Maxm share ad mmax evy far-dvso problems Marco Dall Aglo a,1 ad Theodore P. Hll b,,2 a Dpartmeto d Sceze, Uverstá G. d Auzo, Vale Pdaro 42, 65127 Pescara, Italy b School of Mathematcs, Georga Isttute of Techology, Atlata, GA 30332-0160, USA Submtted by Wllam F. Ames Abstract For far-dvso or cake-cuttg problems wth value fuctos whch are ormalzed postve measures (.e., the values are probablty measures) maxm-share ad mmax-evy equaltes are derved for both cotuous ad dscrete measures. The tools used clude classcal ad recet basc covexty results, as well as ad hoc costructos. Examples are gve to show that the evymmzg crtero s ot Pareto optmal, eve f the values are mutually absolutely cotuous. I the dscrete measure case, suffcet codtos are obtaed to guaratee the exstece of evy-free parttos. Keywords Far-dvso; Cake-cuttg; Maxm share; Mmax evy; Evy-free; Optmal partto; Equtable partto 1. Itroducto The subject of ths paper s far-dvso or cake-cuttg equaltes (cf. [5,6,11]), ad partcular, the relatoshp amog varous otos of optmalty such as maxm share, mmax evy, ad Dubs Spaer optmalty. A cake Ω s to be dvded amog players * Correspodg author. E-mal address hll@math.gatech.edu (T.P. Hll). 1 Partally supported by MURST-COFIN 2000. 2 Partally supported by US Natoal Scece Foudato Grat DMS-9971146 ad Göttge Academy of Sceces (Gauss Professorshp Fall 2000).
whose relatve values v 1,...,v of the varous parts of the cake may dffer. A partto of the cake to peces P 1,...,P s sought so that the resultg values v (P j ) make the mmum perceved share as large as possble, or make the maxmum evy as small as possble. The formal framework s as follows. There are (coutably addtve) probablty measures v 1,...,v o the same measurable space (Ω, F),where Ω represets the cake ad F s the σ -algebra of subsets of Ω whch represets the collecto of feasble peces. For each P F ad each, v (P ) represets the value of pece P to player. (Hece, ths settg, the feasble peces always clude the whole cake, ad are closed uder complemets ad coutable uos; ad the value fuctos are addtve.) Throughout ths paper, Π wll deote the collecto of F-measurable parttos of Ω, that s, { } Π = (P 1,...,P ): P F for all, P P j = f j, ad P = Ω, ad a typcal elemet P Π s the partto P = (P 1,...,P ) represetg allocato of P to player for all = 1,...,. Ths paper s orgazed as follows: Secto 2 cotas deftos ad examples of the value matrx, maxm optmalty ad far parttos, as well as the ma compactess ad covexty theorem for value matrces due to Dubs ad Spaer [6]; Secto 3 cotas the aalogous covexty/compactess result for evy matrces, a proof that eve the mutually absolutely cotuous case, a Dubs Spaer optmal partto eed ot be evyfree, ad several results guarateeg the exstece of quatfably super-far evy-free ad super-evy-free parttos; ad Secto 4 cotas mmax-evy equaltes for geeral measures (cludg measures wth atoms) whose bouds are fuctos of the maxmum atom sze. =1 2. Far ad Dubs Spaer-optmal parttos Deote by M( ) the set of real-valued matrces. Defto 2.1. The value matrx M V (P) of a partto P s the matrx whose etres are the values of the peces of the partto to the respectve players, that s, M V : Π M( ) s gve by ( ) ( ) M V (P) = M V (P 1,...,P ) = v (P j ),j=1, ad the set of F-feasble value matrces M V s gve by { } M V = M V (P): P Π M( ).
