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# Payoff Kinks in Preferences over Lotteries

TL;DR: In this article, the authors identify two distinct types of payoff kinks that can be exhibited by preference functions over monetary lotteries -locally separable and locally nonseparable -and illustrate their relationship to the payoff and probability derivatives of such functions.
Abstract: 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 Frechet differentiable preference functions are found to be incapable of exhibiting locally nonseparable payoff kinks; rank-dependent preference functions are incapable of avoiding them.

## Summary (5 min read)

### 1. INTRODUCTION

• There are several situations where an individual’s preferences over lotteries might be expected to exhibit kinks in the payoff levels.
• 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.

### 2. THE CALCULUS OF PROBABILITIES AND PAYOFFS

• The authors consider the family L of all finite-outcome lotteries P over some real interval [0,M].
• Each such lottery can be uniquely represented by its probability measure '(%), or alternatively, by its cumulative distribution function F(%).
• Throughout, the authors assume that the individual’s risk preferences can be represented by a real-valued preference function V(%) over L, which accordingly assigns the same value to any pair of identified expressions in (2).
• Such as income and substitution effect analysis.

### 2.1 Payoff Changes vs. Probability Changes

• Ultimately, there is no real difference between changing the payoffs of a given lottery and changing its probabilities.
• On the other hand, many economic situations – such as portfolio choice, insurance and contingent production/exchange – involve optimization and/or equilibrium with respect to the payoff levels over some fixed set of states of nature, in which case working with payoff changes would be most natural.
• A situation where both types of changes come into play is Ehrlich and Becker’s (1972) analysis of an agent facing both self insurance options (activities that can mitigate the magnitude of a potential disaster, though not its likelihood) as well as self protection options (activities that can mitigate the likelihood of the disaster, though not its magnitude).
• The equivalence of (3) and (4) implies that an individual’s risk preferences can be completely represented by either their attitudes toward probability changes or their attitudes toward payoff changes – e.g., by either the probability or payoff derivatives of their preference function V(%).

### 2.2 Payoff Derivatives

• The authors can also consider the effect of shifting just a part of xi’s probability mass, say some amount .
• The reason is that the payoff derivatives (5) – (7) represent movements along three different paths in the underlying space of measures over [0,M],7 so they will not satisfy (8) – (10) without additional smoothness on V(%) that links its responses to movements along these distinct paths.

### 2.3 Probability Derivatives and Local Utility Functions

• The reason for this is that, in contrast with the payoff derivatives (5) – (7), the six individual derivative terms in (17) and (18) all represent the effect of moving, though at different speeds, along the same path in the underlying space of measures over [0,M].9.
• To see this, observe that every FSD shift P+ P* can be built out of two-outcome FSD shifts, each of which moves some amount 4p of probability mass from a payoff level x" up to some higher level x!, so that the variables prob(x"), prob(x!) undergo the equal and opposite changes –4p, +4p.
• This use of local utility functions (probability derivatives), termed generalized expected utility analysis, has been applied to generalize additional results of expected utility analysis – including aspects of the Arrow-Pratt characterization of comparative risk aversion, RothschildStiglitz comparative statics of risk, insurance theory and state-dependent preferences – to probability-smooth non-expected utility preference functions.

### 3. LOCALLY SEPARABLE vs. LOCALLY NONSEPARABLE KINKS

• It is well-known that calculus can also be used for the exact analysis of nondifferentiable functions, as long as they are not too nondifferentiable.
• More generally, a function ƒ(%) will satisfy the Fundamental Theorem of Calculus as long as it is absolutely continuous over the interval in question.
• As mentioned, this paper does not consider all types of kinks , and in particular, there is little theoretical, empirical or intuitive reason to expect that agents exhibit “Cantor-type” preferences over monetary lotteries.
• The distinct features of the two functions’ kinks at (1,1) (and elsewhere), with their respective implications for the applicability of calculus, are laid out in Sections 3.2.

### 3.1 Piecewise-Linearity and Local Piecewise-Linearity

• Figures 1a and 1b illustrate the piecewise-linear structures of the functions S(%,%) and L(%,%) about the point (1,1), by indicating the formulas they take over different regions in their domain R+ 2.
• The authors will ultimately concentrate on the same type of preference functions as in his analysis, namely those whose first order approximations (in their case, either linear or piecewise-linear) are strictly increasing.

