Stochastic dominance and Friedman-Savage utility functions

305

Mathematical Social Sciences 16 (1988) 305-317 North-Holland

STOCHASTIC DOMINANCE UTILITY FUNCTIONS *

AND FRIEDMANSAVAGE

Dominique THON Bode School of Business, N-8016 Msrkved, Bod0, Norway

Lars THORLUND-PETERSEN Norwegian School of Economics and Business Administration, Norway

N-5035 Bergen-Sandviken,

Communicated by B. Dutta Received 27 September 1986 Revised 5 May 1988

Two-way stochastic dominance is defined as the ordering corresponding to the unanimous ranking given by all risk-averse and risk-loving agents. This new ordering of random incomes is shown to have an intuitive economic interpretation and to be a potential substitute for first degree dominance. Two-way dominance is shown to be closely related to a class of utility functions which has been studied by Friedman and Savage. The analysis draws on the geometry of cone orderings, a technique which yields rather directly the price characterization of random variables efficient with respect to two-way dominance. Key words: Cone ordering; F-functions; stochastic dominance.

1. Introduction Suppose that random income y is, in an expected utility framework, preferred to random income x by both all risk-averse and all risk-loving agents whose utility function is increasing. It is rather straightforward to show that this is not sufficient to guarantee that all agents whose utility function is increasing will prefer y to X. Results from majorization theory indicate that the class of utility functions corresponding to this stochastic dominance situation, called two-way dominance, equals the class of functions which can be written as the sum of an increasing concave and an increasing convex function. In this paper we provide a characterization of the underlying economic behavior. The class of utility functions corresponding to two-way dominance is shown to have an intuitively clear economic interpretation in terms of acceptance of a well-defined type of transactions which can be reduced * Useful comments acknowledged.

from SW.

Wallace, R.J-B. Wets and an anonymous

0165-4896/88/$3.50 0 1988, Elsevier Science Publishers B.V. (North-Holland)

referee are gratefully

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to elementary insurances, gambles and income increases. The paper thus provides a class of utility functions which allow both insuring and gambling. The problem of characterizing such a class of utility functions was discussed by Friedman and Savage in their well-known 1948 paper. Although we use quite different methods, the class of functions characterized through our stochastic dominance approach turns out to be closely related to theirs. We shall generally be concerned with comparisons of random variables which are vectors in Euclidean n-space R”, n r 2; thus i E N= { 1, . . . , n} represents a state of the world. Without loss of generality, any random variable x=(x,, . . . ,x,,) is supI wn}; note that we do not make the restrictive posedtobeinD={wER”Iw,c... assumption that x be interior to D. Let R = (n,, . . . , n,) be a strictly positive probability measure on 2N. It facilitates the exposition to assume equiprobability: rr = n* where for all i, n:= l/n. No real loss of generality is entailed and we shall briefly show how our results can be extended to a general probability measure R. Consider x, yeD and an open interval Z such that xi, YicZ, all i; thus x, yeDft I”. Let U’ be the set of (weakly) increasing functions on Z and let U2 [U*] be the concave [convex] members of U’. For any prescribed subset US U’, consider the condition:

(1) which expresses that the expected utility of y is not less than that of x. For II = U I, U2, U, condition (1) defines very neat orderings on D n I”. For U= U ‘, (1) is equivalent to y dominating x by the first degree, written x5 ’ y, defined by xilyi, i= 1, . . . . n. Furthermore, familiar results in majorization theory show that (1) holds for U= U2 if and only if y dominates x by concave dominance, (second degree stochastic dominance), written xc 2y, meaning that: i~,Xj~j~i~,Yi~j*

k=l,--*,n*

(2)

Likewise, (1) holds for U= U2 if and only if y dominates x by convex dominance, xs 2y, meaning that: n

C XjRiS i

i=k

i=k

_YiRi,

k= l,...,n.

