Cutting Cakes Correctly

Cut ting Cakes Correctly by Th eo do re P . H ill Sc hoo l of M ath em at ics, Ge or gia In sti tute of Tec hn olo gy , A tlan ta, G A 3 03 32- 016 0 hill @m a th. gate ch .ed u The ar tic le “B ette r W a ys to C ut a C ak e” [ BJ K] by Br am s, J on es and K lam le r in the D ece m ber 200 6 is su e of th es e N otic es is rec eiv ing wi de spr ea d m ed ia a tten tio n i n S cie nti fic A m eric a n Sc ie nce Ne ws , Sc ien ce D aily , an d th e Dis cov er y C ha nne l , a m ong oth er s, as w el l as in our ow n So cie ty’s pro mo tio na l site A MS in the N ew s . U nf ortu n atel y, [ BJ K] co nta ins ser io us ma the m atic al e rro rs, som e o f w hi ch wi ll be su m ma riz ed h er e. F ur the r de ta ils m a y be fo un d in [ TP H] . In t he firs t se c tion of [ B JK] , t he a ut hor s s tate (p 13 14) th at i n p rob lem s o f f air div is ion of a div isib le g o od, “the w ell- kn ow n 2- per so n, 1 -c ut c ak e-c utti ng pr oce du re ‘I cu t, y ou ch oos e’” is Pa ret o-op tim al , tha t i s, “T he re i s n o o the r a lloc ati on tha t is be tter for on e per son a nd a t le as t as goo d f or t he oth er .” C ut- and -ch oo se is n ot e ve n P ar eto op tim al a m ong ( n -1 )-cu t p roc ed ure s, a we ake r f orm of Pa re to o pt im ality ca lled “C -ef fic ient ” [B T p 1 49- 150 ], a s th e foll ow ing sim p le exa m ple s h ows . Co un te rex am pl e 1 . Th e “c ak e” is the un it s qu are , an d p la yer 1 valu es on ly t he top ha lf o f th e cak e a nd pla yer 2 o n ly th e bot tom ha lf (an d on t hos e p ort ion s, th e valu es ar e un if orm ly dis trib ute d) . If play er 1 i s th e c ut ter, an d c uts ve rtic ally , h is u niq ue ly o p tim al c ut- and -c hoo se sol utio n i s to b isec t th e c ak e e xac tly, in wh ich ea ch pla ye r re ce ive s a por tio n h e va lu es e xa ct ly ½. Or if p lay er 1 c uts ho riz on tally , h is uni que ly opt im al r isk -ad ve rse cut -an d-c ho os e po int is the lin e y = ¾ , in wh ich ca se h e rec eiv es a p orti on he val ues at ½ th e c ake , a nd pla yer 2 c ho os es t he bo tto m por tio n an d r ec eiv es a po rt ion he v a lue s a t 10 0% of th e c ake . B ut an allo ca tion o f th e t op h alf of the ca ke t o p laye r 1, a nd the bo tto m h alf to pla ye r 2 is a t le as t as go od for pl aye r 2 in bo th cas es, and is stri ctl y b ette r f or p lay er 1, so c ut- an d-c ho ose is no t P are to o pt ima l in ei the r d ire ction . If th e cak e i s th e u nit in terv al a nd the va lue m ea sur es a re all con tin uo us a s we ll a s m utu al ly abs olu tely c ont inuo u s, th en cu t-an d- cho os e i s Pa re to o ptim a l. T ha t co nc lusi on ma y f ail if t he me as ure s ar e no t c on tinu ou s, a s is e asil y s een by lo ok ing at the ca se whe re al l pl aye rs p la ce all t he valu e of t he cak e o n t he sam e sin gle po int (a nd h enc e ar e m utu all y ab so lute ly c on tinu o us) . A nd if all the m ea sure s a re c ont inu ou s b ut n ot m utua lly ab so lute ly c on tinu ou s ( wh ich w as n ot as sum ed any wh ere in [B T] ), th e state m ent qu ote d f rom [B T ] in [B JK , fo ot note 3 p 1 31 8] “a n e nv y-f re e allo cat ion tha t u ses n - 1 p ar alle l c uts is a lw ay s ef fic ien t [i. e. , Pa re to o ptim a l]”, a s w ell as the cor re spon di ng P ro po sit ion 7 .1 of [B T, p 15 0] , a re n ot true . Co un te rex am pl e 2 . T h e ca ke is the un it i nte rva l; p laye r 1 va lu es i t u nifo rm ly, an d p lay er 2 v alu es onl y t he le ft- an d r igh t-m ost qu art ers of the in ter val, an d v al ues the m equ all y a nd un ifo rm ly. (In othe r wor ds, th e pr o bab ility de nsi ty f un ctio n ( pd f) r epr ese n ting pla ye r 1 ’s v alue is a. s. co n stan t 1 on [0, 1], a n d th at o f p lay er 2 i s a .s. c on sta nt 2 on [0 , ¼ ] an d o n [3/4 ,1] , a nd zer o o ther w ise .) I f play er 1 i s t he c ut ter, hi s un iqu e cut p o int i s a t x = ½, an d e ac h p lay er w ill re ce ive a p