### 3.2 Local Separability vs. Local-Nonseparability

• In particular, while S(%,%)’s kinks are amenable to the local and global calculus of directional derivatives, L(%,%)’s are not.
• In addition, as long as the authors account for directions, the Fundamental Theorem of Calculus still applies to all of S(%,%)’s line and path integrals, even integrals along its horizontal and vertical lines of kink points in Figure 1a.
• It is this property of local additivity in 27 Although additivity does hold when k1, k2 have opposite signs, it fails for negative k1, k2.

### 3.3 Ordinal Implications of Locally Separable vs. Locally Nonseparable Kinks

• Preference functions over lotteries, like elsewhere in consumer theory, represent an individual’s preferences over the objects of choice by mapping them to unobservable “preference levels.”.
• 18 dV(P2)/d2!2 =2(P) > 0 at a given P.31 About any such P, the observable function 2(%) again inherits the general smoothness /kink structure of V(%) in any given set of variables (payoffs and/or probabilities), including its local separability/ nonseparability properties.

### 4. EXPECTED UTILITY AND FRÉCHET DIFFERENTIABLE PAYOFF KINKS

• The authors examine the nature and prevalence of such payoff kinks in expected utility, Fréchet differentiable and rank-dependent risk preferences.
• 32 We shall also compare the global patterns of kinks implied by the expected utility versus rank-dependent forms.the authors.the authors.