(3)

The equivalences between (1) and (2), (3) are discussed by Cheng (1977) and Marshall and Olkin (1979). For II = II *, these results are increasing variants of a famous theorem by Hardy, Littlewood and Polya; see Schmeidler (1979). By definition, for x, y E D, y dominates x by two-way dominance if (2) and (3) hold. This is denoted XI: y. It is now easy to see from the above that (2) and (3) hold if and only if (1) holds for all functions which can be written as a sum of an increasing concave and an increasing convex function. Let Ts; U’ be the set of such functions, called T-functions. We summarize the above facts as a theorem:

D. Than, L. Thorlund-Petersen

Theorem 1. If x, y~DflI”,

/ Friedman-Savage

then x~$y

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if and on/y if (1) holds for U= T.

It is easy to find an example where XI: y but not XI ’ y. Consider for 0<6~+, x= (0, 1,2) and y = (6, 1 - 6,2 + 6). If n = 2, then xc i y is equivalent to XC ’ y. The economic meaning of 5: is not obvious; the definition of a T-function does not readily provide a description of the attitude to risk of agents endowed with such a utility function. $2 provides the preliminaries by applying the geometry of cone orderings to two-way dominance. In $3, we give the economic interpretation and relate our results to earlier work. In $4, we study the price characterization of efficient random variables by Peleg and Yaari (1975) from the point of view of a cone ordering and extend the characterization to efficiency with respect to two-way dominance.

2. Dominance and cone orderings The dominance orderings _( ‘, I 2, s2 and 5: are all induced by cones: if C is a closed convex pointed cone, and BGR”, then the cone ordering s induced by C on Bisdefined byxlyify-x~C.Thepo/arconeC*={p~R”(pzrO,allz~C} of C is the cone of supporting hyperplanes for C. Here we consider only finite cones, bounded by a finite number, M, of hyperplanes, C= {z E R n )AZ zz0}, for some M x n matrix A. The edges (or extreme directions) of C will be called critical transactions, due to their economic interpretation. The set of critical transactions can be identified with the columns of the polar matrix of A; if r is the least positive integer such that some n x r matrix P satisfies C*={q~R”lqP?0}, then P is called a polar matrix of A. Note that a polar matrix P is uniquely determined up to permutation and positive scalar multiplication of columns. If A is invertible, then its inverse is a polar matrix of A but otherwise it is quite difficult to find a polar matrix.

I

Fig. I.

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It is convenient in the analysis of this section first to take n = n*; this assumption is removed in Theorems 2, 3 and 4 below. Let C’, C2, C,, Ci be the cones which induce the orderings 5 ‘, zz2, s2, 5: on Dn I”. Clearly C’ is given by the n x n identity matrix and C’ is given by A*= (ai,), where uij= 1 if irj; a,=0 otherwise. Let P* be the inverse of A’. The transposes of A*, P2 are denoted AZ, P2, and A2 determines C2. The cone Ci is given by the (2n - 2) x n matrix A:, the first n - 1 rows of which equal the first n- 1 rows of A2 and the last n- 1 rows of which equal the last n - 1 rows of A,. We display A*, P2 and Ai for n = 5.

AZ=

10000 11000 1 0 11110 11 1 1 1 1 01

I1 ;

1

0

0

0

0

o-1 1 0 o-1

0 1

0

10000 11000 11100 11110 01111 00111 00011 00001

P2=

0

The columns of P* provide the standard interpretation of increasing concave utility functions in terms of fair insurances (. . . , 1, - 1, . . . ) and a pure income increase (... , l), see the discussion of order preserving functions preceding Theorem 3 below. A polar matrix Pg of Ai is more difficult to obtain. Fig. 1 illustrates how C: looks for the case n = 3. From Fig. 1 we see that in R3, Ci is generated by the unit vectors e’, e2, e3 and - 1, 1). The extension from C’ to Ci can be seen to be quite by et-e2+e3=(l, significant. In general, the characterization of Ci is as follows. Lemma 1. The direction generated by the vector v, C y=, vi = 1, is extreme in Ci if and only if there exists a subset Mz; N, M= {i,, ...,i2k+l} for some integer k, OSkl(n1)/2, such that for v= 1, . . ..2k+ 1:

“i=(-l)“+’

if i=i,EM

and Di=O

if i@M.