Example 2.2. Let (Ω, F) = ([0, 1], Borels), = 2, v 1 = uform dstrbuto o [0, 1], ad v 2 = probablty measure o [0, 1] wth dstrbuto fucto F 2 (x) = x 2,0 x 1. The for P 1 = ([0, 1/2), [1/2, 1]) ad P 2 = ([0,( 5 1)/2), [( 5 1)/2, 1]), ( 1 1 ) ( 5 1 3 5 ) M V (P 1 ) = ad M V (P 2 ) =, 2 1 4 2 3 4 2 3 5 2 2 5 1 2 ad a easy calculato shows that {( ) } x 1 x 2 M V = :0 x 1, (1 x) 2 y 1 x. 1 y y Example 2.3. Let (Ω, F) = ([0, 1], Borels), = 2, v 1 = v 2 = δ (1/2), the Drac pot mass at {1/2},ad let P 1, P 2 be as Example 2.2. The ( ) ( ) 0 1 1 0 M V (P 1 ) =, M V (P 2 ) =, 0 1 1 0 ad {( ) } x 1 x M V = : x = 0or x = 1. x 1 x The ext result, a cosequece of Lyapouov s covexty theorem due to Dubs ad Spaer, s oe of the ma tools measure-theoretc far-dvso problems, ad s recorded here for ease of referece. (Recall that a measure v s atomless f for every P F wth v(p) > 0, there exsts a set A F, A P wth 0 <v(a) < v(p ); for Borel measures o the real le, ths s equvalet to v({x}) = 0for every x R.) Proposto 2.4 [6]. Fx 1 ad v 1,..., v probablty measures o (Ω, F).The () M V s compact (as a subset of real matrces); ad () f each v s atomless, the M V s covex. Remarks. Note that the measures Example 2.2 are atomless, ad hece that the set of feasble value matrces M V s covex. I Example 2.3, o the other had, v 1 ad v 2 are purely atomc, ad M V s far from covex. It s also easy to check that M V may be covex eve f {v } are atomc; for example, by takg = 2ad v 1 = v 2 defed by v 1 ({x}) = x for x = 2, = 1, 2,..., ad v 1 (x) = 0 otherwse, whch case {( ) } x 1 x M V = :0 x 1. x 1 x Defto 2.5. A partto P = (P 1,..., P ) s far f v (P ) 1/ for all ;s equtable f v (P ) = v j (P j ) for all, j ;s maxm optmal f m 1 v (P ) m 1 v (Pˆ) for all Pˆ = (Pˆ1,..., Pˆ) Π ;ad s Dubs Spaer optmal (DS optmal) f (v 1 (P 1 ),..., v (P )) (v 1 (Pˆ 1 ),..., v (Pˆ )) for all Pˆ = (Pˆ1,..., Pˆ) Π,where v (P ) are the creasg order statstcs of the {v (P )} (.e., v 1 (P 1 ) v 2 (P 2 )
v (P )), ad s the real lexcographc order. I other words, P s DS optmal f the smallest share m 1 v (P ) s as large as possble amog all possble parttos, ad amog all parttos attag that maxm, the secod smallest share s as large as possble, ad so forth. Remarks. As show [6], t follows from Proposto 2.4() that maxm-optmal ad DS-optmal parttos always exst; ad from Proposto 2.4() that f the {v } are atomless, that far equtable parttos always exst, ad that every DS-optmal partto s far. Wthout the assumpto of atomless measures, DS-optmal parttos may ot be far, as s easly see Example 2.3. 3. Evy-mmzg parttos A recet alteratve to the objectve of maxmzg oe s ow share v (P ), s the objectve of mmzg oe s evy of other s shares v (P j ) v (P ) (cf. [3 5,14]). Clearly the two objectves are related, but as the ext example pots out, players tryg to mmze evy would sometmes reject a gve partto favor of oe whch gves every player a much smaller share. I ths example, the players would reject a equtable partto whch allocates each player very early 50% of hs ow value of the cake (but wth a accompayg mscule amout of evy) favor of a evy-free partto whch allots each player a pece he feels s worth exactly 1% of the total value. I partcular, the example shows that the evy-mmzg objectve s ot Pareto optmal. Example 3.1. Let (Ω, F) = ([0, 100], Borels), = 100, let v be uform o [ 1, + 1) for = 1,..., 99, ad let v 100 be uform o [99, 100] [0, 1). Let P = (P 1,P 2,...,P 100 ) be gve by P =[ + 0.0001, + 1.0001), = 1,..., 98, P 99 = [99.0001, 100] [0, 0.0001),ad P 100 =[0.0001, 1.0001);ad let Pˆ = (Pˆ1, Pˆ 2,..., Pˆ 100 ) be P ˆ 99 = k= 0 [k + ( 1)/100,k + /100), = 1,..., 100. It s easly checked that, for each, v (P ) = 0.49995 ad the evy of player (see Defto 3.2 below) s 0.00005 for each. O the other had, wth partto Pˆ, each player receves a pece worth exactly v (Pˆ) = 0.01, but o player values ay other pece more tha hs ow. Thus players seekg to mmze evy would choose P 2 over P 1 ad reduce ther shares uformly by early a factor of 50. I the above example, however, t s easy to see that there s a partto (amely P = [ 1,) for all ) whch s smultaeously evy-free, DS optmal, equtable ad far, ad whch assgs each player a share he values exactly 50% of the cake. It s the purpose of ths secto to record several basc propertes of evy, to vestgate the terrelatoshp amog these varous otos of optmalty, ad to derve several geeral equaltes for upper bouds o evy. Defto 3.2. The evy of a partto P to player, e (P),s e (P) = max 1 j v (P j ) = v (P );the maxmum evy of P, e max (P), s e max (P) = max 1 e (P); the evy matrx of P, M E (P) s the elemet M( ) wth (, j)th etry e,j = v (P j ) v (P );ad the
set of F-feasble evy matrces M E s the subset of M( ) gve by M E ={M E (P): P Π }. (Note that the defto of evy here s the egatve of that [14]; here postve evy reflects valug aother s pece more tha oe s ow, ad the objectve s to mmze evy.) Example 3.3. () For the problem Example 2.2, ( ) ( ) 0 0 0 2 5 M E (P 1 ) =, M E (P 2 ) =, 1 0 2 5 0 2 ad {( ) } 0 1 2x 2 M E = :0 x 1, (1 x) 2 y 1 x. 1 2y 0 () For the problem Example 2.3, ( ) ( ) 0 1 0 1 M E (P 1 ) =, M E (P 2 ) =, 1 0 1 0 ad {( ) } 0 1 2x M E = : x = 0or1. 2x 1 0 Lemma 3.4. () dm(m V ) = dm(m E ); () the fucto from M V M E defed by M V (P) M E (P) s oe-to-oe, oto, ad affe. Proof. Cocluso () s a drect cosequece of (). To see (), ote that {v (P j )},j=1 clearly determes {v (P j ) v (P )},j= 1 ; coversely, the sum of the evy etres the th row, so ( ) v (P j ) v (P ) = v (P j ) v (P ) = 1 v (P ), j=1 j=1 v (P j ) = v (P j ) v (P ) + v (P ) ( ) ( ) = v (P j ) v (P ) + 1 1 v (P j ) v (P ). j=1 The ext theorem s a drect aalog of the ma compactess covexty result for value matrces gve Proposto 2.4. Theorem 3.5. Fx 1, ad v 1,v 2,..., v probablty measures o (Ω, F).The () M E s compact; ad () f each v s atomless, the M E s covex.
