk
Expected Value (EV) is the product of the probability (risk) term and reward (payout) term in a given gamble•A 20% chance of winning $100 is equivalent to $20•An 80% chance of winning $25 returns the same $20 in EV
Expected Value
Classically, two gambles are set against one another •Gambles of the same EV• Isolate and define risk preferences independent of reward
•Gambles of different EV• Isolate and define risk preferences in terms of reward
Risk Aversion
Expected Utility Theory (EU) and Subjective Expected Utility Theory (SEU) propose that decision makers consistently choose the more likely option•These theories give short shrift to decisions made under ambiguity and…•Avoid the problem of ambiguity by assuming that every decision can be fitted to a probability distribution drawn from the decision maker’s experience
Risk Aversion
Ellsberg’s Urns (1961)•Subjects are offered the choice between picking a winning color from one of two urns•Urn 1 contains 50 red and 50 black balls•Urn 2 contains 100 balls in unknown quantity•EU and SEU would predict that there should be no preference between the two urns• Individuals consistently chose Urn 1 over Urn 2
Ambiguity Aversion
Urn 1 Urn 2
Prospect Theory
(Kahneman & Tversky, 1979)•Defines a risk aversion curve•Nonlinear function•For gains, as probability become relatively small and payouts become relatively large, risk taking increased•Risk aversion with the goal of maintaining gains is dominant
•For losses, the curve is similar but reversed•Risk taking to minimize loss is dominant
Ambiguity Aversion
• EU, SEU, and Prospect Theory, the risk term is the dominant factor in decision making
• These theories tend not to examine decision making under conditions of outcome ambiguity
Ambiguity and Risk
Most examinations have varied only risk or payout•Terms have been of a fixed value•Gambles can be thought of as being static
Few examinations have sought to introduce true uncertainty into the gamble•Some have fixed a range of payouts•These models begin to identify dynamic gambles
Uncertainty Allocation
In this experiment I aim to directly investigate the influence of uncertainty and ambiguity in decision making
Ambiguity Allocation
A 50% chance of winning $12
A 20% chance of winning $30
Suppose Two Gambles:
• Both gambles have an initial EV of $6• Which gamble will the subject choose• Which term will the subject choose to
change?
• SEU and EU would propose that subject should never choose to change the probability term
• Prospect Theory would predict that the probability term would only be changed where the initial payout of the chosen gamble is large relative to the payout of the spurned gamble
Ambiguity Allocation
• EU and SEU make no prediction as to ambiguities in payout
• Prospect Theory would suggest that the payout term would be changed when the probability difference is large
Ambiguity Allocation
Following from the literature, I predicted that:•Subjects will choose to change the payout term when initial probability is high•As probabilities and rewards in both gambles approach equality, allocation of ambiguity will be less consistent
Expected Results
• Subjects are presented with 45 Gamble pairs via E-Prime• Pairs are drawn randomly from a list of 90 possible
gamble pairings
• Gamble pairs are constructed with equal initial EV• Risk1 x Value1 = Risk2 x Value2
• The probability term in G1 is always higher than in G2
• The range of probabilities is limited from 10% to 90% on 10% increments
• The reward term in G2 is always higher than in G1
• The reward terms are the product of the Risk and EV Value
Experimental Design
• Subjects are presented with one gamble pair at a time
• Subjects must first choose which gamble to play by entering 1 or 0 as they correspond to the two presented gambles
• Subjects see their chosen gamble and decide which term to change by pressing 1 to change the risk term or 0 to change the value term.
• Subjects are then presented with their new gamble, designated as “your Gamble” along side the unchosen gamble, designated as “My Gamble”
• The gambles are calculated, and wins are added to the player’s bank or to the opponent’s bank; ties result in no score, and losses are not taken from a bank
Methods
1) 90% x $6.66
0) 10% x $60.00
Please Choose Between:
Press SPACEBAR to Continue
90% X $6.67
Please Choose to Change:
Press 1 to Change Press 0 to Change
Press SPACEBAR to continue
Your New Gamble:
10% x $6.67
My Gamble:
10% x $60.00
SHOWDOWN!!!