(4)

This lemma is a consequence of a more general result proven in $4, Lemma 2. It shows how a polar matrix Pf of Ai looks like. To illustrate, one has for n = 5 that r= 16 and

P;=

100001111 0 1 0 0 o-1-1-1 0 0 10 0 1 0 0 0 0 1 0 0 1 0000100101011111

1 0 0 o-1-1-1-1 0 1 0

0 1

0 1

1

o-1

1 0 0

0 1 0 -1-l

0 1 o-1 1 1 -1

Let us now reintroduce a general probability measure n, something which is very straightforward. For any rr, let Z7be the n x n diagonal matrix with diagonal 71.Then

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the matrices defining the three orderings I*, I*, 5; become A’Z7, AZZ7 and ,4:Z7. The matrices Z7-’ P2, I7-l P2 and Z7-‘Pj are readily seen to be corresponding polars. The last polar, for example, is obtained by dividing row i of P: by Xi. Then the next theorem follows directly from Lemma 1. Theorem 2. For 71a probability measure on N, a critical transaction for two-way dominance is a vector of the form:

( . . . , l,...,

-l,...,

l,.........,

-l,...,

l,... )W.

Any utility function u in T implies preference for y over x whenever y two-way dominates x. One naturally wonders whether there are functions outside T having this property. As we now show, if two-way dominance is preserved for aN n, then u necessarily must be a T-function. Let I be a cone ordering on BS R”; the real function 4 on Bpreserves I if xzz y implies o(x) I Q(y) forx, YE B. If B is convex and has non-empty interior, then @preserves I if and only if @at any x interior to B is (weakly) increasing in any extreme direction of C, Marshall et al. (1967; Theorem 2). In particular, if @ has a gradient V@, this means that the system of inequalities V#(x)PzO holds for x interior to B, where P denotes a polar matrix corresponding to C. For P= Pi we have: Theorem

3. Suppose @ is defined on Dfl I” by @(x) = I:=, U(Xi)n; where u has a derivative u’ on I. Then @Ipreserves I : if and only if for all subdivisions x, < e-e< x~~+~, OrkS(n1)/2, of I: u’(x2)

+ *** + U’(X2,)I

u’(x,)

+ -*- + U’(X2k+

,),

(5)

where the I.h.s. of (5) equals zero if k = 0. Proof. This follows from Marshall et al. (1967; Corollary

4) and Theorem 2.

It is straightforward that (5) is always satisfied by an increasing function u which is either concave or convex or which is first concave and then convex a la Friedman-savage. On the other hand, condition (5) for a fixed n does not imply that u is a T-function except in some special cases. For example, consider a function u which is first convex, then concave. Then (5) is equivalent to the requirement that the derivative u’ satisfies u’( y2) I u’( y,) + u’( ys) for all yI ~~21~3. Thus if (5) holds for n = 3, then (5) holds for all nz3. Consequently, if u’ is continuous on the closure of I, then u is a T-function, see Thon and Thorlund-Petersen (1986). Actually, u E T if and only if u’(c) s u’(a) + u’(b) where a, b, c are the lower bound of Z, the upper bound of I and the point of inflection, respectively. Thus the convex-concave increasing functions are not all in T. For example, consider u(t) = Arctan defined on I=] -b, b[ for b> 0. Marginal utility is u’(c) = (1 + <*)-’ hence u E T if and only if b I 1.

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In general one has the following characterization way dominance.

utility functions

of Tin terms of preserving two-

Theorem 4. Suppose @ is given by Q(x) = Cy=, U(X;)ni on D f-t I”. If for ali n, @ preserves s I for II = R *, then u E T. Proof. See Appendix. The four dominance orderings corresponding to U’, U*, U2, T all have the remarkable property of being induced by cones on Dn I”, a fact which greatly simplifies their characterization. Unfortunately no other commonly used stochastic dominance concept, such as dominance of third or higher degree, has this property. The relationship between third degree stochastic dominance and cone orderings has been studied in Thorlund-Petersen (1987).

