the devil you don’t know 2

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


• 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


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!!!


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



50% X $16.00




Your New Gamble:

50% x $100.00

My Gamble:

60% x $13.33

SHOWDOWN!!!


I Won:

$0

So Far I’ve Won:

$5.33

You Won:

$42


$42

Results:


• 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 $##



10% X $100.00




Your New Gamble:

50% x $100.00

My Gamble:

80% x $12.50

SHOWDOWN!!!


I Won:

$0

So Far I’ve Won:

$0

You Won:

$40


$40

Results:


• 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


20%

40%

60%

80%

100%

53.11%46.89%

59.67%

40.33%


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)


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


10%

20%

30%

40%

50%

60%

70%


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


Higher Initial Risk/ Change Value

Polynomial (Higher Initial Risk/ Change Value)


$0-1$ $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


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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References

the devil you don’t know 2

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