Press SPACEBAR to continue
I Won:
$5.33
So Far I’ve Won:
$5.33
You Won:
$0
So Far You’ve Won:
$0
Results:
Press SPACEBAR to play next round
1) 60% x $13.33
0) 50% x $16.00
Please Choose Between:
Press SPACEBAR to Continue
50% X $16.00
Please Choose to Change:
Press 1 to Change Press 0 to Change
Press SPACEBAR to continue
Your New Gamble:
50% x $100.00
My Gamble:
60% x $13.33
SHOWDOWN!!!
Press SPACEBAR to continue
I Won:
$0
So Far I’ve Won:
$5.33
You Won:
$42
So Far You’ve Won:
$42
Results:
Press SPACEBAR to play next round
• All change terms were randomly drawn from the available R x V = EV matrix to maintain the numeric distribution inherent in the gamble set
• All gambles were conducted in terms of gains, and no penalty was assessed for losing a match.
• Player and opponent banks were kept to encourage task involvement by providing apparent competition
• No actual currency was exchanged
Some Other Design Considerations
Results from Experiment 1
%H->% %H->$ %L->% %L->$-10%
0%
10%
20%
30%
40%
50%
60%
6.46%
43.94% 41.52%
8.08%
Strategy Breakdown
%HRChosen %LRChosen %ValueChange %RiskChange0%
20%
40%
60%
80%
100%
50.40% 49.60% 52.02% 47.98%
Simple Choice Breakdown
Results from Experiment 1 Ctd.
$0-1$ $1-$2 $2-$3 $3-$4 $4-$5 $5-$10
$10-$15
$15-$20
$20-$25
$25-$30
$30-$40
$40-$50
$50-$60
$60-$70
$70-$80
$80-$90
0%
10%
20%
30%
40%
50%
60%
70%
80%
f(x) = − 4.08622619020021E-06 x⁴ + 0.00094939537029753 x³ − 0.0234727041307605 x² + 0.140490447204093 x + 0.421533674658672R² = 0.784834084282139f(x) = − 7.84942695094949E-06 x⁴ − 0.00049639458496311 x³ + 0.0132583229863769 x² − 0.0429423712654378 x + 0.21337742118992R² = 0.962220721386088
Proportion of Strategy Choice by Initial Value Dif-ference
Higher Initial Risk/ Change Value
Polynomial (Higher Initial Risk/ Change Value)
Higher Initial Value/ Change Risk
Polynomial (Higher Initial Value/ Change Risk)
• N = 21
• Crossover interaction at 20-25$ of Value Difference is the most intriguing feature of this initial analysis
• Analysis of choice strategy by Risk Difference shows no pattern
• Difference in rates of two main strategies were not significant
Summary Conclusions from Exp. 1
• Because I am investigating ambiguity, specifically outcome ambiguity, I decided to mask all Value terms with greater than $25 difference at initial presentation
• This manipulation attempts to move the crossover point found in experiment 1 to a lower range of values
• This experiment was designed to influence strategy exploration by increasing outcome uncertainty
Experiemnt 2
1) 80% x $##
0) 10% x $##
Please Choose Between:
Press SPACEBAR to Continue
10% X $100.00
Please Choose to Change:
Press 1 to Change Press 0 to Change
Press SPACEBAR to continue
Your New Gamble:
50% x $100.00
My Gamble:
80% x $12.50
SHOWDOWN!!!
Press SPACEBAR to continue
I Won:
$0
So Far I’ve Won:
$0
You Won:
$40
So Far You’ve Won:
$40
Results:
Press SPACEBAR to play next round
• Aside from the masking manipulation, all other parameters remained the same between Experiment 1 and Experiment 2
Design Considerations
Results from Experiment 2
%H->% %H->$ %L->% %L->$0%
10%
20%
30%
40%
50%
60%
3.33%
49.78%
37.00%
9.89%
Strategy Breakdown
%HRChosen %LRChosen %ValueChange %RiskChange0%
20%
40%
60%
80%
100%
53.11%46.89%
59.67%
40.33%
Simple Choice Breakdown
Results from Experiment 2 ctd.