3. Economic interpretation For interpretative purposes, it is convenient to rely on the case R = n* and first consider the extreme increasing concave and convex functions. For given numbers b, c define the following increasing functions of the real variable <, called concave and convex angles respectively: ({ - c)- = b + min{ < -c, 0); (< - c)+ = b + max{r - c, 0). Any increasing concave (convex) function can be approximated uniformly on closed bounded subintervals of Z by a finite positive combination of concave (convex) angles, an important fact in stochastic dominance analysis. Consider, for the sake of interpretation, a random income x which is interior to D; thus for any direction z, x+62 is in D if 6>0 is sufficiently small. Thus suppose there is some E>O such that Xi+l-XiTE, i=l,...,n-1. Granted that no riskaverse agent strictly accepts without compensation an elementary fair gamble, i.e. a transaction (-E, E) covering two states of the world, one may ask what compensation is needed to induce all risk-averse agents to accept such a gamble. By ‘compensation’ we mean a payment under some state(s) of the world. Considering the concave angles, it is easy to see that no compensation will do, which is not paid in lower states of the world than the two ones covered by the fair gamble and that, if concentrated at a single state, the compensation must have size at least E, giving 6% -E, E). Also, consideration of the angles shows that, if further elementary gambles were added, covering higher states of the world than the ones in which the compensation is paid, they must be of size (-E, E) at most, for the resulting transaction to be accepted by all risk-averse agents. In this way we obtain the critical transactions for the T-functions: (E, -E, E, -a, E, . . . ). Put in other words, all risk-averse agents can be be induced to accept any number of elementary fair gambles if a compensation of at least E is paid in a lower state than the ones covered by those gambles. By a symmetrical argument, to induce all risk-loving agents to accept any

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number of elementary fair insurances requires a compensation E to be paid in a higher state and this leads to the same critical transactions. This coincidence allows one to interpret those critical transactions either as compensated gambles acceptable to all risk-averse agents or compensated insurances acceptable to all risk-loving agents. To look at the matter in a slightly different way: no risk-averse agent is willing to accept the less-than-fair insurance (-g, E, -E) if the pure premium g>O, paid in a low state, is larger than E. Insurance taking is thereby associated with a fear of becoming poor. By a symmetrical interpretation, the class of risk-loving agents exhibits a propensity to gamble in order to become rich: no risk-loving agent is willing to accept the less-than-fair gamble (-E, E, -e) if the premium Q, paid in a high state, exceeds E. Characterizing a strict subset of the class of increasing utility functions requires putting some restriction on the type of transaction accepted by the corresponding agents. The approach taken in the earlier literature, see below, consists typically in postulating a shape for the utility function in terms of it having successive concave and convex segments. This approach is tantamount to disallowing fair gambles or fair insurances over certain ranges of the random variable. The class of T-functions, on the other hand, provides a global characterization. The motivation for using such a demanding criterion as first degree dominance in economic analysis is no doubt that one is at times unwilling to assume globally risk-averse behavior and the most casual evidence suggests that the same agents do indeed insure and gamble. Using two-way dominance recognizes this fact as well but gives a stronger ordering. Its suitability in economics is partly an empirical issue; the description of the corresponding critical transactions makes it possible, at least in principle, to test whether expected utility maximization with T-functions constitutes a reasonable behavioral assumption. Agents who gamble at some income level and also insure over some range of a random income have been considered by Friedman and Savage (1948). They propose two classes of utility functions, the first one of which is the class of functions which are first concave then convex. It follows from Theorem 3 that such functions always are T-functions and that y two-way dominates x if and only if y is preferred to x by all agents having such utility functions, as T equals the convex hull of U2 and Uz. The set T is thus arguably the most natural candidate among the proper subsets of U1 to be associated with this set of Friedman-Savage utility functions on I. The second class of utility functions proposed by Friedman and Savage comprises all increasing functions which are successively concave-convex-concave. The convex hull of this set is dense (in the sense of pointwise convergence) in U’. Therefore, the corresponding stochastic dominance concept is first degree dominance. In general, functions having at least one convex to concave inflection point may or may not be T-functions. If the slope at such an inflection point is very steep, then the function will not be in T, see (5) for n = 3. The classes of utility func-

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tions considered for example by Markowitz (1952) and by Fishburn and Kochenberger (1979) are not contained in T and they both correspond to first-degree stochastic dominance.