• It is clear that an expected utility preference function VEU(P) # $ni=1U(xi)%pi will exhibit payoff kinks if and only if its von Neumann-Morgenstern utility function U(%) is kinked at one or more payoff levels. • 33 Kinked von Neumann-Morgenstern Utility Function and its Indifference Curves (indifference curves are kinked along dashed lines) Figures 2a and 2b illustrate a strictly risk averse von Neumann-Morgenstern utility function U(%) with a kink at payoff level x = 100, along with its indifference curves in the HirshleiferYaari diagram,34 for fixed state probabilities –p1, –p2. • Any larger or smaller load factor from C will lead to a partial insurance optimum, at either a tangency or kink point, located strictly southeast of the certainty line. • 36 Actuarially unfair insurance induces a budget line from C to a point on the certainty line with lower expected value. ### 4.2 Local Separability of Fréchet Differentiable Payoff Kinks • As with expected utility preference functions, Fréchet differentiable preference functions are also locally payoff-separable. • By (29) this implies that the left side of (63) converges to 0, and thus so does the right side (and does so uniformly in$(x1…,xn) – (–x1,…, –xn)$), which establishes (60) and hence the property of local separability in (individual or joint) regular and whole probability payoff changes. • That is, the marginal effect of a joint payoff shift equals the sum of the marginal effects of its individual component shifts. • In the latter case, these forms will: continue to have welldefined local utility functions, satisfy the generalized expected utility results of Section 2.3, satisfy the directional payoff derivative formulas (64) – (67), and satisfy the first order and first order conditional risk aversion properties and results of the following section. ### 4.3 First Order and First Order Conditional Risk Aversion • Segal and Spivak (1990) have defined and characterized a particular sense in which risk preferences about piecewise-linear payoff kinks can be qualitatively different from smooth preferences:. • This notion is not limited to preferences about certainty. • Many insurance contracts explicitly specify such events, and may or may not refund the original contract price if they occur. • Similarly, if VFR(%)’s local utility functions are twice differentiable (and U(x;P), U"(x;P), U!(x;P) are continuous in P), V(%) exhibits second order risk aversion at all sure wealth levels. ### 5. RANK-DEPENDENT PAYOFF KINKS • As it turns out, one of the most important non-expected utility preference functions has probability derivatives that are amenable to generalized expected utility analysis, but payoff kinks that are not (nor to the calculus of directional derivatives). • This form, proposed by Quiggin (1982),43 is known as the “rank-dependent expected utility” or rank-dependent form. • When two or more of P’s payoff values are equal, ties in the definition of x̂i, p̂i can be broken in any manner. • Having noted this, and in order to concentrate on the additional kinks inherent in the rank dependent form, the authors henceforth assume that R(%) and G(%) are both smooth (continuously or up to infinitely differentiable) with R"(%), G"(%) >. ### 5.1 Rank-Dependent Probability Derivatives and Local Utility Functions • Chew, Karni and Safra (1987) have shown that the rank-dependent form VRD(%) does not / can not satisfy the strong smoothness condition of Fréchet differentiability (eq. (29)), except for its special case VEU(%). • But it also follows that URD(%;P) generally has kinks at the boundaries of these intervals, that is, at each of P’s payoff values x̂1,…,x̂n, with distinct left/right directional derivatives at x̂i given by 45 As observed by Chew, Karni and Safra (1987), the result for infinite-outcome distributions is less straightforward. • 46 As with all derivations of probability derivatives/local utility functions, the authors invoke the discussion preceding (15). • The authors know that the expected utility characterizations of first order stochastic dominance preference, risk aversion, and even certain aspects of comparative risk aversion48 do not require differentiability of the von Neumann-Morgenstern utility function U(%), and this is also true for the generalized expected utility characterizations of these properties in terms of not-necessarilypayoff-differentiable local utility functions U(%;P). ### 5.3 Rank-Dependent Payoff Derivatives and Payoff Kinks: General Formulas & Properties • The authors continue to assume that R(%) and G(%) are smooth (continuously or even infinitely differentiable) with R"(%), G"(%) >. • Since it includes each type of payoff shift as a special case, (90) allows for the derivation of the regular, whole-probability and partial probability payoff derivatives for the rank-dependent form. ### 6.1 Modeling Departures from Separability: Two Approaches • But say the evidence suggests that individuals’. • %) is not exactly linear over any region, it is everywhere locally linear, and its indifference curves gradually move from flatness to steepness across the domain. ### 6.2 Are Observed Risk Preferences Kinked in the Payoffs? • Payoff kinks were not among the empirical phenomena reported or modeled in the classic expected utility analysis of Friedman and Savage (1948), or in its modification by Markowitz (1952) who defined the function U(%) over changes from current wealth and also observed that individuals are generally averse to small symmetric gambles about current wealth. • Indeed, as noted in the discussion of Figure 3b, a kinked utility function U(%) implies that the Engel curve for insurance can (in whole or part) take an unusual V-shaped form. • An alternative and perhaps more parsimonious way to reconcile the above three phenomena – both with each other and also with the hypothesis of risk aversion – might be to attribute them to heterogeneity of individuals’ subjective beliefs. • Figure 8b shows how optimistic beliefs can lead a risk averter to purchase zero insurance, even when it is better than actuarially fair. ### 6.3 Induced Payoff and Probability Kinks • Most economically important situations of choice under uncertainty (e.g., agriculture, insurance, real investment) involve delayed-resolution risk or uncertainty. • Researchers such as Markowitz (1959, Ch.11), Mossin (1969), Spence and Zeckhauser (1972), Kreps and Porteus (1979), Machina (1984) and Kelsey and Milne (1999) have examined how agents’ induced preferences over such temporal lotteries – that is, the preferences obtained by maximizing out the auxiliary decision(s) – can systematically differ from their underlying risk preferences. • Depending upon the decision, the auxiliary variable 2 could either be continuous or discrete. • When the auxiliary variable 2 can only take discrete values, the induced preference function W(%) is in general only regionwise smooth, which as seen in Section 6.1, implies it will generally exhibit locally nonseparable payoff kinks on the boundaries of these regions. Did you find this useful? Give us your feedback Content maybe subject to copyright Report UC San Diego Recent Work Title Payoff Kinks in Preferences Over Lotteries Permalink https://escholarship.org/uc/item/7vn7d2hs Author Machina, Mark J Publication Date 2000-09-11 eScholarship.org Powered by the California Digital Library University of California 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.

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##### Frequently Asked Questions (1)
###### Q1. What are the contributions in "Payoff kinks in preferences over lotteries" ?

In this paper, the authors identify two distinct types of payoff kinks in preference functions over monetary lotteries, namely  locally separable '' versus  local nonseparable '' kinks.