Proof. Cocluso () follows from Lemma 3.4 ad Proposto 2.4, sce M E s a cotuous mage of the compact set M V, ad () follows smlarly sce the atomless measure case, M E s the mage of the covex set M V uder a affe trasformato. Note that Example 3.3, M E s covex case (), ad ot covex (); both cases t s compact. Defto 3.6. A partto P Π s evy-free f e max (P ) 0; s mmax evy optmal f e max (P ) = m{e max (P): P Π };ad s DS mmax evy optmal f t attas the mmum, lexcographcally, of the set of feasble ordered evy vectors {(e 1 (P),...,e (P)): P Π } (cf. Defto 3.2). Example 3.7. The partto P 2 Example 2.2 s the uque (up to sets of measure zero) DS-mmax-evy-optmal partto ad s also evy-free (see Example 3.3()); every partto Example 2.3 s DS mmax evy optmal wth maxmum possble evy +1 for oe of the players, ad o partto s evy-free. Theorem 3.8. Fx 1, ad v 1,...,v probablty measures o (Ω, F).The () Mmax-evy-optmal ad DS-mmax-evy-optmal parttos always exst; () If a partto s evy-free, the t s far; () If {v } 1 are atomless, the evy-free parttos always exst; (v) If {v } 1 are atomless ad learly depedet, the super-evy-free parttos (e max < 0) always exst. Proof. Cocluso () follows easly from Theorem 3.5() sce the mappg M E [ 1, 1] gve by M E (P) e max (P) ( (e 1 (P),...,e (P)), respectvely) s cotuous, so ts mmum s attaed; () s trval sce v (P j ) v (P ) 0for all, j mples that v (P ) 1/ for all ; () follows by Theorem 3.5() by cosderg the parttos P 1 = (Ω,,..., ), P 2 = (,Ω,,..., ),..., P = (,...,,Ω), ad otg that j=1 M E(P j ) s the zero matrx; ad (v) s the ma result [3]. Cotrary to a clam [15], the ext example shows that eve for three mutually absolutely cotuous measures v 1,v 2,v 3, a DS-optmal partto eed ot be evy-free. (Recall that Example 3.1, P was strctly better value-wse for each player tha the evyfree partto Pˆ,but P was ot DS optmal.) Example 3.9. Let (Ω, F) = ([0, 3], Borels), = 3, ad (lettg I(a,b) deote the dcator fucto I (a, b)(x) = 1f a<x<b,ad = 0 otherwse) let v 1,v 2,v 3 be the cotuous dstrbutos wth desty fuctos f 1,f 2,f 3, respectvely, gve by f 1 = 0.4I(0, 1) + 0.1I(1, 2) + 0.5I(2, 3), f 2 = 0.3I(0, 1) + 0.4I(1, 2) + 0.3I(2, 3), f 3 = 0.3I(0, 1) + 0.3I(1, 2) + 0.4I(2, 3).
