Ordinal One-Switch Utility Functions

Transcription

1 Ordinal One-Switch Utilit Functions Ali E. Abbas Universit of Southern California, Los Angeles, California 90089, David E. Bell Harvard Business School, Boston, Massachusetts 0163, Abstract We stud the problem of finding ordinal utilit functions to rank the outcomes of a decision when preference independence between the attributes is not present. We propose that the next level of complexit is to assume that preferences over one attribute can switch at most once as another attribute varies from low to high. We refer to this propert as ordinal one-switch independence. We present both necessar and sufficient conditions for this ordinal propert to hold and provide families of functions that satisf this new propert. 1. Introduction An ordinal utilit function reflects a decision maker s rank order for the consequences of a decision, as described b a set of attributes, X,..., 1 X n. For example, a function u( x, ) over two attributes X and Y is an ordinal utilit function when it returns a higher value for a more preferred prospect and returns equal values when two prospects are equall preferred, i.e. u(x 1, 1 ) > u(x, ) Û (x 1, 1 ) (x, ) and. An ordinal utilit function is sufficient to determine the best decision alternative if there is no uncertaint about the outcomes. The best decision alternative corresponds to the prospect with the highest ordinal utilit. When uncertaint is present, a cardinal utilit function is needed, and the best decision alternative is the one with the highest expected utilit. Identifing a suitable utilit function to determine the best decision alternative (whether ordinal or cardinal) can be challenging. The standard approach is to decompose a multiattribute utilit function into lower-order components b finding appropriate simplifications. A wealth of literature has provided conditions for preferences over lotteries to characterize the functional form of the cardinal utilit function. See for example Pfanzagl (1959), Bell (1988) and Abbas (007) for conditions on univariate lotteries, and Farquhar (1975), Fishburn (1974 and 1

2 1975), Keene and Raiffa (1976), Bell (1979), Abbas (009), and Abbas and Bell (011 and 01) for conditions on multivariate lotteries. Much less literature has provided ordinal conditions that characterize the utilit function. The best-known condition on ordinal preferences is based on the idea of preferential independence (Debreu 1960), which requires that ordinal preferences for consequences of a decision characterized b an subset of the attributes do not depend on the levels at which the remaining attributes are fixed. If three or more attributes satisf this ordinal propert, then the ordinal utilit function must be a monotone transformation of an additive function of the attributes, i.e. n u( x1,..., xn ) g ui ( xi ), i1 where n 3, g is a monotonic function and u, i 1,..., n are arbitrar functions. i The condition of mutual preferential independence does not determine the ordinal functions u, i 1,..., n, but it does decompose the functional form into univariate assessments over each i of the individual attributes, which reduces the search space for the structure of the ordinal utilit function significantl. Debreu s result onl works if there are three or more attributes. For n= there are classic conditions which dictate the additive form, for example Karni and Safra (1998). One of the contributions of this paper is to derive the general form of preference independence for two attributes. While ordinal preferences described b mutual preferential independence ma suffice in a variet of problems, it is natural to consider how to proceed with the construction of the utilit function if it does not hold. Keene and Raiffa (1976) introduced the notion of utilit independence, where preferences for lotteries over a subset of the attributes do not depend on the levels of the remaining attributes. The derived a famil of cardinal utilit functions that satisfies this propert. In Abbas and Bell (011, 01), we generalized the notion of utilit independence for cardinal utilit functions. The idea is to consider how man times preferences over pairs of gambles on one attribute can switch

3 as another attribute ranges from small to large. That is, for an two lotteries x 1,x in X, how often can the difference in expected utilities of the lotteries, u(x 1, ) - u(x, ), switch sign? Utilit independence assumptions correspond to no switch. In our work, we proposed a functional form that allows for preferences over lotteries to switch at most once, and we also generalized the result to functional forms that allow for preferences to switch an number of times. But if pairwise preferences can switch multiple times as varies one has to wonder whether the attributes have been chosen appropriatel. The purpose of this paper is to provide new conditions on ordinal preferences to help the analst identif suitable ordinal utilit functions that ma be used when preferential independence conditions are not present. We start our discussion with the case of a single attribute, and discuss the implications of ordinal one-switch and zero-switch independence for wealth. Next, we consider the case of two attributes. We consider the functional form of an ordinal utilit function where onl one attribute is preferentiall independent of the other. We then consider our new ordinal propert. We provide families of functions that satisf this ordinal one-switch propert and also discuss conditions (Theorem 5.1) to identif when this propert does not hold. We conclude with a new formulation (Theorem 7.3) that can be used to construct ordinal one-switch utilit functions using two curves on the surface of the ordinal function, subject to reasonable conditions on its derivatives. Throughout the paper we will assume utilit functions are continuous, and on occasion, differentiable.. Basic Notation and Definitions To simplif the exposition, we start our analsis b considering the case of a one-attribute utilit function, u( x w), where w is initial wealth and x is the increment to be considered. In our exposition, we first consider lotteries on X, which we write as x, denoting the expected utilit of those lotteries as u(x + w). A utilit function, u( x w) is cardinal zero-switch if for an two lotteries over X, sa x 1 and x, the difference in expected utilities D(w) = u(x 1 + w) - u(x + w) does not change sign as w varies. 3