$0-1$
$1-$2
$2-$3
$3-$4
$4-$5
$5-$10
$10-$15
$15-$20
$20-$25
$25-$30
$30-$40
$40-$50
$50-$60
$60-$70
$70-$80
$80-$90
0%
10%
20%
30%
40%
50%
60%
70%
f(x) = − 0.000018462631256 x⁴ + 0.0008728190617 x³ − 0.0153730700229 x² + 0.0828765509544 x + 0.4296271436896R² = 0.752748421565484
f(x) = 0.000099744957604 x⁴ − 0.0031620506205 x³ + 0.03083847928041 x² − 0.0654502345449 x + 0.1567770770896R² = 0.848048006865791
Proportion of Strategy Choice by Initial Value Difference
Higher Initial Risk/change Value
Polynomial (Higher Initial Risk/change Value)
Higher Initial Value/ Change Risk
Polynomial (Higher Initial Value/ Change Risk)
• Masking of high initial value differences seems to make no difference in choice strategy
• No discernable pattern for choice strategy by risk difference
• Value information plays only a secondary role in decision making strategy
Summary Conclusions from Exp. 2
Experiment 1 vs. Experiment 2
%H->% %H->$ %L->% %L->$0%
10%
20%
30%
40%
50%
60%
Strategy Breakdown
GainsMasked
%HRChosen %LRChosen %ValueChange %RiskChange0%
10%
20%
30%
40%
50%
60%
70%
Simple Choice Breakdown
GainsMasked
Experiment 1 vs. Experiment 2
$0-1$ $1-$2 $2-$3 $3-$4 $4-$5 $5-$10
$10-$15
$15-$20
$20-$25
$25-$30
$30-$40
$40-$50
$50-$60
$60-$70
$70-$80
$80-$90
0%
10%
20%
30%
40%
50%
60%
70%
80%
f(x) = − 4.08622619020021E-06 x⁴ + 0.000949395370297535 x³ − 0.0234727041307605 x² + 0.140490447204093 x + 0.421533674658672R² = 0.784834084282139f(x) = − 7.84942695094949E-06 x⁴ − 0.000496394584963114 x³ + 0.0132583229863769 x² − 0.0429423712654378 x + 0.21337742118992R² = 0.962220721386088
Proportion of Strategy Choice by Initial Value Difference
Higher Initial Risk/ Change Value
Polynomial (Higher Initial Risk/ Change Value)
Higher Initial Value/ Change Risk
$0-1$ $1-$2 $2-$3 $3-$4 $4-$5 $5-$10
$10-$15
$15-$20
$20-$25
$25-$30
$30-$40
$40-$50
$50-$60
$60-$70
$70-$80
$80-$90
0%
10%
20%
30%
40%
50%
60%
70%
f(x) = − 1.84626312561034E-05 x⁴ + 0.000872819061727284 x³ − 0.0153730700229182 x² + 0.0828765509544117 x + 0.429627143689644R² = 0.752748421565484
f(x) = 9.97449576042179E-05 x⁴ − 0.00316205062050214 x³ + 0.030838479280413 x² − 0.0654502345449282 x + 0.156777077089577R² = 0.848048006865791
Proportion of Strategy Choice by Initial Value Difference
Higher Initial Risk/change ValuePolynomial (Higher Initial Risk/change Value)Higher Initial Value/ Change RiskPolynomial (Higher Initial Value/ Change Risk)
Gains
Masked
• Introduce losses
• Mask Risk information
• Extend the length of trial blocks
• Continue to investigate the role of outcome in complex choice and decision-making
Directions for Future Study
Questions?
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