The, as wll be proved the ext theorem, the partto P = (P 1,P 2,P 3 ) = ([0, 1), [1, 2), [2, 3]) yelds the uquely maxm-optmal vector (v 1 (P 1 ), v 2 (P 2 ), v 3 (P 3 )) = (0.4, 0.4, 0.4), but P s ot evy-free. Thus every evy-free partto s strctly suboptmal the maxm crtero, ad hece also strctly suboptmal the DS crtero. Theorem 3.10. () If = 2 ad v 1,v 2 are atomless, the every maxm-optmal partto s evy-free; ad () for each 3, there exst mutually absolutely cotuous atomless measures v 1,..., v such that o maxm-optmal partto s evy-free. ( ) ( ) 10 01 Proof. To see (), ote that ad are M V (takg P 1 = (Ω, ), P 2 = (,Ω)), ( 10 ) 01 1/2 1/2 so by Proposto 2.4, 1/2 1/2 M V, ad thus every maxm-optmal partto P satsfes v 1 (P) 1/2, v 2 (P) 1/2. By addtvty, ths mples that v 1 (P 1 ) v 1 (P 2 ) ad v 2 (P 2 ) v 2 (P 1 ), ad hece that P s evy-free. To see (), ote that [6, last remark o p. 17] for ay, whe the measures are mutually absolutely cotuous the DS-optmal soluto s equtable. Therefore, all maxmoptmal solutos are DS optmal ad equtable. For = 3, cosder the measures v 1, v 2,ad v 3 of Example 3.9. The set of all possble parttos of [0, 3] ca be descrbed as follows: the terval [0, 1) s dvded to three parts wth player 1 (respectvely, player 2) recevg a pece of legth p 1 (respectvely, p 2 )ad player 3 gettg the rest,.e., 1 p 1 p 2. Smlarly, [1, 2) s splt to three parts of legth q 1, q 2,ad1 q 1 q 2, respectvely, ad [2, 3] s parttoed as r 1, r 2,ad1 r 1 r 2. Every equtable partto s obtaed as a soluto of the followg system of lear equatos ad equaltes: 0.4p1 + 0.1q 1 + 0.5r 1 = α, 0.3p 2 + 0.4q 2 + 0.3r 2 = α, 0.3(1 p 1 p 2 ) + 0.3(1 q 1 q 2 ) + 0.4(1 r 1 r 2 ) = α, p 1,p 2,q 1,q 2,r 1,r 2 0, p 1 + p 2 1, q 1 + q 2 1, r1 + r 2 1, ad the largest value of α s sought that keeps ths system admssble. The correspodg solutos for the p s, q s, ad r s descrbe all possble maxm-optmal solutos. Solvg the frst equato for p 1 terms of α, q 1,ad r 1, ad the secod for q 2 terms of α, p 2,ad r 2, ad substtutg these expressos the thrd equato yelds 0.3p 2 + 0.9q 1 + 0.1r 1 + 0.7r 2 = 4 10α. (3.1) If α> 0.4, Eq. (3.1) has o soluto wth oegatve varables. If, stead, α = 0.4, (3.1) admts oly the soluto p 2 = q 1 = r 1 = r 2 = 0. (3.2) (Thus, the segmet [2, 3] s gve ts etrety to player 3, who reaches hs quota of 0.4 ad has o terest the other parts of the cake.)
Hece, sce α = 0.4, p 1 + p 2 = 1 ad q 1 + q 2 = 1. Ths fact ad (3.2) mply that p 1 = q 2 = 1, whch shows that P = ([0, 1), [1, 2), [2, 3]) s, up to sets of Lebesgue measure zero, the oly mmax-optmal (ad DS-optmal) soluto. But ( 0 0.3 0.1 ) M E (P) = 0.1 0 0.1, 0.1 0.1 0 whch shows that ths partto s ot evy-free. Ths completes the case = 3 (ad establshes the clam Example 3.9). For > 3, let (Ω, F) = ([0,], Borels) ad fx ε wth 0 < ε< 1. Cosder the cotuous dstrbutos v 1,...,v wth desty fuctos f 1,...,f, respectvely, that have costat values each terval [ 1,), = 1,...,, wth values show the followg table: ε ε ε ε ε ε ε f 1 0.4(1 ε) 0.1(1 ε) 0.5(1 ε ) /( 3) /( 3)... ε/( 3) [0, 1) [1, 2) [2, 3) [3, 4) [4, 5)... [ 1,] f 2 0.3(1 ε) 0.4(1 ε) 0.3(1 ε ) ε /( 3) ε /( 3)... ε/( 3) f 3 0.3(1 ε) 0.3(1 ε) 0.4(1 ε ) ε /( 3) ε /( 3)... ε/( 3) f 4 ε/( 1) /( 1) ε /( 1) 1 εε/( 1)... ε/( 1) f 5 ε/( 1) /( 1) ε /( 1) /( 1) 1 ε... ε/( 1).......... f ε/( 1) /( 1) ε /( 1) /( 1) ε /( 1)... 1 ε As the = 3 case, the dstrbutos are mutually absolutely cotuous ad all mmaxoptmal solutos are DS optmal ad equtable. Deote by p,j (, j = 1,...,) the legth of the part of [ 1,) assged to player j wth the usual costrats p,j 0ad j=1 p,j = 1for all. The partto, defed by p, = 1for all, has 0.4(1 ε) as ts lowest value, so the mmax-optmal value caot be smaller tha ths value. Cosder ow the frst three players oly. Sce the mmax value s at least 0.4(1 ε) ad sce 3 f dv = ε, = 1, 2, 3, each of the players 1, 2, 3 must receve somethg worth atleast0.4 1.4ε from the terval [0, 3), so the followg system of equaltes must be satsfed by ay mmax-optmal soluto: 0.4p 1,1 + 0.1p 2,1 + 0.5p 3,1 β + O(ε), 0.3p 1,2 + 0.4p 2,2 + 0.3p 3,2 β + O(ε), 0.3p 1,3 + 0.3p 2,3 + 0.4p 3,3 β + O(ε), β 0.4 + O(ε). A smple cosequece of the ormalzg costrats for the p,j s that (3.3a) (3.3b) (3.3c) (3.3d) p,3 1 p,1 p,2, = 1, 2, 3. (3.4) Ths, wth (3.3c), mples that 0.3(1 p 1,1 p 1,2 ) + 0.3(1 p 2,1 p 2,2 ) + 0.4(1 p 3,1 p 3,2 ) β + O(ε).
