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).