4 Throughout the paper, when we sa that a function does not change sign, we mean that it is either alwas positive, alwas negative, or alwas zero. As has been shown b Pfanzagl (1959), the cardinal zero-switch condition is satisfied for all lotteries if and onl if the utilit function is a linear transform of either a linear or an exponential utilit function, i.e. the utilit function is one of the forms u( x) a be cx or u(x) = a + bx. Having discussed the condition of zero-switch utilit functions over lotteries, we now define the corresponding propert for ordinal utilit functions over known consequences. Definition.1 A utilit function, u( x w) is ordinal zero switch if for an two fixed values x x of attribute X the difference ( w) u( x1 w) u( x w) does not change sign as w varies. 1, Theorem.1 A utilit function ux ( ) is ordinal zero-switch if and onl if it is strictl monotone. The proof is evident. As we have seen, the equivalent cardinal propert for zero-switch utilit functions requires the linear or exponential functions (both of which are monotone functions). The ordinal zero-switch propert provides less specificit than the cardinal zero-switch propert (requiring the function onl to be monotone and not necessaril linear or exponential). This is a general theme that we shall observe throughout this paper: ordinal utilit functions provide less specification than the corresponding properties over lotteries as the impose milder conditions. However, the are also easier to assert than preferences over lotteries. What if ordinal preferences for increments of X can switch, but at most once? Once again let us first start with the cardinal case. A utilit function, u( x w) is a cardinal one-switch utilit function over wealth if for an two lotteries, x 1 and x, over X the difference in expected utilities, D(w) = u(x 1 + w) - u(x + w), is either never zero, is zero for all w, or is zero for at most one w. In this and later switching definitions, the one-switch condition will include the zero-switch condition as a special case, so that zero-switch utilit functions will automaticall qualif as oneswitch functions. 4

5 Bell (1988) showed that the onl utilit functions that satisf this cardinal one-switch condition are the four functions: u( x) ax bx c; ( ) ( ) cx cx u x a bx e d; u( x) ax be d; and bx dx u( x) ae ce f. We now turn to the ordinal case. Definition. A utilit function u(x+w) is ordinal one switch if for an pair of X consequences either one is preferred to the other for all w, or the are indifferent for all w, or there exists a unique wealth level above which one consequence is preferred, below which the other is preferred. Evidentl the ordinal one-switch condition implies that ( w) u( x1 w) u( x w) is either alwas zero, never zero, or zero for exactl one w. In particular it excludes cases where two values of X are indifferent on an interval but strict preference holds on another, for if x1 ~ x at an two wealth levels then our definition requires that x1 ~ x for all w. Theorem. A utilit function ux ( ) is ordinal one-switch if and onl if it is unimodal (i.e. has at most one turning point, which ma be infinite). Again the proof is evident. The ordinal zero-switch condition required strictl monotone functions, and the ordinal one-switch condition requires unimodal functions. Theorem. also illustrates the generalit of the ordinal one-switch condition in comparison to its corresponding cardinal form. Note that the four cardinal one-switch utilit families shown above are all unimodal functions. 3. Two Attributes: Ordinal Zero-Switch Independence So far we have considered ordinal and cardinal switching properties for a single attribute. Specifing similar properties for more than one attribute enables a decomposition of multiattribute utilit functions that simplifies their assessment. For the sake of completeness we review preference independence. 5

6 Definition 3.1: Attribute X is preferentiall independent of attribute Y if the preference ordering of an two levels of X is independent of the fixed level of Y, that is, the difference ( ) u( x, ) u( x, ) has constant sign as varies. As we noted, Debreu (1960) discussed the implications of mutual preferential independence for three or more attributes, where ever subset of the attributes is preferentiall independent of its complement. Surprisingl, we have not found an explicit discussion of this ordinal propert for two attributes. The following theorem characterizes this propert. Theorem 3.1 X is preferentiall independent of Y if and onl if u( x, ) v ( x), a strictl monotonic function of v 1 for all. where is Note that the function v ( ) 1 x itself need not be monotone. For example u( x, ) e sin( x) satisfies the condition of X being preferentiall independent of Y. The following equivalent definition of preferential independence, using the notion of zero-switch preferences, serves to underline our view of one-switch as a natural extension of preference independence. Definition 3. An attribute X is ordinal zero-switch independent of attribute Y, if for an two values of X, sa x 1 and x, the difference ( ) u( x1, ) u( x, ) does not change sign as varies. Proposition 3.1 X is zero-switch independent of Y if and onl if it is preferentiall independent of Y. Proposition 3.1 and Theorem 3.1 impl that X is zero-switch independent of Y if and onl if 1 u( x, ) v ( x),. The equivalent (stronger) condition for lotteries is the notion of utilit independence (Keene and Raiffa 1976) where preferences for lotteries over X do not change for an value of Y and so the difference D() = u(x 1, ) - u(x, ) does not change sign with. Keene and Raiffa (1976) show that this condition implies that u( x, ) g0( ) g1( ) v( x), 1 6