Rearragg (3.3a) (3.3c) yelds p 1,1 5 2 β 1 4 p 2,1 5 4 p 3,1 + O(ε), (3.5a) p 2,2 5 2 β 3 4 p 1,2 3 4 p 3,2 + O(ε), (3.5b) 0.3p 1,1 + 0.3p 1,2 + 0.3p 2,1 + 0.3p 2,2 + 0.4p 3,1 + 0.4p 3,2 1 β + O(ε), (3.5c) β 0.4 + O(ε). Substtutg (3.5a) ad (3.5b) to (3.5c) yelds (3.5d) 0.3p 1,2 + 0.9p 2,1 + 0.1p 3,1 + 0.7p 3,2 4 10β + O(ε), (3.6a) β 0.4 + O(ε). (3.6b) These mply that 0.3p 1,2 + 0.9p 2,1 + 0.1p 3,1 + 0.7p 3,2 4 10β + O(ε) = O(ε). (3.7) Sce all varables (3.7) are oegatve, they all satsfy p 1,2 = O(ε), p 2,1 = O(ε), p 3,1 = O(ε), p 3,2 = O(ε). (3.8) From (3.5a) ad (3.5d), ad the fact that p 2,1 ad p 3,1 are O(ε), t follows that p 1,1 = 1 + O(ε), ad hece that p 1,3 = O(ε). (3.9) Smlarly, (3.5b), (3.5d), ad (3.8) mply that p 2,2 = 1 + O(ε),so p 2,3 = O(ε). (3.10) Fally, (3.5c), (3.5d), (3.9), ad (3.10) mply that p 3,3 = 1 + O(ε). (3.11) Thus, player 1 s evaluato of hs ow share ay mmax-optmal soluto s v 1 (P 1 ) = 0.4p 1,1 + 0.1p 2,1 + 0.5p 3,1 + O(ε) = 0.4 + O(ε). Player 1 s evaluato of player 3 s share, o the other had, s v 1 (P 3 ) = 0.4p 1,3 + 0.1p 2,3 + 0.5p 3,3 + O(ε) = 0.5 + O(ε), where the last equalty follows from (3.11). Therefore, e 1,3 = v 1 (P 3 ) v 1 (P 1 ) 0.1as ε 0, so asymptotcally, every maxm-optmal partto, player 1 eves player 3 s share by a amout arbtrarly close to 0.1. The fal theorem ths secto gves sharp bouds for faress of evy-free parttos ad mmax evy, the case where the measures are atomless ad have kow upper ad lower bouds, respectvely.
For measures v 1,..., v, the fucto = 1 v : F [0, 1], called the maxmum of {v 1,..., v }, s the smallest set fucto whch domates each of the {v }; = 1 v s the aalogous mmum. It s easy to check that both v ad v are also coutably addtve measures o (Ω, F), ad lettg v,v deote the total masses of v, v, respectvely, that v 1 v, wth equalty f ad oly f v 1 = v 2 = =v.(whe {v } are absolutely cotuous wth destes {f }, v s smply the total area uder the outer evelope max 1 f of {f },ad v s the area uder m 1 f.) I far-dvso problems, v represets the cooperatve value of Ω, that s the total value to the coalto of all players f each pece s gve to the player who values t most, ad these values are added together. Smlarly, v represets the worst-case allocato f the values are added (cf. [8,10]). Example 3.11. For the measures Example 2.2, v = 5/4ad v = 3/4; Example 2.3, v = 1 = v. Theorem 3.12. Fx 1 ad v 1,..., v atomless probablty measures o (Ω, F). The there exst parttos P (1), P (2), P (3), P (4) Π such that (1) () P (1) s evy-free ad v (P ) = ( v + 1) 1 for all ; (2) () P (2) s evy-free ad v (P ) = ( + v } 1) 1 for all ; { v 1 () e max (P (3) ) m 0, { v +1 ; ad +v 3} (v) e max (P (4) ) m 0,, +v 1 ad these bouds are best possble. Recall that v = 1 f ad oly f v = 1 f ad oly f v 1 = = v, so the bouds ( v + 1) 1 ad ( + v 1) 1 () ad () are strctly bgger tha 1/ wheever the {v } are ot detcal. Thus, that case, () ad () guaratee the exstece of evy-free super-far parttos, wth super-faress quatfably greater tha 1/. Smlarly, for v suffcetly large, or v suffcetly small (v > 1, v < 3 ), () ad (v) guaratee the exstece of super-evy-free parttos wth evy quatfably strctly egatve (cf. [6] ad [3] for oquatfable super-far ad for super-evy-free parttos, respectvely). Proof of Theorem 3.12. Let µ = = 1 v.every v s absolutely cotuous wth respect to µ, so, by the Rado Nkodym theorem, there exsts a fucto f, called the desty fucto of v, such that v (A) = A f dµ for all A F. To prove (), let P = (P 1,...,P ) be the partto of Ω whch assgs each elemet of Ω to the player whose desty s hghest that pot. I case of tes, the pot s allocated to the player detfed by the lowest umber. More formally, let { } P = x Ω: f 1 (x) = max f m (x) 1 m
