Bell, D.E.

International Institute for Applied Systems Analysis (IIASA)

A Group Utility Function by Optimal AllocationBell, D.E.IIASA Working PaperWP-74-0551974 Bell, D.E. (1974) A Group Utility Function by Optimal Allocation. IIASA Working Paper. WP-74-055 Copyright © 1974 by the author(s). http://pure.iiasa.ac.at/116/ Working Papers on work of the International Institute for Applied Systems Analysis receive only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work. All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage. All copies must bear this notice and the full citation on the first page. For other purposes, to republish, to post on servers or to redistribute to lists, permission must be sought by contacting repository@iiasa.ac.at A GROUP UTILITY FUNCTION BY OPTIMALALLOCATIONDavid E. BellOctober 1974 WP-74-55Working Papers are not intended fordistribution outside of IIASA, andare solely for discussion and infor-mation purposes. The views expressedare those of the author, end do notnecessarily reflect those of IIASA.Consider an imaginary research institute where theidirector has to make ､ ･ ｣ ｩ ｳ ｩ ｯ ｮ ｾ under uncertainty concerningthe institute's research funds which when received, must beallocated by him between several projects. The projectleaders have supplied him with their utility functionsu l ' u 2 ' ... u n for their own project funding, that is, projectleader k uses utility function uk in matters concerningfunding for his own project.Now the director decides that, given an amount ofresearch funding x, he will allocate an amount x k to projectk so as to maXlmlze+ U (x )n nsubject to the oonstraint xl + ... + x = xnThe question considered here, posed to me by Ralph L. Keeney,1S, what is the implied utility function of the director?Note that what follows applies also to the case wherethe director chooses to maX1m1zek u (x )n n nfor some positive weights k.1slnce utility function U.1maybe rescaled without loss to k.u ..1 1Let+ U (x )Jn ns . t . + x = xn-2-Result I It is sufficient to consider the case n = 2 sincea general case may be obtained by repeated application ofthe n = 2 case.Proof.We requireu (x) = maxx I +x 2 +x 3 = xbut let us consider first the functionv (y) = !'J.axY2+Y3 = YIt will be sufficieni for the result to show thatButu( x) = maxxI+Y = xmaxxl +y = x- maxxl +y 2 +y 3. IIHence' until further notice this paper will consider theproblemu(x) = maxy( I )For clarification of notation,.) Ii· If(X = dtf'(t) t=x-3-so. for example.d-u(y-x)dxwhere= ｾＨｙＭｘＩＨ､ｙＭｬＩ dxu(y-x) = dU(t)1dtt=y-xH2sult 2 An optimal y exists and 1S unique for (1) ifu and u are strictly concave.1 2Proof First note that ul(y) + u 2 (x - y) 1S strictlyconcave 1n y since u l and u 2 are. and the sum of twoconcave functions is concave.Hence a maximum 15 attained and is un1que.Define y(x) by the relationU\X) ::: U (y(x)) + u,,(x - y(x))1. ,:then result 2 sLows that y(x) is well defined on x.IIResult 3 If u l and u 2 are twice differentiable and strictlyconcave thenI ,u(x) = ul(y(x)) =ProofLetthenHencededy, I= ul(y) - u 2 (x - y)dedy = 0 whereItil (y) = ( 2 )-4-Thid is a maXlmum Slnce 0(y) is strictly concave Ｑ ｾ y.Equation (2) will always be satisfied if it is assumed that, Ithe ranges of u l and u 2 are equal. For most utility functionsthis range 1S - (X) 0), and thus (2) will hold.Using a Taylor expans10n, , ,u(x) + 6x u(x) = ul(y) + 6y ul(y) + u 2 (x-y) + (6x-6y) u 2 (x-y).( 3 )and (1) and (2) glve thatI I Ioxu(x) = ll)y(u l (y)) + (ox-6y) u 2 (x-y) =,6xul (y) IIResult 4 If the total funding x is increased then eachgroup rece1ves an increased allocation.Proof Using a Taylor Expansion for the optimal y = y(x),so thatu(x + 6x)( ) ( I: I:) '( ) (5x-Oy)2 u", (x-y)+ u 2 x-y + uX-uy u 2 x-y + 2 2using (1) and (2) givesu(x+6x) I ｾ " (6x-6y)2 "= u(x) + 6x u(x) + 2 ul (y) + 2 u 2 (x-y).( 4 )Maximizing (4) with respect to 6y implies that" "oy ul(y) - (ox - cSy) u 2 (x - y) = 0that J.S"u 2 (x - y)fJ..x.= > 0ox " "u l (y) + u 2 (x - y)sJ.nce u and u 2 arestrictly concave.1Al::;o" "u l (y) + u 2 (x - y)> 0also so that each group receives a strict J.ncrease inallocation. I ITheorem ｾ Assuming(i) u l and u 2 are strictly increasing, strictly concavewith continuous ::;econd derivativesthenI(ii) u.(x) -+ 0J.