7 where g ( ) 1 does not change sign. It is clear that the functional form corresponding to utilit independence is a special case of that corresponding to preferential independence, where the function is an affine function of vx ( ), i.e. f v, ( ) = g 0 () + g 1 ()v(x). Once again, we observe the generalit of the functional form corresponding to the ordinal propert. Note for example, that the function v, v( x)(1 ) preferential independence of X on Y but not utilit independence. 4. Two Attributes: Ordinal One-Switch Independence e would satisf the condition of We have emphasized the equivalence of preference independence and the ordinal zero-switch propert to suggest that the ordinal one-switch propert that follows is a natural generalization of preference independence. Definition 4.1: X is ordinal one-switch independent of Y, written X 1S Y, if for an pair of consequences in X either one is preferred to the other for all, or the are indifferent for all, or there exists a unique level of (which ma be infinite) above which one of the consequences is preferred, below which the other is preferred. This definition is unchanged even if X is multidimensional. It also extends readil to n-switch independence but we believe that zero-switch and one-switch are the most useful cases. Note that a function that satisfies ordinal zero-switch independence (preferential independence) of X from Y also satisfies the condition of ordinal one-switch independence of X from Y. In correspondence with the single-attribute case, our definition of ordinal one-switch independence excludes the case where a pair of consequences is equall preferred onl on an interval: if two X values are ever indifferent at two different values of Y then the must be indifferent for all values of Y. The remainder of this paper will provide tests to establish whether or not a particular function u( x, ) is one-switch and will provide families that, subject to given conditions, satisf the oneswitch rule. The following examples illustrate the brute-force method of testing for ordinal oneswitch independence for some simple functions. 7

8 Example 4.1 Consider the ordinal utilit function ux, x x For an two values of X, sa x1, x, the difference. ( ) u( x, ) u( x, ) ( x x ) ( x x )( x x ). This function satisfies X 1S Y, because the difference switches sign onl once, when x x. / Example 4. Consider the function The difference u x, x x x ( ) u( x, ) u( x, ) ( x x )( x x ) ( x x ) ( x x ) is quadratic in, and therefore it need not satisf X 1S Y. For example, for x 1 1 and x 0, there are two switches as varies. 5. Sufficient Conditions for Ordinal One-Switch Independence In this section we consider how to test whether a given function u( x, ) satisfies the one-switch condition. Of course, we can alwas tr the brute-force method for particular values of X for simple functions, but more complex functions require additional tools. Test 1: : u( x, ) u( x, ) is strictl monotonic in x x. if ( ) u( x, ) u( x, ) is strictl monotonic then it can cross the x-axis at most once, thus If satisfing the one-switch condition. The function in Example 4.1 is a linear function of and therefore (strictl) monotonic. The function in Example 4. is quadratic and thus not monotonic. Monotonicit of is a sufficient but not a necessar condition for satisfing the ordinal one- x switch condition. To illustrate, the function u( x, ) e satisfies the ordinal one-switch propert. However, if we pick two values of X, sa x 1 and x 1, the difference 8

9 ( ) e e is not monotone because the derivative negative when. ' ( ) is positive when and Test : Constant Sign of the Cross-Derivative (for differentiable functions) : X if the cross derivative u( x, ) x does not change sign. If the cross derivative has a constant sign (either positive or negative), then for an x x,, the difference u( x, ) [ u( x1, 1) u( x1, )] [ u( x, 1) u( x, )] dxd x x x 1 1 does not change sign. This implies that the difference ( ) u( x1, ) u( x, ) is strictl monotone if the cross-derivative is either alwas positive or alwas negative. This condition is sufficient but not necessar. For example u(x, ) = x satisfies X 1S Y but the cross derivative is which changes sign. We note that the condition on the sign of the cross-derivative is a smmetric condition. Therefore, it implies not onl that X 1S Y, but also that Y 1S X. x Example 5.1 Consider the function u x, k x. The cross derivative, u k x has constant sign. This implies that satisfies X 1S Y (and that Y 1S X ). Example 5. Consider the function ux x x u sin x 0 impling that X 1S Y (and Y 1S X). x, sin. The cross-derivative 9