ad { } k 1 P = x Ω: f k (x) = max (x) P, k = 2,...,. k f m m =1 Let M V (P ) = (v (P j )),j=1 be the value matrx assocated wth the partto P.The ad ( ) v P = f dµ = max dµ = max f m dµ = v (3.12) f m m =1 =1 P =1 P Ω ( ) ( ) v P = f dµ = max f m dµ f j dµ = v j P m P P P for all, j = 1,...,. (3.13) Now, for each k = 1,...,, cosder the partto P k = (P k 1,...,P k ) whch assgs the whole set Ω to player k,.e., { k Ω f j = k, P j = (3.14) otherwse. Clearly, the value matrx M V (P k ) satsfes { ( ) k 1 f j = k, v P j = for all = 1,...,. 0 otherwse, Sce the v are atomless, Proposto 2.4() mples that M V s covex. Therefore, for +1 ay choce of β 1,...,β,β +1 wth β 0for all = 1,...,+ 1ad =1 β = 1, there (1) (1) exsts a partto P (1) = (P,...,P ) such that 1 M V (P (1) ) = β k M V (P k ) + β +1 M V (P ). k=1 Defe the coeffcets {β } as follows (cf. [12,13]): 1 v k (P k ) 1 β k =, k = 1,...,, ad β +1 =. v + 1 v + 1 The {β } are all oegatve sce v, ad satsfy β = 1 by (3.12). The elemets of M V (P (1) ) satsfy ( ) (1) ( ) ( ) ( ) v P k P j = β k v j + β +1 v P j = β j + β +1 v Pj k=1 1 v j (P j ) + v (P j ) 1 ( (1)) = = v P. (3.15) v + 1 v + 1 The equalty (3.15) follows by (3.13), wth the roles of ad j reversed. Therefore, P (1) s evy-free ad allots the value ( v + 1) 1 to each player. m
The proof of () also requres the followg verso prcple (cf. [10, Proposto 2.3]): M V M V (1 M V )/( 1) M V, (3.16) where 1 s the matrx whose elemets are all 1 s. Ths tme, the partto P, whch assgs each pot to the player wth the lowest desty, s { } P 1 = x Ω: f 1 (x) = m f m (x) m ad { } k 1 P k = x Ω: f k (x) = m f m (x) P, k It s easy to see that ad v (P ) = v =1 1 M V (P ) M V (P (2) ) = βˆkm V (P k ) + βˆ+1, 1 k=1 m = 2,...,. v k (P k ) 1 βˆk =, k = 1,...,, ad βˆ+1 =. + v 1 + v 1 +1 It s easy to check that βˆ 0, ad, by (3.17a), =1 βˆ = 1. From (3.17b) t follows that =1 v (P ) v j (P ) for all, j = 1,...,. (3.17a) (3.17b) By (3.16), (1 M V (P ))/( 1) M V ad, therefore, by Proposto 2.4(), there exsts (2) (2) a partto P (2) = (P,...,P ) whose value matrx satsfes 1 where coeffcets βˆ are gve by ( (2) ) ( ) 1 v (P j ) 1 v (P j ) v P j = βˆkv Pj k + βˆ+1 = βˆj + βˆ+1 1 1 k=1 v j (P j ) + 1 v (P j ) 1 ( (2) = = v ) P, (3.18) + v 1 + v 1 so P (2) s evy-free ad allots the value ( + v 1) 1 to each player. Statemets () ad (v) are a drect cosequece of () ad (), respectvely. I partcular, to prove (), aga cosder the partto P (1). It was show () that ths partto s evy-free, so e max (P (1) ) 0. Also, by (3.15), v (P (1) ) = ( v + 1) 1 ad ( (1)) ( (1)) v v P j 1 v P = for all j. v + 1
Therefore ( (1)) ( (1)) v 1 v P j v P for all j =, v + 1 whch completes the proof of (). Smlarly, to obta (v), ote that e max (P (2) ) 0 ad, by (3.18), so ( (2)) ( (2)) + v 2 v P j 1 v P = for all j, + v 1 ( (2)) ( (2)) + v 3 v Pj v P for all j. + v 1 Example 3.13. For the measures Example 2.2, Theorem 3.12() guaratees the exstece of a evy-free partto wth equtable share (2 v + 1) 1 = 4/7 = 0.57 for each player, whereas for these partcular measures eve more s possble (amely ( 5 1)/2 = 0.61, see Example 2.2). Smlarly, Theorem 3.12() guaratees the exstece of a super-evyfree partto wth maxmum evy (2 v 1)/(2 v + 1) = 1/7, whereas eve smaller maxmum evy 2 5 s possble (cf. Example 3.3). 4. Mmax-evy equaltes for measures wth atoms For atomless measures, far ad evy-free parttos always exst (cf. Theorem 3.12), as a cosequece of the covexty of the value ad evy matrx rages, respectvely (Proposto 2.4, Theorem 3.5). For measures wth atoms, however, geeral the sets of F-feasble value matrces ad evy matrces are ot covex, ad ether far or evy-free parttos exst (cf. Examples 2.3 ad 3.3()). It s the purpose of ths secto to establsh bouds o the ocovexty, ad upper bouds o evy based o the mass of the largest atom, aalogous to the bouds foud [7] for value matrces. The uderlyg tuto s smply that f the atoms are all very small, the the evy-matrx rage must be early covex, ad hece early evy-free parttos must exst. For α (0, 1), let P(α) deote the set of value fuctos wth o atom mass greater tha α. That s, { P(α) = v: v s a probablty measure o (Ω, F ) } wth v(a) α for all v-atoms A F. The ext theorem gves a upper boud o how far from covex the set of feasble evy matrces ca be as a fucto of the maxmum atom sze ad the umber of measures. Here co(s) deotes the covex hull of the set S. Theorem 4.1. Fx 1 ad α (0, 1), ad let v P(α), = 1,...,. The for every C = (c j ),j= 1 co(m E) there exsts P Π wth 3/2 e,j (P) c,j α(2) for all, j = 1,...,.
