A functional equation arising in multi-agent statistical decision theory

Using recent results of J~rai we show that the measurable solutions of the functional equation f ( x l y 1 . . . . . x,,y.)f((1 x0(1 Y0 ..... (1 x.)(l y.)) =f(xl(1 Yl) ..... x,(1 y.))fOq(l x 0 ..... y,(1 x.)), where f : (0, 1)"--~(0, oo) and 0 < x~,y~ < 1, i = 1 ..... n, are of the form f ( x l . . . . . x ) = cexp ai(x i -x2)) x b,, \ i = 1 i = 1 where c > 0, a 1 .... , a and b~ .... . b are arbitrary real constants. This result enables one to characterize certain independence-preserving methods of aggregating probability distributions over four alternatives. W h e n the p r o b a b i l i t y d i s t r i bu t i ons o f several exper t s a re to be agg rega t ed in to a single " c o n s e n s u a l " d i s t r ibu t ion , i t is of ten r e c o m m e n d e d tha t this be d o n e in such a way as to p rese rve any a g r e e m e n t r e g a r d i n g the i n d e p e n d e n c e of va r ious events. F o r p r o b a b i l i t y spaces c o n t a i n i n g a t least five poin ts , however , G e n e s t a n d W a g n e r [ to a p p e a r ] have s h o w n that , subjec t to a mi ld a d d i t i o n a l res t r ic t ion , the on ly i n d e p e n d e n c e p r e s e r v i n g a g g r e g a t i o n m e t h o d s are those which endo r se the o p i n i o n of a single exper t . W h e n jus t four a l t e rna t ives are present , on the o the r hand , the same c o n d i t i o n s a l low a r ich var ie ty of n o n d i c t a t o r i a l a g g r e g a t i o n me thods . Indeed , fo l lowed by a su i t ab le n o r m a l i z a t i o n , any func t ion f : ( 0 , 1)" ~ ( 0 , ~ ) sat isfying the func t iona l e q u a t i o n . f ( x l Y l . . . . . . ~c,y,)f((t -x 0 ( l -Yl) . . . . . (1 -x , ) ( l -y , ) ) = f ( x l ( 1 -Yl) . . . . . x ( 1 -y , ) ) f ( y ~ ( 1 -x l ) . . . . . y,( t -x ) ) , (1) AMS (1980) subject classification: Primary 39B40. Secondary 62CO5. Manuscript received January 31, 1986, and in final form, May 21, 1986. * Research supported in part by the National Science Foundation. 32 Vol. 32. 1987 A functional equation arising in multi-agent statistical decision theory 33 for 0 < x i, yl < 1, i = 1 . . . . . n, yields an independence-preserving method for combining the probabi l i ty assignments of n experts. The measurable solutions of (1) are of the form f ( x l , . . . . x,) = c exp ai(x i -x~) x~ , \ i = 1 i = 1 (2) where c > 0, a t . . . . . a , and b l , . . . , b are arbi t rary real constants, a result established by Abou-Za id [1984]. In what follows the R o m a n capital letters, R, S, T, U, V, X, and Y denote ndimensional vectors, 0 and 1 the n-dimensional vectors (0 . . . . . 0) and (1 . . . . . 1), respectively, and all opera t ions and relations are taken coordinatewise. With this convention we may, for example, abbreviate formula (1) by f ( X Y ) f ( ( 1 X )(1 Y)) = f ( X (I Y))f(Y(1 X)), where Q < X, Y < ! . Abou-Zaid ' s solution of the above equat ion involved the solution of a much more general functional equat ion by a painstaking induction on n, the cases n = 1,2 having been treated by K a n n a p p a n and Ng [1980]. At the Twentyth i rd In ternat ional Symposium on Funct ional Equat ions (Gargnano [1985]) Wagner posed the prob lem of deriving (2) directly from (t) by an a rgument of reasonable length. In response, Jfirai suggested that the following result might provide the first step of such a derivation: LEMMA (Jfirai [1985]), Every measurable solution of (1) is infinitely differentiable. Proof Lett ing T = XY, the vector abbrevia t ion of (1) becomes f ( T ) = f ( Y T) + f ( ( T / Y ) T) f ( ! + T Y (T/Y)), with Q < T < Y < i . Hence f satisfies the hypotheses of Theorem 1.3 (Jfirai [1986]), which imply that f is infinitely differentiable. In this note we use J/trai's L e m m a to offer a new proof of the THEOREM. The measurable solutions o f ( l ) are given by (2). Proof Let F = log f, where f satisfies (1). Then F(RS) + F ( ( I R)(1 S)) = F ( R ( I S)) + F(S(1 R)), (3) 34 C A R L S U N D B E R G and C A R L W A G N E R AEQ. MATbL where 0 < R,S < 1_. Let t ing U = RS, V = R ( I _ S), X = ( ! R)S and Y = (1 R) ( ! S), (3) becomes F (U) -F(V) = F(X) -F(Y), (4) where U, V, X, Y > 0, U + V < 1, and X + Y < 1. We note for future reference that U, V, X, and Y are related by the formulas X U m U = X + Y X X U + V U (5) Y V V = Y Y = V. X + Y U + V It follows from (4) that 02/Ox~Oyi(F(U) F(V)) = 0, i, j = 1 . . . . . n, the requisi te differentiabil i ty of F fol lowing from Jfirai 's Lemma. Expand ing the above (with F i and Fil denot ing, respectively, the par t ia l der ivat ive with respect to the i-th var iable and the second par t ia l with respect to the i-th and j th variables) we ob ta in t~ul t3u i ~2u s Fu(U~)-4~-x + Fj(U) ~ . t% (6) c x i cy j •XiCT j Ovi Ovj ~32vj -Fu (V)0x i ~yj F j (V ) ox~Oy.i = O. We first t rea t the case n = 1, for which equa t ion (6) becomes . OuOu , O2u . OvOv F (U)~xfffy + F ( u ) o ~ y F ( V ) ~ x ~ y C o m p u t a t i o n s based on (5) establ ish that , ¢32v r ( V ) ~ ~ = 0 . (7) cxoy Ou/Ox = [v/(u + v)(1 u v)] 1 Ou/Oy = u / ( u + v)(l u v) O2u/~x Oy = ( u v ) / ( u + v ) ( 1 u v) 2 avl~x = -vl(u + v ) ( l u v ) c~v/Oy = [u/u + V)(1 -u -V)] -1 O2V/Ox Oy = (V -u)/(u + v)(1 -U -v) 2, (8) Vol. 32, 1987 A functional equation arising in multi-agent statistical decision theory 35 and (7) a n d (8) yield F"(u)[u (u + v)Z]u + F ' ( u ) [ u z v z] = F"(v) [v (u + v) 2] + F'(v)[v 2 u 2] (9) Se t t ing G(u) = uF'(u), e q u a t i o n (9) b e c o m e s G'(u)[u (u + v) z] + G(u)[2u + 2v 1] = G'(v)[v (u + v) 2] + F(v)[2u + 2v 1]. (10) va l id for all u, v such tha t u > 0, v > 0, a n d u + v < 1. F o r fixed smal l pos i t ive v, (10) is a first o r d e r l i nea r d i f ferent ia l e q u a t i o n for G(u). M u l t i p l y i n g (10) by [u (u + v) 2] z a n d i n t e g r a t i n g with respec t to u yields G(u) (u + v)G'(v) G(v) + + C ( v ) . ( 1 1 ) u (u + v) z u (u + v) z u (u + v) z Set t ing u = v ( a s s u m i n g v < 1/2, so tha t u + v < 1) in (I 1) a n d so lv ing for C(v) yie lds C(v) = 2G' (v ) / ( l 4v), and subs t i t u t i ng this exp re s s ion in (11) a n d s impl i fy ing yields G'(v)[u 2u z] G'(v)[v 2v z] G(u) = + G(v) 1 4 v 1 4 v for 0 < u < 1 v. Since v m a y be chosen a r b i t r a r i l y smal l pos i t ive , G mus t t ake the fo rm G(u) = a ( u 2u z) + b for s o m e c o n s t a n t s a a n d b. Since G(u) = uF'(u), F mus t the re fore t ake the fo rm F(u) = a(u -u z) + b log u + c (12) for c o n s t a n t s a, b, and c, which yields (2) for f = e x p ( F ) when n = 1. If n > 1, c o n s i d e r e q u a t i o n (6) for i = j. The a b o v e a r g u m e n t for n = 1 then shows tha t 36 CARL SUNDBERG a n d CARL WAGNER AEQ. MATH. F(U) = ai(u 1 . . . . . a i . . . . . u,,)(u i u 2) + b i ( u l , ' " , ul . . . . . u ) log u i + ci(ul . . . . . ul . . . . . u,) for i = 1 . . . . . n, where "^" indicates omission of the corresponding variable. Thus F must have the form F(U) = ~ A ......... ~o~,(Ux). • • q0~ (u.), (13) I ~S l , . . . ,Sn < 3 where q~l(u) = u u 2, q~z(u) = logu, ~03(u) = 1, and the A ......... are constants. Now equat ion (6) for i # j reads 8ui c3ui V ~vi 8vj (14) Fij(U)~xi-b~ = Fij( )~xi~O),j.