orti on he valu es at ex act ly ½. Th e a lloc ati on o f t he in terv al [ 0, ¼ ] t o p lay er 2 a nd the re st to pl aye r 1, how ev er, gi ves pl aye r 1 a p or tion he va lue s ¾ , a nd pla ye r 2 a p or tion he va lue s ½ a ga in, so cu t- and -c hoo se (w hich is an en vy -fr ee a lloc at ion for 2 pla yer s) i s n ot Pa reto op tim al. 1 The ne w cak e- cutt ing pr oc edu re d es crib ed in [B JK ], S ur plus P roc ed ure (S P) , is n o t we ll d efi ne d. The in tegr al s de fin ing S P [B JK p 13 15] ne ed no t ex ist , if , fo r e xam p le, t he me as ure s do no t h ave den sit ies. A lso, if a p lay er’s va lue me as ure do es no t ha ve a u niq ue m edia n, the de fin ing cu t-p oin t in S P is n ot u n ique . S im ilar ly, w i tho ut add itio nal as sum pt ion s, t he n ew E qu itab ilit y P roc ed ure (E P) in [ BJ K] is n ei ther we ll d ef ine d n or con str uc tive , s ince th e u nd erly in g s yste m s of n- 1 i nte gra l equ ati ons in n- 1 u nkn ow ns m ay n ot ha ve s ol utio ns, a nd t hos e t hat do m ay no t hav e so luti ons in clos ed fo rm (c f. [T PH ]) . The au tho rs cla im tha t b oth EP an d S P are Pa re to o ptim a l [B JK , pp 1 31 8, 1 32 0], bu t ne ith er pro ce dur e is Pa re to o ptim a l as de fin ed i n [ BJ K, (2) p 1 31 4] an d as de fin ed in [B T, p ag e 4 4]; for cou nte rex am ple s, see [T PH ]. T h e un de rly ing re aso n is th at b ot h EP a nd SP al loca te c o ntig uo us por tio ns t o e ac h p lay er, a n d as no ted in [B T, p 1 49] , “sa tis fyin g c on tig uity m ay be inc on sis ten t wit h sa tis fy ing e ff icie ncy [ Pa reto op tim ali ty] ”. The au tho rs [B JK , 13 16] d efin e an a llo ca tion pr oce dur e t o b e s tra te gy-v u lner abl e if a “p lay er c an , by mi sre pre sen ting i ts v alue fu nc tion , a ssu red ly do bet ter , w ha teve r t he v al ue f unc tio n of t he o th er play er ”; a nd o the rw ise the p roc ed ure is c al led str ate gy -pr oof . T he s eco n d co nc lus ion of [B JK ,Th eor em 1] sta tes “an y p roc ed ure tha t m ak es e th e c ut- po int is s trat eg y vu ln era ble ”. T hat con clu sio n is fa lse , as th e n ext ex am ple sh ow s. Co un te rex am pl e 3. Su ppo se pla ye r 1 has tr ue val ue m eas ure v . E ve ry pro ced ur e a lloc ate s d isjo in t sub se ts o f t he c ake , o ne to eac h p laye r, and if the p laye rs’ va lues ha pp en to be i de ntic al, at lea st one pl aye r r ece ive s a por tio n h e v alu es no m ore tha n 1/n . T hat p la ye r co uld be p lay er 1 , s o he w ill not do “as sur edly be tte r” th an hi s fa ir s ha re o f 1/n . He nc e e ve ry f air all oca tio n pr oc ed ure is stra teg y-p ro of, inc lu ding the o ne tha t m ak es e th e c ut-p o int, co ntra dic tin g th e c on clu sio n o f The or em 1 ( an d s how ing th at t he firs t c on clu sio n of [B JK T he ore m 1] a nd [B JK , Th eo rem 2 ] ar e triv ial) . The f ifth sen ten ce in t he pr oof of [B JK , T heo rem 3 ] sa ys “By m ov ing all pla yer s’ m a rks righ tw ar d … on e c an g iv e e ac h pla ye r a n e qua l a mo un t gr ea ter t ha n 1 /n”. Thi s i s fa lla ciou s ( for cou nte rex am ple se e [ TP H] ). T he res t o f th e pr o of o f [ BJ K, The or em 3] i s i nco m ple te, s inc e i t o nly pro ves a c lai m ab out the m ovi ng- kn ife pro ced ur e, w he re as t he d e sire d c onc lu sio n co nc ern s E P. Ac kn ow led ge me n t. Th e a uth or i s gr ate fu l to Pr ofe sso r Ke nt E . M orr is on and tw o ano nym o us coll eag ue s f or v alu ab le com m en ts a nd sug ges tio ns . Re fer en ce s [B JK ] B ram s, S . J ., J one s, M. A. , a nd Kla m ler, C . (2 006 ), B e tter wa ys to c u t a c ak e, N ot ice s o f the AM S 5 3, 13 14- 132 1. [B T] B ra m s, S . J. and T aylo r, A. D. (1 996 ), F a ir-D iv isio n: Fro m Ca ke -Cu ttin g t o D isp ute Res ol utio n , Ca m brid ge U ni ve rsity P re ss, U K . [TP H ] H ill, T. P . ( 200 8) , C ou nter ex am ple s in C ake -C uttin g , ar Xi v:0 807 .2 277 v1 [m ath .H O] . 2