Proof. Fx C = (c,j ),j=1 co(m E). By Lemma 3.4, there exsts D = (d,j ),j =1 co(m V ) such that c,j = d,j d, for all, j = 1,...,. (4.1) Sce v 1,..., v P(α), by a theorem of Allaart [2, Theorem 2.11()], the Hausdorff eucldea dstace betwee M V ad ts covex hull s o more tha 2α 3/2,sothere exsts M = (m,j ),j=1 M V wth [ ] 1/2 2 3/2 (m,j d,j ) 2α. (4.2),j=1 Sce M M V, there exsts a partto P = (P 1,..., P ) Π wth v (P j ) = m,j for all, j = 1,...,. (4.3) m Sce max{ a 2 1,..., a m } ( k=1 a k ) 1/2, (4.2) ad (4.3) mply that 3/2 v (P j ) d,j 2α for, j = 1,...,. (4.4) By defto of evy, e,j (P) = v (P j ) v (P ), so (4.1) mples that e,j (P) c,j = v (P j ) v (P ) (d,j d, ) 3/2 3/2 v (P j ) d,j + v (P ) d, 2 2α = α(2), where the last equalty follows by (4.4). Allaart has also foud the sharp boud for the Hausdorff dstace betwee the partto rage ad ts covex hull [1, Theorem 2.5] terms of α, whch has drect applcato to maxm-share but ot to mmax-evy equaltes. The ext result s a example of a applcato of Theorem 4.1 to establsh the exstece of evy-free parttos some fardvso problems wth atoms. Recall that v ad v are the total masses of the smallest measure domatg, ad the largest measure domated by, respectvely, all the measures v 1,..., v (cf. Example 3.11). Theorem 4.2. Fx 1 ad α (0, 1), ad let v P(α) for all = 1,...,.Thef ether ( +v +1 () α< ) ( v +1 (2) 3/2 or v +3 ) () α< (2) 3/2, +v 1 the there exsts a super-evy-free partto P Π. Proof. To see (), assume wthout loss of geeralty, that ( + v + 1)> 0, for otherwse the cocluso s trval. Elargg Ω f ecessary (e.g., replacg A by A [0, 1] for every v-atom A F), t may be assumed wthout loss of geeralty that there exsts a σ -algebra Fˆ F, ad atomless measures u 1,..., u o (Ω, Fˆ ) such that
u = v ad u (P ) = v (P ) for all P F. (4.5) Lettg {( ) } Mˆ E = u (Pˆj ) : Pˆ1,..., Pˆ Fˆ, Pˆ = Ω, Pˆ Pˆj = f = j,,j=1 t follows by the defto of {u } ad Fˆ that ˆM E = co(m E ). (4.6) By (4.5) ad Theorem 3.12(), there exsts a partto Pˆ = (Pˆ1,..., Pˆ) wth Pˆ Fˆ for all, ad satsfyg u 1 u (Pˆj ) u (Pˆ) for all = 1,...,, j. (4.7) u + 1 By (4.6), (u (Pˆj )),j= 1 co(m E), so by Theorem 4.1 there exsts a partto P Π wth ( ) ( ) 3/2 + v + 1 3/2 3/2 e,j (P) u (Pˆj ) u (Pˆ) α(2) < (2) (2) v + 1 + v + 1 =, = j, v + 1 so by (4.7) ad the fact that u = v, + v + 1 e,j (P)< + u (Pˆj ) u (Pˆ) 0, v + 1 so e max (P)< 0ad P s super-evy-free, whch proves (). The argumet for () s smlar, usg Theorem 3.12(v). Example 4.3. Suppose that v 1 ad ( v 2 are probablty measures wth v = 5/4. If o atom v +1 v ) 1 or v 2 has mass greater tha 2+ 2 v (4) 3/2 +1 = 1/56, the there s a super-evy-free partto. (Compare wth Example 3.13, where v 1 ad v 2 are atomless wth the same outer measure v = 5/4.) The ext proposto, whch s recorded here for ease of referece, gves the sharp guarateed maxm share as a fucto of maxmum atom sze ad umber of measures; t wll be used here to establsh upper bouds o maxmum evy also as a fucto of atom sze ad umber of measures. Defto 4.4. V : [0, 1] [0, 1] s the uque ocreasg fucto satsfyg V (x) = 1 k( 1)x for all x [(k + 1)k 1 ((k + 1) 1) 1,(k 1) 1 ], k = 1, 2,... Proposto 4.5 [9]. Fx 1 ad let v 1,..., v P(α). The there exsts a partto P = (P 1,..., P ) Π satsfyg v (P ) V (α) for all = 1,...,, ad ths boud s attaed.
