Certainty Equivalent and Stochastic Preferences

June 2006FUR 2006, Rome

Pavlo Blavatskyy Wolfgang Köhler

IEW, University of Zürich

Introduction

Most decision theories are deterministic although observed choices are stochastic.

Most (economic) theories of decision making do not explain failure of procedure invariance.

Instead: strong assumptions on reasoning abilities of individuals.

Our model has two ingredients:

- Individuals have stochastic preferences.

- Individuals solve only binary decision problems. If faced with a complex

decision problem, they split it into a sequence of binary decision problems.

We are interested in the interpretation of observed choices.

We consider two types of decision problems:

- binary choice

- the determination of certainty equivalents (matching).

We assume that subjects have correct incentives to reveal their preferences.

(Introduction)

The Model

Let X be a convex and bounded subset of R.

A lottery L is defined on a finite subset

Let and

Let ≿ be a preference relation defined on the set of lotteries.

Assumptions: is: complete≿

transitive

monotone

satisfies Convexity

Let P be the set of preferences relations.

Let be a probability measure on P.P.

}min{ LL Xx

.XX L

}.max{ LL Xx

Binary choice:

Individual draws ≿ according to and chooses accordingly.P

Determination of Certainty Equivalent

Different elicitation methods in experiments.

Here: Consider situation where subjects are asked to state certainty equivalent.

Let be the elicited certainty equivalent.

From Convexity follows that ).,( LLL xxCE

LCE

Binary choice:

Individual draws ≿ according to and chooses accordingly.P

(Determination of certainty equivalent)

Idea: Individuals split determination of certainty equivalent into sequence of binary decisions. I.e., individuals compare some amount to lottery. Depending on whether amount or lottery is preferred, individuals adjust amount and make new comparison, and so on.

Formally:

1. Step Draw

2. Step Draw ≿ according to and compare L and .

3. Step If ~ L then If L (if L) then is replaced by

and step 2 is repeated.

If the preferred alternative switches, is equal to average of the

last two amounts to which lottery has been compared.

).,( LL xxx

P

x.xCEL x x

x

)( x

LCE

(Determination of certainty equivalent)

Idea: Individuals split determination of certainty equivalent into sequence of binary decisions. I.e., individuals compare some amount to lottery. Depending on whether amount or lottery is preferred, individuals adjust amount and make new comparison, and so on.

x x

To relate results to empirical findings, we need two technical assumptions on the distribution of preferences.

A5 (Symmetry): ≿ ≿ LEVN (( LNL (()) ))LEV

(The Model)

To relate results to empirical findings, we need two technical assumptions on the distribution of preferences.

A5 (Symmetry): ≿ ≿

If let n be the unique integer s.t. and

and similar for

A6 (Preferences sufficiently stochastic): If , then

≿ and similar for

LEVN (( LNL (()) ))LEV

2LL

L

xxEV

LL EVnx )5.0(

LL EVnx )5.1( .2

LLL

xxEV

n

j

LN1

((1 5.0)) jxL

2LL

L

xxEV

.2

LLL

xxEV

(The Model)

Theorem 1 If A1-A6 are satisfied then for any L with

If , then .2

LLL

xxEV

5.0}Pr{ LL EVCE

.5.0}Pr{ LL EVCE2

LLL

xxEV

Theorem 1 refers to a situation where subjects are asked to state certainty equi-valent (e.g., willingness to pay under BDM-mechanism).

(The Model)

Lemma 1 Suppose A1-A6 are satisfied. If the auction is ascending and starts at

then If the auction is descending and starts at ,

then

Elicitation via second-price auction

Price increases/decreases with step-size .

Lx

.5.0}Pr{ LL EVCE Lx

.5.0}Pr{ LL EVCE

Other elicitation procedures

Lemma 2 If A1-A6 are satisfied and if subjects start with one of the choices at

random and then solve adjacent choice problems, then if

and if

Elicitation via a sequence of observed choices

Suppose that amounts are equally spaced with distance , that is one of the

amounts, that ~ and that a computer program prevents

inconsistent choices.