10 Test 3: Monotone Ratio: u( x1, ) X 1S Y if the ratio is strictl monotone in for x x. u( x, ) If u( x1, ) and u( x, ) are equal at both 1 and then their ratio is 1 at those two values and x thus the ratio cannot be strictl monotone in. To illustrate, if u( x, ) e, then the ratio u( x1, ) ( x1 x) e is strictl monotone and therefore this function satisfies the ordinal one-switch u( x, ) condition. Test 4: Local and Boundar Double Switches If u( x, ) does not satisf X 1S Y then it must have a pair, x 1 and x, that switches twice, i.e. x 1 x " < 1 ; x x 1 " 1 < < ; and x 1 x " > But finding such pairs x 1 and x might be difficult. To help identif such pairs, we introduce a test based on two new definitions: local and boundar double-switches. A local double switch is one where x 1 and x lie in a local neighborhood of each other and switch twice. Definition 5.1 u( x, ) has a local double switch at x 1 if, for all small enough values of d 0, we have x1 ~ x1 d at two values of, but not for all. A local double switch ma be found quite easil b identifing an x such that sign twice as varies. u( x, ) x changes Definition 5.. X has a boundar double switch in X if x1, x s.t. x1 ~ x at both boundar values of Y on which u is to be defined but there exists at least one value of within the boundar for which x1, x are not indifferent. This kind of double switch is also relativel eas to test for. If 1 and are an two values of, in particular those at the boundaries, then we can assess, plot, or calculate u( x, 1) and u( x, ) and then check directl whether these two curves cross twice. If the do, we then verif strict inequalit of the curves u( x1, ) and u( x, ) for at least one value of. 10

11 Clearl for u to be one-switch it is necessar that it have neither a local nor a boundar double switch. But when is the lack of local or boundar double switches sufficient to impl X 1S Y? Theorem 5.1 If u( x, ) has neither a local double switch nor a boundar double switch in X, and if for all fixed values of Y, u( x, ) is unimodal in x, then X 1S Y. The proof is length, but the intuition is simple. If u( x, ) has a double switch with x1 ~ x at both 1 and, then we look for a nearb double switch involving a pair of x s that are closer than x 1 and x. In general this is not alwas possible (as in Example 5.3 below). However if the curves u( x, ) are unimodal in x, then a sequence of ever closer x s can be constructed until either the x s are arbitraril close (a local double switch) or the s reach the boundar (a boundar double switch). The proof, along with others, is in an appendix. The following example shows that a general u can have a double switch but neither a localdouble switch nor a boundar double switch. Example 5.3 Consider the function 1 ( m ( ) 1) u x x x m m x 3 3 (, ) ( )( ( ) 1), where m ( ) e e e. To test for local double-switches we take the partial derivative u( x, ) ( x m( ))( x m( ) 1). x The partial derivative is zero if and onl if x m( ) or x m( ) 1. Therefore, u( x, ) has no local double switches. To test for boundar double-switches, we have x x u( x, ) x 3 and x 3x u( x, ) x 3. 11

12 Two values x 1 and x are indifferent at if and onl if x x x x 3x x at if and onl if x x x x 1 9x x 1 we see that an solution must satisf x1 x and 1. Comparing these two expressions and x x x x, which are not 1 3 simultaneousl possible for different x s. Hence there are no boundar double switches. However u does have a double switch in this example, and thus X is not 1S Y. For example x.5, x 1.75 are indifferent at m.5 and.75, corresponding to 1.55 and.75 respectivel. Note that u( x, ) is not unimodal in x as it is a cubic function. Based on our difficult in constructing Example 5.3, we believe that as a matter of practice the absence of local and boundar double switches can be an important indication that X 1S Y even when the marginals are not unimodal. Milgrom and Shannon (1994) developed a single crossing test for three dimensional utilit functions that will be useful to us later. Similar to the Debreu result, their result does not appl to the case n=. 6. Families that Satisf One-Switch Independence In our prior work, we applied the one switch independence idea to cardinal utilit functions (Abbas and Bell 011) and showed that the most general such function is u(x, ) = g 0 () + g 1 ()[ f 1 (x) + f (x)w()], where g ( ) 1 does not change sign and w() is a monotone function. This function necessaril also satisfies ordinal one switch (because a sure thing is a special case of a gamble). We have ( ) u( x, ) u( x, ) g ( ) f ( x ) f ( x ) w( )[ f ( x ) f ( x )], which, since g ( ) 0, changes sign at most once when w() = f (x ) - f (x ) f (x ) - f (x 1 ). In Abbas and Bell (011), we illustrated how to assess such functions and in Abbas and Bell (01) we provided a variet of special cases that the analst ma wish to choose from since the also satisf the ordinal condition. Note that this cardinal form does not necessaril satisf 1