Theorem 4.6. Fx 1 ad α (0, 1) ad let v 1,..., v P(α). The there exst parttos P (1), P (2) Π satsfyg () e max (P (1) ) α(2) 3/2 ; ad () e max (P (2) ) 1 2V (α). Proof. Let u 1,..., u, F,ad ˆ Mˆ E be as the proof of Theorem 4.2. Theorem 3.8() mples the exstece of a evy-free partto Pˆ for u 1,..., u, ad va correspodece (4.6), ths mples that there s a elemet C = (c,j ),j=1 co(m E) wth c,j 0for all, j = 1,...,. Cocluso () the follows mmedately from Theorem 4.1. To see (), let P = (P 1,..., P ) Π be as Proposto 4.5. By addtvty of the measures {v }, v (P j ) 1 v (P ) for all j, so v (P j ) v (P ) 1 2v (P ) 1 2V (α). Example 4.7. Let α = 0.01, that s, o partcpat values ay crumb more tha oe hudredth of the total value of the cake. If there are two players, the boud Theorem 4.6() s 0.08 ad checkg that V 2 (0.01) = 50/101, the boud () s 1/101, whch s sharper. If there are three players, the the boud () s (0.01)6 3/2 = 0.1470, ad that () (checkg that V 3 (0.01) = 33/101) s 35/101, whch ths case s substatally weaker tha the boud gve by (). Refereces [1] P. Allaart, A sharp ocovexty boud for partto rages of vector measures wth atoms, J. Math. Aal. Appl. 235 (1999) 326 348. [2] P. Allaart, Bouds o the o-covexty of rages of vector measures wth atoms, Cotemp. Math. 234 (1999) 1 11. [3] J. Barbael, Super evy-free cake dvso ad depedece of measures, J. Math. Aal. Appl. 197 (1996) 54 60. [4] S. Brams, A. Taylor, A evy-free cake dvso protocol, Amer. Math. Mothly 102 (1996) 9 18. [5] S. Brams, A. Taylor, Far Dvso: From Cake-Cuttg to Dspute Resoluto, Cambrdge Uv. Press, Cambrdge, UK, 1996. [6] L. Dubs, E. Spaer, How to cut a cake farly, Amer. Math. Mothly 68 (1961) 1 17. [7] J. Elto, T. Hll, A geeralzato of Lyapouov s covexty theorem to measures wth atoms, Proc. Amer. Math. Soc. 296 (1987) 297 304. [8] J. Elto, T. Hll, R. Kertz, Optmal parttog equaltes for oatomc probablty measures, Tras. Amer. Math. Soc. 296 (1986) 703 725. [9] T. Hll, Parttog geeral probablty measures, A. Probab. 15 (1987) 804 813. [10] T. Hll, A sharp parttog-equalty for o-atomc probablty measures based o the mass of the fmum of the measures, Prob. Theory Related Felds 75 (1987) 143 147. [11] T. Hll, Parttog equaltes probablty ad statstcs, IMS Lecture Notes 22 (1993) 116 132. [12] J. Legut, Iequaltes for α-optmal parttog of a measurable space, Proc. Amer. Math. Soc. 104 (1988) 1249 1251. [13] J. Legut, M. Wlczysk, Optmal parttog of a measurable space, Proc. Amer. Math. Soc. 104 (1988) 262 264. [14] J. Robertso, W. Webb, Cake-Cuttg Algorthms, A.K. Peters, Natck, MA, 1998. [15] D. Weller, Far dvso of a measurable space, J. Math. Ecoom. 14 (1985) 5 17.