5.0}Pr{ LL EVCE

5.0}Pr{ LL EVCE

LEV

LN (( 0))LEV

2LL

L

xxEV

.

2LL

L

xxEV

(Other elicitation procedures)

Explanation of empirical observations

Fourfold pattern of risk-attitudes

Tversky and Kahneman (1992), Cohen et al. (1985): Subjects make choices between lottery and list of amounts for certain.

Find fourfold pattern:

- most decisions are riskaverse if likely gain or unlikely loss

- most decisions are riskseeking if unlikely gain or likely loss

Explanation of empirical observations

Fourfold pattern of risk-attitudes

Likely gain or unlikely loss

Unlikely gain or likely loss

Hence Lemma 2 implies that elicitation via list of observed choices generatesthe fourfold pattern of risk-attitudes.

Tversky and Kahneman (1992), Cohen et al. (1985): Subjects make choices between lottery and list of amounts for certain.

2

LLL

xxEV

Find fourfold pattern:

- most decisions are riskaverse if likely gain or unlikely loss

- most decisions are riskseeking if unlikely gain or likely loss

2

LLL

xxEV

(fourfold pattern of risk-attitudes)

Harbaugh et al. (2003): six lotteries (low, medium, and high probability of gain/loss)

1) pricing task: elicitation of certainty equivalents via BDM-procedure

2) choice task: subjects choose between lottery and its expected value

(fourfold pattern of risk-attitudes)

Harbaugh et al. (2003): six lotteries (low, medium, and high probability of gain/loss)

1) pricing task: elicitation of certainty equivalents via BDM-procedure

2) choice task: subjects choose between lottery and its expected value

Find fourfold pattern only in pricing task but not in choice task.

Choice behavior is statistically indistinguishable from risk-neutrality. Only 4 of 64

subjects choose according to predictions of fourfold pattern.

Shows difference between (single) choice and elicitation of certainty equivalent

Our model: predicts fourfold pattern in pricing task (Theorem 1)

predicts stochastic choice in choice task but no systematic bias

Preference Reversal

Tversky et al. (1990) use ordinal payoff schemes.

For each lottery pair, fix amount X (equal or slightly smaller than expected values).

Two Tasks:

1) binary choice between $-bet vs. P-bet, $-bet vs. X, and P-bet vs. X.

2) state certainty equivalent for $-bet and P-bet.

45% of response patterns are standard preference reversals.

4% of response patterns are non-standard preference reversals.

Standard preference reversal: and

Non-standard preference reversal: and

betbetP $betbetP CECE $

betbetP $betbetP CECE $

(Preference Reversal)

Pattern percent Diagnosis

10.0 Intransitivity

65.5 Overpricing of $-bet

6.1 Underpricing of P-bet

18.4 Over- and Underpricing

X$ PX

Distribution of response patterns for standard reversals in Tversky et al.(for decisions with and )$P PCEXCE $

X$PX $X

XP $X

XP procedureinvariance

(Preference Reversal)

Pattern percent Diagnosis

10.0 Intransitivity

65.5 Overpricing of $-bet

6.1 Underpricing of P-bet

18.4 Over- and Underpricing

X$ PX

Distribution of response patterns for standard reversals in Tversky et al.(for decisions with and )$P PCEXCE $

X$PX $X

XP $X

XP procedureinvariance

1) Since we assume that preferences are stochastic, our model predicts that some fraction of choices violates transitivity.

2) Our model explains why procedure invariance is violated (e.g., Theorem 1 predicts that subjects are likely to overprice the $-bet and to underprice the P-bet).

3) Over-/underpricing explains why standard reversal observed more frequently.

Conclusion

Model has two ingredients:

- Stochastic preferences

- Complex decision problems are split into sequence of binary decision problems

Binary choice can immediately infer preferences

Complex decision problem cannot directly infer preferences, details of

decision problem matter

Model offers explanation for failure of procedure invariance (e.g., preference

reversal).