4. Prices and efficient random incomes Peleg and Yaari (1975) have given a complete price characterization of a random income efficient with respect to concave dominance. The reader is referred to Peleg (1975) and Dybvig and Ross (1982) for related results. The cone ordering approach employed in this paper yields the Peleg-Yaari result in a very direct manner, as well as the corresponding result for two-way dominance. We follow Peleg and Yaari and consider a closed convex set XcR”. For UC II ‘, the income XE X is U-efficient if for all y E X, i$i

U(Xi)Tcjs

i$l U(yj)ni

for all UE U,

implies X=_Y.

(6)

Without loss of generality, assume that XED. If x is interior to D, then it follows by (2), (3) and (6) that x is (/*-efficient or T-efficient if and only if (X- {x}) n C* = 0 or (X- {x}) fl Ci = 0, where 0 denotes the empty set. However, as x may be a boundary point of D, slightly more complicated cones than C2 and Ci are necessary to characterize efficiency with respect to concave and two-way dominance. The required definitions are given in the following and the preliminary results are summarized in Lemmas 2 and 3. We commence with two-way dominance and assume throughout this section that n = rr*, except that a general rr is reintroduced without further comments in the statement of the final results, Theorem 5 and 6. Suppose BcR” is a closed, polyhedral convex set with interior points; thus B is the intersection of a finite number of halfspaces. For any x in B, the convex cone F(x, B) = {z E R” lx+ 6z E B for O< 6 sufficiently small) is readily verified to be closed. Let I be a cone ordering on B induced by the cone C. Then the cone K(x, B) = C n F(x, B) is called the cone of preferred directions for I at x in B. If x is interior to B, then obviously K(x, B) = C. In particular, define K*(x, B) = C* fl F(x, B) and Ki(x, B) = Cz n F(x, B). Suppose that z is a direction such that any agent with utility function in U* or in T prefers x+z to XED. There exists a permutation o on N such that both vectors (Xo(i))ieN and (X0(i)+&,(i))iE~ belong to D provided that 6>0 is sufficiently small. For XED, let C(x) denote the set of permutations cr on N such that (Xq(i))ipN ED and let D, denote the image of D under the permutation cr. Thus any preferred direction (with respect to U* or T) at x in R” will point into some D,, Q in C(x). For U= I/*, T the efficiency criterion (6) can now be reformulated. The income XE D is efficient if and only if no preferred direction in R” points into X. Define the two cones

D.

Than.

P(x)=

L.

Thorlund-Petersen

IJ K2(x,D,); UEL(x)

/ Friedman-Savage

L;(x)=

u

UEr(X)

utility functions

Ki(x, D,).

313

(7)

In general, we conclude that xrzXnD will be U2- or T-efficient if and only if (X-{x})OP(x>=0 or (X-{x})n~;(x,=0. The non-zero vector p E R” is a system of X-efficient prices at XE X if the linear function determined by p attains its maximum over X at x. It is easily verified that L’(x) is a pointed convex cone. Thus by the separating hyperplane theorem, if x is CJ2-efficient, then there exists a X-efficient price system p which belongs to the polar cone of E2(x). Conversely, if the interior of this polar cone contains an Xefficient price system, then x is U2-efficient. A similar argument is valid for Et(x); in order to determine E:(x), we need to find all the extreme directions of the cone: KZZ(X,D)={~ER~IA:Z~O,Z~~Z~+I

if xi=Xi+I,i=l,...,n-I).

(8)

ForxED, a set E={i,..., i+ k) C N is a right-plateau if xi= **-=xi+k and Xi_ 1
and only if supp(v) = M, where M= E, U -a- U EZk+, is a T-set satisfying: if i, j E E, for some V, then Vi=Vj andfor every V, CiEE, Vi=(-I)“+‘. Proof.

See the Appendix.