13 Tests 1,, 3 or 4. For example, ( ) need not be strictl monotone because g ( ) 1 can be an non-negative function, such as +sin(). As we have seen, ordinal functions satisfing preferences over consequences are more general than equivalent cardinal functions and so it is not surprising that there are even more general functions satisfing this ordinal one switch propert. The following theorem provides a famil of such functions. 0 i i i i1 Theorem 6.1 The sum of functions, u x v x k v x w satisfies X 1S Y if each derivative term monotonic. k has constant sign for all and if ' ' i i i is strictl u x Note that v x w Σ k ' ' i i i has constant sign if is monotonic but not otherwise, a further demonstration that the cross derivative condition is a sufficient but not necessar condition. Example 6.1 Consider the function, u x v x x e e e e 0 x x This function satisfies the format of Theorem 6.1 because the term is respectivel and which all have the same sign and. Example 6. Consider the function, w w ae be w1 w is zero if e ( ) ' ' monotone, that is, if w w 0 1 u x v x e v x e. The difference w w e w1 w b / a. This alwas has at most one solution if. Note that and do not have to be monotone individuall. So has but and., which changes sign uniquel when is but changes sign when x 3 e. 7. Ordinal Utilit Functions Satisfing One-Switch Independence 13

14 It will be convenient to adopt the notation v1 ( x) u( x, 1), v( x) u( x, ) where 1, are the boundaries of an interval of interest. Theorem 7.1 If X 1S Y then there exists a function such that u( x, ) v ( x), v ( x), for 1. where v1 ( x) u( x, 1), v( x) u( x, ), is strictl monotone in v 1 and in v, and the function satisfies the boundar conditions,,,,, v v v v v v. 1 1 Theorem 7.1 provides a necessar condition for ordinal one-switch functions. Compare this formulation to that of preferential independence in Theorem 3.1, where onl one univariate function of x was required. Theorem 7.1 shows that ordinal one-switch independence is a generalization of preferential independence requiring two functions v1( x), v( x ). Example 7.1 Consider the cardinal one-switch form with and. As we have seen, this function satisfies X 1S Y because it satisfies the cardinal condition, but note that it can also be written as u( x, ) v ( x), v ( x),, where is linear in both v ( x), v ( x ). Therefore it satisfies the necessar condition for ordinal one-switch independence, as expected., u x g g v x w v x 0 1 Example 7. Consider the function u( x, ) x x x x(1 ) x. This function can be expressed in terms of the representation u( x, ) v ( x), v ( x), and v x ux x v1 x u x,0 x,1 then 1 u( x, ) v ( x), v ( x), v ( x)(1 ) v ( x).. If we define However, as we showed in Example 4., this function does not satisf the one-switch rule. 14

15 Our focus will now be on twice continuousl differentiable functions. For expositional purposes it will be convenient to define DR( v1, v, ) as the ratio of partial derivatives v, v, / v to 1 v, v, / v, i.e. v, v, / v DR( v1, v, ) v, v, / v 1 Theorem 7. Monotone Tradeoffs: Given u( x, ) v ( x), v ( x), monotone in 1, where is strictl v v and satisfies the boundar conditions of Theorem 7.1 then if v ( x), v ( x), has a local double switch then DR( v1( x), v( x), ) is not strictl monotone in. Put the other wa, this result shows that if DR( v1, v, ) is strictl monotone in then the corresponding u( x, ) cannot have a local double switch. Note that the structure of ensures that it also cannot have a boundar double switch. For if x1 ~ x at both 1 and, then automaticall makes them indifferent for all, which is not a double switch. An function v, v, can be used to generate utilit functions u( x, ) b replacing v 1 b v ( x) u( x, ) and 1 1 v b v( x) u( x, ). Milgrom and Shannon (1994) proved an important result for three dimensional ordinal utilit functions that we can appl directl to to guarantee that the corresponding u( x, ) satisfies X 1S Y. Theorem 7.3 The function v, v, satisfies X 1S Y over an interval ( 1, ) for an choice of functions v 1 = u(x, 1 ) and v = u(x, ) iff is strictl monotone in v 1 and v and DR( v1, v, ) is strictl monotone in. Example 7.4 Consider the case u( x, ) f1( x) f( x). This alwas satisfies X 1S Y. We ma construct the corresponding as follows, with, for example, 1 1 and 1. We have 15

16 v1 v v1 ( x) u( x, 1) f1( x) f( x), v( x) u( x,1) f1( x) f( x), so that f1 and f v v v v v u v. 1 ielding Checking the conditions of Theorem 7.3 we note this is strictl monotonic in v 1 and v over the 1 range (-1,1) and DR( v1, v, ) which is strictl monotonic over the same range. 1 While we know from Theorem 7.1 that ever u(x,) satisfing X 1S Y can be expressed as v ( x), v ( x), and we see from Theorem 7.3 that ever v ( x), v ( x), obeing its conditions satisfies X 1S Y, we see from the following example that not ever one-switch v ( x), v ( x), satisfies the conditions of Theorem 7.3. Example 7.5 The function v1( x), v( x), u( x, ) e e x x does not satisf the conditions in Theorem 7.3. satisfies X 1S Y but the corresponding To see that u is one-switch note that if a>b are two values of X then the cross when e e e b b e e a a, which has a unique positive solution in because e is monotone for >0. However e is not monotonic when <0. The corresponding to u satisfies the conditions of Theorem 7.3 when >0, but not alwas when <0. The problem arises due to the fact that although u does allow switching, it does so onl when >0, but the violations of ϕ occur when <0 (ϕ does not violate the conditions if is restricted to the positive range). Corollar 7.4 The function v, v, satisfies X 1S Y over the interval ( 1, ) for an choice of functions v 1 = u(x, 1 ) and v = u(x, ) if is strictl monotonic in v 1 and v and if 0 v 1 and 0. v 16

