1. Ascertaining certain certainties in choices under uncertainty
Daniel R. Cavagnaro
Clintin P. Davis-Stober
Presented at the 56th Edwards Bayesian Research Conference; March 2-3, 2018

2. Choices under uncertainty
Many real-life decisions must be made with uncertainty.
Insurance
When will we need to use it?
How much would we save by having it?
Evacuation
How long will we need stay away?
What is the likelihood of being in the direct path?
Diet
When will we lose those 10 extra pounds?
How likely is it to be successful?
How will this make me feel?

3. Type of uncertainty Objective risk
Lotteries with known probabilities
A $100 wager on the flip of a fair coin
Non-probabilistic risk (“ambiguity”)
Uncertainty about the relative likelihoods of events
A $100 wager on the outcome of the Super Bowl
Ellsberg urn
Others
Next-day or standard shipping?
$5000 cash or 10oz of gold?
Almost by its very nature, the phenomenon of uncertainty is ill-defined. Here are three different types of uncertainty, which

4. Background Fragmented literature
People tend to prefer known risks over unknown risks: Ambiguity aversion
No universally accepted formal definition
Experimental evidence is mixed
Numerous econometric models
Rarely tested empirically
Many degrees of freedom in designing an experiment
Different types and source of uncertainty
Different display formats
Different elicitation procedures

5. Goals of the project
Identify behavioral properties (axioms) that hold empirically across a broad range of contexts.
Determine whether there can exist any interval-scale utility representation to describe preferences under uncertainty.
Uncover the underlying structure to utility representations.
Method: conjoint measurement theory

6. Models of choice under ambiguity
Subjective Expected Utility (SEU)
Choquet Expected Utility / Rank Dependent Utility (Schmeidler, 1989)
Minimax expected utility (Gilboa and Schmeidler, 1989)
Alpha Expected utility (Ghirardato et al., 2004)
Smooth Ambiguity Preferences Model (Klibanoff et al., 2005)
Contraction model (Gajdos et al., 2008)
Vector Expected Utility (Siniscalchi, (2009)
Etc…

7. Separable representation
For a special case, many (but not all) of these models reduce to a common representation:
Biseparable utility (Ghirardato and Marinaci, 2001)
A binary prospect yielding x if uncertain event E occurs, and y otherwise, with x ≻ y is evaluated by:
𝑉 𝑥,𝐸,𝑦 =𝑊 𝐸 ∗𝑈 𝑥 + 1−𝑊 𝐸 ∗𝑈(𝑦)

8. Separable representation
When y=0, the binary prospect is called unitary, and the representation is “separable” 𝑉 𝑥,𝐸,0 =𝑊 𝐸 ∗𝑈 𝑥 also known as multiplicative conjoint

9. Additive conjoint measurement (ACM)
Axioms of ACM for a separable representation
Monotonicity (MO)
Stochastic Dominance (SD)
Transitivity (TR)
Double cancellation (DC)
These axioms can be tested empirically!
If they hold, then it supports the existence of an interval-scale utility representation.
Rejection of any of these axioms casts doubt on models that imply a separable representation for unitary gambles.

10. Why has this not been done before?
Curley and Yates (1989) used ACM to discriminate among polynomial utility representations of preferences under uncertainty.
“Conjoint measurement has a major weakness. The theory basically does not accommodate any errors. Even one violation of a property, in principle, is to be treated as a failure of the property and any model for which the property is necessary. In applying the theory to empirical data, a softer hand, one which allows errors, must be developed … there is no currently accepted approach for handling errors”

11. Axioms of ACM
ACM concerns how two attributes (row and column) relate with a third attribute (cells). B1 B2 B3 A1 A2 A3

12. Axioms of ACM Uncertain unitary prospects (ambiguity)E1 E2 E3
$10
$50
$75
U($50 if E3, otherwise $0)

13. Axioms of ACM Uncertain unitary prospects (risk)
.10 .5
.75
$10
$50
$75
U(.75 chance of winning $50, otherwise nothing)

14. Axioms of ACM Preferences can be represented with arrows
.10 .5
.75
$10
$50
$75
(0.5,$10) ≻ (0.1,$50)

15. Axioms of ACM
We are concerned with whether a preference relation satisfies the axioms of ACM.
.10 .5
.75
$10
$50
$75

16. Axioms of ACM Monotonicity (MO):
Consistent ranking of the rows, independent of the column, and of the columns, independently of the rows B1 B2 B3 A1 A2 A3
Consistent ranking of the rows, independently of the column, and of the columns, independently of the rows.

17. Axioms of ACM Monotonicity (MO):
Consistent ranking of the rows, independent of the column, and of the columns, independently of the rows
.10 .5
.75
$10
$50
$75
Consistent ranking of the rows, independently of the column, and of the columns, independently of the rows.