By Lemma 2 and (7), L:(x) equals the pointed convex cone generated by those directions v = (... , 1, . . . , - 1, . . . , 1, . . . ) the support {i,, . . . , ix+ ,} of which satisfies case xi = --. =x,, L:(x) equals the usual nonxil < “.
*.’
implies

pi, + **-+p&+, Zpi,+ *a*+piu,

with pi,+ em*+pi, =0 if k=O. This yields the price characterization with respect to two-way dominance:

(9) of efficiency

Theorem 5. If x E X n D is T-efficient, then there exists a system of X-efficient prices p satisfying (9). Conversely, ifxe Xn D and the r.h.s. of (9) holds with strict inequality for some X-efficient p, then x is T-efficient. (The theorem remains true for a general n if every pi in (9) is replaced by pi/ni).

By way of comparison, we now establish the corresponding result for U2efficiency by an analogous argument, again drawing on the cone ordering property.

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For XED, we define a class of non-empty subsets of N, to be called steps: the set S={i,..., i+k}GN, llksn-1, is aproperstep if for some h, Xi=

“.

=Xj+h_I
and the set R={i,...,

“.

=Xi+&r

(10)

n} 5;N is a terminal step if Xi= -.. =x,,.

Lemma 3. The operation of taking support of x establishes a one-to-one correspondence between the extreme directions of K’(x, D) and the steps in N. Suppose that v is such a direction. If v, + ... + v, = 0, then supp(v) is a proper step as in (10) and vi= **. =~;+h_l>O; vi+h=“’ =vi+kO, then supp(v) is a terminalstep R={i,...,n} where vi= ..a =v,. Proof. See the Appendix. By Lemma 3 and (7), it follows that the polar cone L*(x)* equals those p E R” such that for all i, jEN: XiCXj

implies

Orpjlpi.

(11)

Theorem 6. If x E X is U2-efficient, then there exists a system of X-efficient prices p satisfying (11). Conversely, if XE X r7 D and (11) holds with strict inequalities at the r.h.s. for some X-efficient vector p, then x is U2-efficient. (The theorem remains true for a general n if every pi in (11) is replaced by pi/xi). The proof given here of Theorem 6 substantially differs from that of Peleg and Yaari. They note that if the boundary of X is a differentiable manifold, then their proof can be shortened. Our proof would not benefit from such a differentiability assumption. On the contrary, our approach shows how Theorems 5 and 6 can be strengthened by X being a polyhedral set: in this case, there exists an X-efficient price vector p such that (9) or (11) will hold with strict inequalities on their r.h.s. for x being T- or U2-efficient in X. Our proofs of Theorem 5 and 6 indicate that the efficiency results for concave and two-way dominance owe their relative simplicity to the cone ordering properties involved: once the relevant cone has been identified, the results follow immediately. As mentioned at the end of $3, stochastic dominance of higher degree is not induced by a cone ordering on D fl I” and there is accordingly no reason to believe that the Peleg-Yaari result has any simple generalization to, for example, third degree dominance. On the other hand, Theorems 5 and 6 have obvious generalizations in another direction. So far, we have studied efficiency with respect to an additive function 4, 4(x) = Ii u(Xi)ni. Suppose for simplicity that x=x*. Then UE I/’ if and only if @ is increasingly Schur-concave on I” and u E T if and only if 4 equals the sum of an increasing Schur concave and an increasing Schur-convex function on I” (see Mar-

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shall and Olkin, 1979). It is now a consequence of the theory of cone ordering preserving functions [Marshall et al. (1967)] that Theorems 5 and 6 remain true if efficiency is defined in (6) by the wider class of such possibly non-additive functions: @ being the sum of an increasing Schur-concave and an increasing Schur-convex function in Theorem 5 and 4 being increasingly Schur-concave in Theorem 6. Such generalizations are of potential interest in connection with the rapidly growing literature on attitudes to risk without the complete set of von Neumann-Morgenstern axioms, in which case fl consequently need not be additive.

Appendix 4. Consider a function u as in Theorem 4. We now show that u E T by drawing on a condition in Thon and Thorlund-Petersen (1986). Suppose that the interval [a, b] C I is subdivided according to Proof of Theorem

a
6>0.