17 The cross-derivative conditions impl that DR( v1, v, ) is strictl monotone decreasing so that Theorem 7.3 applies immediatel. We highlight this result because the conditions on in this result seem highl intuitive and transparent. Generating ordinal utilit functions using the famil v, v, with appropriate restrictions seems to us to be a good wa to generate a wide range of one-switch utilit functions. 8. Conclusions The concept of zero-switch preference, expressed ordinall as preference independence and cardinall as utilit independence, has a long pedigree in the literature. The ordinal propert in the univariate case corresponds to a monotone function of wealth, and to the notion of preferential independence for multiple attributes. Cardinall, this zero-switching propert corresponds to the more specific linear and exponential utilit functions and to utilit independence for the case of multiple attributes. While the concept of zero-switching preferences leads to simple functional forms, it is hard to proceed if the do not appl. We propose that the natural extension of these concepts is that preferences can switch but at most once. In the ordinal case, we have shown that this propert corresponds to the notion of a unimodal function, a natural extension of monotone functions in terms of switching preferences. The extension of the ordinal propert of preferential independence to one-switch preferences has been the focus of this work. We defined the notion of ordinal one-switch utilit independence and focused on two main problems. The first was whether a given utilit function satisfies this ordinal propert. We defined easil-discoverable local and boundar double switches and showed when testing for them was sufficient to prove X 1S Y. Though the absence of local and boundar switches is not, in general, a definitive indication that there are no double switches, we believe that it is a ver strong indication. The second focus was on how to generate one-switch functions. We showed that ordinal oneswitch utilit functions ma be represented in the form u( x, ) v ( x), v ( x), and, subject to 17

18 reasonable restrictions on ϕ, that an function of the form v ( x), v ( x), will be a one-switch utilit function. We believe that, despite exceptions (Example 7.5), most one-switch functions can be generated in this wa. Consequentl we propose the functional form u( x, ) v ( x), v ( x), as the ordinal generalization of our recent work on one-switch independence (Abbas and Bell 011, 01). References Abbas, A. E Invariant utilit functions and certain equivalent transformations. Decision Analsis. 4(1) Abbas, A.E Multiattribute utilit copulas. Operations Research. 57 (6) Abbas, A. E. 01. Utilit Copula Functions Matching All Boundar Assessments. 36() Operations Research 61(), Abbas, A. E. and D. E. Bell One-Switch Independence for Multiattribute Utilit Functions. Operations Research. 59 (3) Abbas, A. E. and D. E. Bell. 01, One-Switch Conditions for Multiattribute Utilit Functions. Operations Research. 60 (5) Bell, D. E Multiattribute utilit functions: decompositions using interpolation. Management Science. 5 () Bell, D. E One-switch utilit functions and a measure of risk. Management Sci. 34(1) Debreu, G Topological Methods in Cardinal Utilit Theor. In K. Arrow, S. Karlin and P. Suppes (Eds.) Math. Methods in the Social Sciences. Stanford Univ. Press, Stanford, CA Farquhar, P.H A Fractional Hpercube Decomposition Theorem for Multiattribute Utilit Functions. Operations Research, 3, Fishburn, P. C von Neumann-Morgenstern utilit functions on two attributes. Operations Research,, Fishburn, P. C Approximations of two-attribute utilit functions. Mathematics of Operations Research,, Karni, E. and Safra, Z Hexagon condition and additive representation of preferences: An algebraic approach. Journal of Mathematical Pscholog, 4, Keene, R. L. and H. Raiffa Decisions with Multiple Objectives. Wile. New York, NY. 18

19 Milgrom, P. and Shannon, S Monotone Comparative Statics. Econometrica, 6 (1) Pfanzagl, J A general theor of measurement. Applications to utilit. Naval Res. Logist. Quart. 6, von Neumann, J. and O. Morgenstern Theor of games and economic behavior. Princeton, N.J. Appendix Proof of Theorem 5.1 For a given that does not satisf X 1S Y, let be the set of all that switch more than once, where. We will sa that a pair is dominated if with. We will sa that a function changes sign twice if as increases, it is strictl positive, then strictl negative, then strictl positive, or the same statement with positive and negative reversed. Lemma A1 If and if changes sign at least twice, then is dominated. Proof If changes sign twice then, reling on the continuit of u, so does an function of the form if and are sufficientl small. B picking we see that is dominated. QED. Suppose has zeros; i.e. where. We know from Lemma A1 that an undominated u cannot change sign twice. Therefore for at least of the If all of the s are finite then of them involve being -shaped at, that is, the touch the axis at without changing sign. For to have no explicit -shaped tangent to the axis the s would have to be infinite. Lemma A If and forms a -shaped tangent at, then is dominated if all marginals are unimodal. 19

