2000-22

UNIVERSITY OF CALIFORNIA, SAN DIEGO

DEPARTMENT OF ECONOMICS

PAYOFF KINKS IN PREFERENCES OVER LOTTERIES

BY

MARK J. MACHINA

DISCUSSION PAPER 2000-22

SEPTEMBER 2000

PAYOFF KINKS IN PREFERENCES OVER LOTTERIES

Mark J. Machina

Department of Economics

University of California, San Diego

La Jolla, California

92093-0508

September 11, 2000

This paper identifies two distinct types of payoff kinks that can be exhibited by

preference functions over monetary lotteries – “locally separable” vs. “locally

nonseparable” – and illustrates their relationship to the payoff and probability

derivatives of such functions. Expected utility and Fréchet differentiable prefer-

ence functions are found to be incapable of exhibiting locally nonseparable pay-

off kinks; rank-dependent preference functions are incapable of avoiding them.

JEL classification: D8 Keywords: R

ISK

, U

NCERTAINTY

, P

AYOFF KINKS

I have benefited from the comments of Vince Crawford, Valentino Dardanoni, Peter Hammond,

Zvi Safra, Uzi Segal, Ross Starr, Peter Wakker and especially Edi Karni, Joel Sobel and Joel

Watson. This material is based upon work supported by the National Science Foundation under

Grant No. 9870894.

PAYOFF KINKS IN PREFERENCES OVER LOTTERIES

1. INTRODUCTION

2. THE CALCULUS OF PROBABILITIES AND PAYOFFS

2.1 Payoff Changes vs. Probability Changes

2.2 Payoff Derivatives

2.3 Probability Derivatives and Local Utility Functions

2.4 Fréchet Differentiability and the Link between Payoff and Probability Derivatives

3. LOCALLY SEPARABLE VS. LOCALLY NONSEPARABLE KINKS

3.1 Piecewise-Linearity and Local Piecewise-Linearity

3.2 Local Separability vs. Local Nonseparability

3.3 Ordinal Implications of Locally Separable vs. Locally Nonseparable Kinks

4. EXPECTED UTILITY AND FRECHET DIFFERENTIABLE PAYOFF KINKS

4.1 Local Separability of Expected Utility Payoff Kinks

4.2 Local Separability of Fréchet Differentiable Payoff Kinks

4.3 First Order and First Order Conditional Risk Aversion

5. RANK-DEPENDENT PAYOFF KINKS

5.1 Rank-Dependent Probability Derivatives and Local Utility Functions

5.2 Local Nonseparability of Rank-Dependent Payoff Kinks: Illustration

5.3 Rank-Dependent Payoff Derivatives and Payoff Kinks: General Formulas & Properties

6. CONCLUDING TOPICS

6.1 Modelling Departures from Separability: Two Approaches

6.2 Are Observed Risk Preferences Kinked in the Payoffs?

6.3 Induced Payoff and Probability Kinks

REFERENCES

1. INTRODUCTION

The purpose of this paper is to

!" identify

two

distinct

types of payoff kinks

in

preference

functions

over

monetary

lotteries, namely “locally separable” versus “locally nonseparable” kinks

!" illustrate the relationships between a preference function’s (directional) payoff deriva-

tives and its probability derivatives in the presence of these different types of kinks, and

!" compare the expected utility model and two important non-expected utility models with

respect to the types of payoff kinks that they can, cannot, or must exhibit.

There are several situations where an individual’s preferences over lotteries might be

expected to exhibit kinks in the payoff levels. The simplest and undoubtedly most pervasive are

piecewise-linear income tax schedules, which imply that the individual’s utility of before-tax

income will typically have kinks at the boundaries of the tax brackets. Similar instances include

kinks induced by the option of bankruptcy, or the intended purchase of some large indivisible

good. Alternatively, payoff kinks may be an inherent part of underlying attitudes toward risk. We

also briefly consider another source of kinks, namely the phenomenon of temporal risk.

Models of preferences over lotteries, like models of preferences elsewhere in economics,

should be flexible enough to be able to exhibit kinks in situations where they might be expected

to occur, as well as avoid kinks (be globally smooth) in situations where they might not. As it

turns out, three important models – (1) expected utility risk preferences, (2) Fréchet differen-

tiable risk preferences, and (3) rank-dependent risk preferences – all exhibit this flexibility with

respect to the first type of payoff kink (locally separable). However, none of these models are

flexible with respect to the second type: Whereas expected utility and Fréchet differentiable

preferences cannot exhibit locally nonseparable payoff kinks, probability-smooth rank-dependent

preferences cannot avoid exhibiting them at every lottery.

As mentioned, another purpose of this paper is to clarify the relationship between payoff

kinks, payoff derivatives, and probability derivatives of preference functions over lotteries.

Probability derivatives have proven useful in generalizing many of the basic concepts and results

of expected utility analysis to more general non-expected utility preferences. For the expected

utility preference function V

EU

(x

1

, p

1

;...;x

n

, p

n

) #

$

n

i

=

1

U(x

i

)%p

i

, the probability coefficient of a

payoff level x – the coefficient of V

EU

(%) with respect to changes in prob(x) – is simply x’s utility

level U(x). By viewing expected utility results as statements about probability coefficients,

researchers have exploited the natural correspondence between coefficients in linear algebra and

partial derivatives in calculus to generalize many expected utility results to smooth non-expected

utility preference functions V(%). As is usual in the linear algebra

& calculus correspondence,

such “generalized expected utility” theorems include both local and exact global results.

Early work in generalized expected utility analysis imposed the smoothness property of

Fréchet differentiability, which is not satisfied by the rank-dependent form. However, Chew,

Karni and Safra (1987) showed that most of its basic results also hold under weaker notions of

smoothness, and that many indeed apply to the rank-dependent form. This paper helps clarify the

boundaries of this extension, namely that a generalized expected utility result will typically hold

for the rank-dependent form unless it involves its (full or directional) payoff derivatives, in

which case it is usually invalidated by the specific nature of rank-dependent payoff kinks.