(-4.1)

slope-sequence (Sj)j, j = 1, . . . , n is defined by

7j=[u(U+js)-u(a+(j_l)g)1/6.

For m=O, I,2 ,..., consider the set M= {jr, . . . ,jz,+ ,> sN= show that for any such set M, 5j2+ tj4+ *'*+ TjlmI Tj,+ Tj,+ .*'+rjh+i,

64.2)

{ 1, . . . ,n).

We shall (A.31

where the 1.h.s. of (A.3) equals zero for m=O. Define the vectorsy,tcR” byyj=a+(j-1)6 and zj=(-l)“‘*, forj=j,EM, z+=O otherwise; thus z=( . . . . 6 ,..., -6 ,..., 6, . . . ) and supp(z) =M. Now, (A.3) is equivalent to

C dYjlr

jFM(Yj+zj)*

/EM

.

(A-4)

.

Call two elements of A4, jV,jy+ 1, adjacent if j,, , = j,+ 1 and v is odd. Suppose, say, j= 1, j = 2 are adjacent. Then yl + zl =y2 and y2 + z2 = y1 hence inequality (A.4) holds if and only if it holds after deletion of (1,2) from M. More generally, let S G M be the set of elements remaining in M after all pairs of adjacent elements have been deleted. Define 2~ R” by 4 = Zj if j E S, 2’ = 0 otherwise. It follows that (A.4) is equivalent to

C

jeM

u(Yj)5

jFMu(Yj+i,)-

(A.3

Since ZE Ci and y, y + Z-EDfl I”, then (AS) is a direct consequence of cp, defined by G(X) = Cy=, U(Xj), being order preserving for 5;. It has now been proven that for any subdivision (A. l), the slope condition (A.3)

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is satisfied. According to Thon and Thorlund-Petersen that u is a T-function. q

utility funclions

(1986; Lemma 1) this implies

Proof

of Lemma 2. Consider an extreme direction v as in Lemma 2. For h= 1V-m** n - 1 call inequality h and inequality h + n - 1 of the system A;.zr 0 mates; then every such inequality has exactly one mate. If A:tzO, zr + --+ +zh =O, *-+zn= 1 and Z,,=t h + I = (, then c = 0. Since u is extreme, exactly n - 1 of %+I+ the defining inequalities of Ki(x, D), see (8). are binding, Gale (1960; Theorem 2.16). Thus uh=uh+i +O holds for exactly those h such that both inequality h and its mate are non-binding. Suppose without loss of generality that oI, v,>O. Define the increasing sequence of sets E,, Ez, . . . , as follows. Let E, = {I,..., i,) where i, is the largest index such that u1 = --- =ui,. Let E2={Ujz,..., Viz), where j, is the smallest and iz is the largest index such that Vjz= --a = Vi,# vi,, 0; j,> i,, and SO on. By the above result, v;=(-l)“+‘, v=l,.... Thus there must be an odd number, 2k+ 1, of sets CisE, E,. Furthermore, one can easily check that E, is a left-plateau, El is a right-plateau and so on. Therefore, supp(v) = E, U ... U Ezk+, is a T-set having the required property. Conversely, if v is a direction of Ki(x, D) and supp(v) is a T-set, as described in Lemma 2, then v is determined by n - 1 binding inequalities in (8); thus o is extreme. 0

As an illustration of Lemma 2, if n = 6, x, =x,
A2ZZ0, Zi_f5Zi

if

Xi_[=Xi*

(A-6)

First, assume that v1 + --a + v, =O. Let i be the least index with vi>0 and let i+ k + 1 be the least index such that ui+ *a- + Vi+k+t = 0. Since there are exactly n - 1 binding inequalities in (A.6) for v =z, there must be some h such that xi= .** =Xi+h_i 0. Choose the largest i such that Vi_t + --. + v, = 0. Again, applying the fact that there are n - 1 binding inequalities in (A.6) yields Xi= --a =x,. Thus R = {i, . . . , n} is a terminal step. Furthermore, supp(v) = R and vi= -.a = v,. 0

D. Then, L. Thorlund-Petersen

/ Friedman-Savage

317

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