18. Axioms of ACM Stochastic Dominance (SD):
major diagonals (upper-left to lower-right) B1 B2 B3 A1 A2 A3 B1 B2 B3 A1 A2 A3

19. Axioms of ACM Transitivity (TR):
minor diagonals (lower-left to upper-right)* B1 B2 B3 A1 A2 A3 B1 B2 B3 A1 A2 A3
* Provided that Monotonicity and Stochastic Dominance hold

20. Axioms of ACM Transitivity (TR):
do not need to all be in the same orientation B1 B2 B3 A1 A2 A3 B1 B2 B3 A1 A2 A3

21. Axioms of ACM Transitivity (TR):
violations occur when there are preference “cycles” B1 B2 B3 A1 A2 A3

22. Axioms of ACM Transitivity (TR): Is this a violation of TR? B1 B2 B3B1 B2 B3 A1 A2 A3

23. Axioms of ACM Transitivity (TR): Is this a violation of TR? YES! B1 B2B1 B2 B3 A1 A2 A3
YES!

24. Axioms of ACM Double-Cancelation (DC):
Given MO, SD, and TR, the Luce-Tukey conditions concern three pairs.* If the blue conditions hold, then the red consequence needs to be true. B1 B2 B3 A1 A2 A3 B1 B2 B3 A1 A2 A3
* Provided that Monotonicity, Stochastic Dominance, and Transitivity hold

25. Experimental setup
Obtain repeated choices on pairs of stimuli (unitary prospects) from a 4x4 matrix
Design considerations:
Cannot do all pairwise comparisons (120 in a 4x4 matrix)
It’s more interesting if we can induce violations!

26. Experimental setup
Hypothesis: Ambiguity aversion could induce violations of DC (or TR). B1
objective B2 B3
ambiguous B4 A1 A2 A3

27. Experimental setup
Hypothesis: Ambiguity aversion could induce violations of DC (or TR).
0.2
0.4
0.4<p<0.5
0.5
$30
$50
$90

28. Experimental setup
Set A1, A2, A3, B1, B2, and B4 to induce the preference pattern shown below (e.g., by expected value): B1
objective B2 B3
ambiguous B4 A1 A2 A3

29. Experimental setup
Set A1, A2, A3, B1, B2, and B4 to induce the preference pattern shown below (e.g., by expected value):
0.2
0.4 B3
ambiguous
0.5
$30
$12
$15
$50
$10
$90
$18

30. Experimental setup
Monotonicity implies preference according to the blue arrow (set B3 accordingly) B1
objective B2 B3
ambiguous B4 A1 A2 A3

31. Experimental setup
Transitivity implies preference according to the green arrow B1
objective B2 B3
ambiguous B4 A1 A2 A3

32. Experimental setup Ambiguity aversion could lead to a cycle! B1 B2 B3B1
objective B2 B3
ambiguous B4 A1 A2 A3

33. Experimental setup
Ambiguity aversion could induce preference according to the red arrow. B1
objective B2 B3
ambiguous B4 A1 A2 A3

34. Experimental setup This pattern violates double-cancellation! B1 B2 B3B1
objective B2 B3
ambiguous B4 A1 A2 A3

35. Experimental setup Example 1: 0.2 0.4 0.4<p<0.5 0.5 $30 $6 $12
0.2
0.4
0.4<p<0.5
0.5
$30 $6
$12
$12-$15
$15
$50
$10
$20
$20-$25
$25
$90
$18
$36
$36-$45
$45

36. Experimental setup Example 1: 0.2 0.4 0.4<p<0.5 0.5 $30 $6 $12
0.2
0.4
0.4<p<0.5
0.5
$30 $6
$12
$12-$15
$15
$50
$10
$20
$20-$25
$25
$90
$18
$36
$36-$45
$45

37. Experimental setup Example 2: 0.2 0.4 0.2<p<0.7 0.5 $30 $6 $12
0.2
0.4
0.2<p<0.7
0.5
$30 $6
$12
$6-$21
$15
$50
$10
$20
$10-$35
$25
$90
$18
$36
$18-$63
$45

38. Experimental Design 8 for monotonicity 0.2 0.4 0.2<p<0.7 0.5 $30
0.2
0.4
0.2<p<0.7
0.5
$30
$50
$90
$140

39. Experimental Design 8 for monotonicity 6 for double cancellation 0.2
0.2
0.4
0.2<p<0.7
0.5
$30
$50
$90
$140

40. Experimental Design 8 for monotonicity 6 for double cancellation
4 for transitivity
0.2
0.4
0.2<p<0.7
0.5
$30
$50
$90
$140

41. Experimental Design Manipulation of the “level” of ambiguity:
0.4<p<0.5; 0.3<p<0.6; 0.2<p<0.6
0.2
0.4 ?
0.5
$30
$50
$90
$140

42. Upper left (EV) ≺

43. Main diagonal (EV) ≻

44. Monotonicity ≺

45. Transitivity ≺

46. Lower right (Ambiguity aversion)≺

47. Results
TBA

48. Extensions Time preferences delay reward 2 days 5 days 10 days x1 x2
2 days
5 days
10 days x1 x2 x3
reward

49. Extensions Time preferences with uncertainty delay reward 2 days
2 days
3-5 days
7-10 days x1 x2 x3
reward

50. Extensions Choice under risk with incomplete information probability
0.25
0.5
0.75
10 oz silver
1 oz gold
$5000
reward

51. Thank You!