20 Proof It suffices to show that for some the equation has two distinct solutions for some. As a first-order approximation we ma write this as or. We know, b assumption, that the RHS is -shaped in, touching the axis at. In addition, if u is unimodal, then there exist infinitel man pairs such that. Consider the two possible pairings and. One of these makes the LHS positive, the other makes it negative. Since the RHS is -shaped, one or the other of these choices has two solutions for. Since and ma be chosen to be arbitraril small the approximation is inconsequential. QED. So, an undominated element of must have a that changes sign at most once, and makes a tangent at either or, perhaps both, but not between. Lemma A3 If has a that changes sign exactl once, and has no other zeros except at infinit, then it is dominated if all marginals are unimodal. Proof If, consider. Again, because the marginals for fixed are unimodal, we know s.t.. But for sufficientl small, must also cross zero. Hence. Hence is dominated. QED. What remains is that if is undominated, it must have a that is strictl non-zero except at and where it is zero. This is a boundar double switch. To outline the remainder of the proof, starting with an double switch having we ma construct a sequence of double switches b replacing each sequence member with another pair that dominates it. It is apparent from our derivation that each step in the sequence can be assumed to be arbitraril small. Either this sequence converges to a local double switch, or it has a subsequence that converges to a local double switch, in which case we are done, or it 0

21 converges prior to becoming a local double switch. We will show that the sequence ma alwas be restarted from such a situation. Lemma A4 If switch twice where are small, and for all, then switch twice. Proof We know is double switch, but for small this equals switches twice, since. But then also switches twice. QED. which means that We note that if is a sequence of double switches converging to where then if then b selecting some large enough, we know b Lemma A4 that is double switch and, moreover, that dominates since. Proof of Theorem 5.1 Since u is not one-switch, then S has at least one element sa. Construct a sequence ( ) b selecting for an element of that dominates. If converges to a limit (or a subsequence does) where then there are two cases to consider. Either is in S or it is not. If it is, simpl restart the sequence with in place of. If it is not, then Lemma A4 must appl in which case the sequence ma again be restarted. Since the sequence so constructed cannot end finitel, converges to, and the sequence ( ) must contain a subsequence in which and. This point is a local double switch. QED. Proof of Theorem 6.1. Δ ki i i w where v x v x and v x i , Δ w Σk i i i. The derivative with respect to Y gives ' 1 v x. i i i 1

22 This derivative has constant sign for all because each has the same sign as. Therefore, Δ is monotone. Hence. QED. Proof of Theorem 7.1 We start with the following Lemma, where we define v1 ( x) u( x, 1), v( x) u( x, ). Lemma 7.1 If X 1S Y and if there exist an two distinct values x1, x such that then u( x1, ) u( x, ) for all. v1 ( x1 ) v1 ( x) and v( x1 ) v( x) Proof of Lemma. If there exist x1, x that satisf v1 ( x1 ) v1 ( x) and v( x1 ) v( x), then this means indifference of x1, x at two points 1,, which (b our definition) is a violation of X 1S Y unless u( x1, ) u( x, ) for all. Geometricall speaking, Lemma 7.1 implies that if we pick an two curves u( x, 1) and u( x, ) on the surface of a function u( x, ) that satisfies X 1S Y, then there cannot be an two distinct x1, x whose utilit values are equal on these curves unless the are equal on the entire domain u( x, ) u( x, ). A corollar of this result implies that there cannot be an regions of v ( x ) 1 and v ( ) x that are flat for the same values of x. QED. We now prove the Theorem. Proof of Theorem 7.1 Define b the relation ( ) ( ) converts a two dimensional function (, ) v x, v x, u( x, ). This assignment u x into a three-dimensional function v( x), v ( x),. To provide this mapping, ever x must correspond to a unique pair v( x), v ( x ), i.e., this representation fails onl if for some and, we have v x v x, v x v x but 1 1 for some. But this cannot happen if X 1S Y as we have discussed in Lemma 7.1. The condition on the boundar conditions follows from consistenc of both sides of