-+ -00asasJ. = 1,2,() ( I -1 I ) [( 1 + u' -2 1 u' 1 ) -1 ( x )Ju x = u l + u 2 u 2 u lProof We have from (2) thatand sincean inverseorstrictly decreasing and continuous it has,which is valid over the range of ul' so that'-1 'x - y = u 2 u l (y)'-1 '= (1 + u 2 u l ) (y)Since ｾ > 0 anddx-6-ｾ ｳ continuous we have that'-1 '1 + ｾＲ u, ｾ ｳ strictly increasing and continuous so that ittoo has an inverse, hence'\(1 + '-1' ,-1 (x)y = u 2 U l )Sinceu(x) = ul(y) + u 2 (x - y)from ( 5 ) and ( 6 ) we have that'-1 ,u(x) = ( u l + u 2 u 2 ul) (y)( G)[(1 + ｾ［ｬ ｾｬＩＭｬ (x)]( 7 )Note that the formula ｾ ｳ entirely in terms of the knownfunctions ul and u 2 "Examples·11Takeso the problem is, find u(x) such thatu( x) = ma.xy [log (y + a) + log (x - Y + b)]By Qifferentiating, setting to to zero we have1y + a1x - y + b = 0x - y + b = Y + ay = x - a + b2Henceu(x) log( x - a + b) (x - b + a )= 2 + a + log 2 + bNow to demonstrate that the same result may be had uS1ngThus1x + a1x + bNow'-1 (x) 1 bu 2 = x'-1 , 1y(x) = (1 + u 2 u 1 ) (x) = x + - u,u 1 (x)= 2x + a - bHence= 2 1 (x - a + U + a)og 2( X + a + b) .= 2 log as requ1red.2-8-Hence u l = log(x + a) u 2 = log(x + b) yields a utilityfunction of log(x + a + b) for the director.As an exercise for the reader, it may be showneither ｾ ｹ direct route or via (7) that if-cxu (x) = -e1and -dx-e, thenu(x) = -e-cx= -eHencefunction of -ecd- --xc+d-dx= -efor the director.yields a utilityIn these two standard cases u(x) is the same form as thefunctions u l and u 2 •Indeed ifthen-cx= -mecd- --xc+d-e-dx= -nethat lS, the directors weighting of the projects does notaffect his decisions.Theorem 2If' each member of the group has( i ) utility function-ckxa -e( i i ) a utility function log(xor ( iii ) a utility function -(x +k = l, ... , n+ a k ) k = l, ... , nb )-p P > 0, k = l, ... ,nkthen the group utility function u lS independent of theweightings given to the individual members utilities.Proof Since it is merely a matter of solution the proofwill be omitted.be ampl.ified.However the statement of the theorem will( i ) If+W Un nwherethenu(x)where-cxef ( ,.) > 0 and1c(ii) If1+ ..• +cnthenu(x)wherea(iii) If'= a1+ ... +ang(o) > 0thenu(x) h( ) (x + b)-P= - wl '··· ,wn ' PnCo) > :)where b = b l + . .. + b n · 11The importance of this result lies in the observationthat the group decision maker need only know the group-10-members' utilities before making decisions on group funds.He need not decide beforehand how he will weight theimportance of the group members. In particular the roleof group decisiun maker and weighting decision maker maybe divided between two different people (or groups) whoneed not even communicate with each other.The interestingly simple expression for the exponentialcoefficient in Theorem 2 (i) can be generalized.TheorPlli 3 If r(x) is the coefficient of risk for the grouputility function and rk(x) that for the individual membersthen1 1r ( x ) = -r-l-:-(-x-l'7'")1+ r (x )n nwh.::re x k is the optimal allocation to group member k.Freof Hecall that r(x)"= _u(x),u(x)Note that this result ｾ ｳ true whatever the forms of theuk but that this does not imply the proof of a generalform of Theorem 2 because the optimal x k will depend on theweightings.For simplicity's sake we ,.ill. ｲ ｾ Ｇ ｇ Ｑ ｬ ｾ it for the casen = 2.Nowru(x),= u l (y) = (8 )-11-Thus, differentiating with respect to x,"u(x) = ｾｬＨｙＩ £x.dxFrom the proof of Result 4"u,) (x - Y).s!.l. := Ldx " "u1(Y) + 11 ') ( X - Y),:.." ""U1 (Y) u 2 (x - y)u(x)=" "Ul(Y) + u 2 (X - Y)that lS1 1 1-11---"+"u(x) u1(Y) u 2 (x - Y)Using ( 8 ) once more we have that,u(x)- -,,--u(x)=ｷｨｩ｣ｾ gives the result.trivial using Result 1.The extension to a general n is-12-The following is a characterization (perhaps not complete)of functions having the property of Theorem 2.Theorem 4 If all members of the group have a utilityfunctionthen u(x) the group utility function, 1S independent of theweightings assigned to the group members if( i ) ＸｶＨｾＩ = v(x + f(8)) -1(where f exists)or if (ii) v(8x) = v(x) + g(8)F'rc,o f Note that it 1S sufficient for (i) to ｴ ｡ ｫ ｾ b k = 0and for (ii) to take a k = 1.We will ｡ ｬ ｾ ｯ just prove it for the case n = 2.(i) At optimality,D.lGv(uly) =or using property (i)'-ISince v existsa (x - y)2andyD. 2x-f(6)= a l + a 2u(x)·-13-.vhich is independent of the veighting 8.(ii) At optimalityIv(b 2 + x - y)No .....I8v(x) I X= v(-)8 from (ii)henceor y= b,... + x - yc...1 + 8vhich again 1S independent of the veightings. IIConcluding Remarks-14-Results have been presented when agroup decision maker allocates resources uS1ng the criterionof max1m1z1ng total utilities. This criterion has not beenjustified, although Theorem 2 suggests to me that it maywell be reasonable and certainly simple to use.Additivity is not essential for many of the results.For example if the criterion used isu (x )n nMany of the results are actually simpler because the problemof weightings does not arise. The results of this paper maybe applied very straightforwardly to the situationlog u(x) = maxxl + .. • + x n = x

A Group Utility Function by Optimal Allocation

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