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

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?

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

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

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

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…

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)

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

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.

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”

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

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

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

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

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

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.

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.

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

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

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

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

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

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

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

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!

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Upper left (EV) ≺

Main diagonal (EV) ≻

Monotonicity ≺

Transitivity ≺

Lower right (Ambiguity aversion) ≺

Results TBA

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

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

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

Thank You!