23 the equation ( ) ( ) v x, v x, u( x, ). Lemmas 7. and 7.3 below show the necessit of strict monotonicit with v 1 and v. Lemma 7. If X 1S Y, and if u( x, 1) u( x1, 1) and u( x, ) u( x1, ) then necessaril u( x, ) u( x, ). 1 Proof of Lemma 7. Note that if u( x, 1) u( x1, 1) and u( x, ) u( x1, ) but there exists 3 ( 1, ) such that u( x, 3) u( x1, 3), then this would impl that preferences for x and x 1 have switched more than once, which is a violation of ordinal one-switch independence. The case of equalit at 3 is also excluded because it would impl that the difference changes sign from positive to zero and back to positive which is a violation of the one-switch condition. QED. Lemma 7.3 If X 1S Y, and if u( x, ) v () x, v (), x, where v1( x) u( x, 1) and v ( x) u( x, ) then must be strictl monotonic with respect to v 1 and v where defined. Proof of Lemma 7.3 This Lemma translates the results of Lemma 7. into the formulation of v1, v, : If v1 ( x) v1 ( x1 ) and v( x) v( x1 ), then x is preferred to x 1 at both 1 and, and so it is preferred for all 1 d 0, d 0 and d1d 0. QED. Hence v d v d v, v, 1 1,, 1 (, ) if Proof of Theorem 7. If there is a local double switch then x1 ~ x1 d at 1 and sa. This means that for the corresponding v ( x), v ( x), we have dv1 ( x) dv( x) 0 v dx v dx 1 at the two values of. Thus DR( v1, v, ) equals v' / v 1' at two values of and is therefore not strictl monotonic. QED. 3

24 Proof of Theorem 7.3 In our notation, Milgrom and Shannon (1994) showed that attributes v 1 and v will cross at most once as varies if v is never zero, if DR( v1, v, ) is strictl monotonic in, and if an two indifferent pairs of v values at a given, sa * v and * w at 1, are alwas connected b a path of indifferent points also at. Though their result concerned 1 three attributes we can appl their reasoning to, which has three attributes, and thereb obtain a result for two attributes. As our conditions are slightl different we exhibit the proof. Since is strictl monotonic in v 1 and v it is clear from the geometr that, for an fixed, its isopreference curves are connected. More specificall however, suppose * v and * w are two values of (v 1,v ) that are indifferent for some 3, where 1 3 <. Note that because of strict monotonicit in v 1 and v we must have (v 1 *-w 1 *)(v *-w *)<0 and this is true for an pair of points that is indifferent for a fixed. Suppose that v 1 *>w 1 * and v *<w * and consider a new point (v 1 *-d 1 (v 1 *-w 1 *), v *-d (v *-w *)) where d 1, d >0 are to be chosen. We can alwas find d 1 and d so that this new point is indifferent to v* (and to w*) at 3. This is because (v 1 *-d 1 (v 1 *-w 1 *), v *) is strictl decreasing in d 1 and (v 1 *, v *-d (v *-w *)) is strictl increasing in d. So we can alwas find d 1, d >0 to ensure (v 1 *-d 1 (v 1 *-w 1 *), v *-d (v *-w *))= (v*,w*). In this wa we can construct a series of points v(k) such that v(1)=v*, v(n)=w* and v 1 (k)>v 1 (k+1), v (k)<v (k+1). Now suppose that DR(v*,) is strictl monotone decreasing in. Note that DR cannot be infinite or zero, except perhaps at the boundaries of Y. (If it were to be infinite at an interior point it could not then be decreasing before that point, if zero it could not be decreasing afterwards). Following Milgrom and Shannon we now observe that d 1 (v 1 *-w 1 *) +d (v *-w *) v 1 v =0 at 3 so that d 1 (v 1 *-w 1 *)DR(v*, 3 ) +d (v *-w *)=0 at 3 and therefore b strict monotonicit of DR we have d 1 (v 1 *-w 1 *)DR(v*, 4 ) +d (v *-w *)<0 at an larger value of, sa 4. 4

25 Therefore d 1 (v 1 *-w 1 *) +d (v *-w *) v 1 v <0 at 4 or v(1)<v(). But since DR( v1, v, ) is non-negative and cannot be zero except at the boundaries then for ever v(k)~v(k+1) at 3 we have v(k)<v(k+1) at 4. Hence we cannot have v*~w* at 4. To show the other direction of the proof we will suppose a violation occurs at some point (v 1 *, v *, *) and then select functions v ( x ) and v ( ) x to exploit the violation. If is not strictl monotone in v 1 at this point, select v 1 (x) and v (x) so that v ( x ) v, * * 1 1 v ( x ) v, * * v ' 1 (x * ' ) > 0, andv (x * ) = 0. Then for small d, we have x*+d preferred to x* at the lower boundar (where v 1 is defined), x*+d<=x* at * (because is not monotonic in v 1 at that point) and x*+d~x* at the upper boundar (where v is defined). If DR(v 1 *, v *, ) is equal at 1 and then as before, select functions v ( x ) and v ( x ) so that * * v ( x ) v, v ( x ) * * DR( v1, v, 1). QED v ( x ) ' * ' * v ( x ) v * *, and 5