Equidistribution in Sharing Games

Transcription

1 Ope Joul of Dscete Mthemtcs, 4, 4, 9-8 Publshed Ole Juy 4 ( Equdstbuto Shg Gmes Clos D Ade, Emlo Gómez Deptmet d Àlgeb Geomet, Uvestt de Bcelo, Bcelo, Sp Deptmet of Mthemtcs, Uvesty of Clfo, Bekeley, Bekeley, USA Eml: cdde@ubedu, emgomez@mthbekeleyedu Receved August 3, 3; evsed Septembe 5, 3; ccepted Octobe, 3 Copyght 4 Clos D Ade, Emlo Gómez Ths s ope ccess tcle dstbuted ude the Cetve Commos Attbuto Lcese, whch pemts uestcted use, dstbuto, d epoducto y medum, povded the ogl wok s popely cted I ccodce of the Cetve Commos Attbuto Lcese ll Copyghts 4 e eseved fo SCIRP d the owe of the tellectul popety Clos D Ade, Emlo Gómez All Copyght 4 e guded by lw d by SCIRP s gud ABSTRACT Gmes ofte povde good toducto to teestg pheome mthemtcs I ths ote, we exme thee vtos of tetve shg gme plyed oud ccul (o ot so ccul tble Moe pecsely, fo ech vto, we study the tedecy towd equl dstbuto mog the plyes I the fst vto, the plyes hve dscete mouts t ech step The secod vto emoves ths estcto, d the thd oe cosdes ftely log tble wth fte umbe of plyes KEYWORDS Equdstbuto; Iteto; Le Tsfomto; Lmt Dstbuto Shg oud Tble wth Dscete Amouts Suppose you hve pesos seted oud ccul tble, ech hvg eve umbe of dmes Numbe them fom to couteclockwse, d deote by the (eve umbe of dmes tht the -th peso hs, =,, The gme s plyed by epetg the sme two steps ove d ove, s follows: fst, ech peso gves oe hlf of he dmes to the peso sttg o he ght I symbols, +, =,,, ( wth the coveto = Note tht the ew s e ot ll ecessly eve The secod step, whch llows us to epet (, s to dd oe dme to ech odd : +, =,, ( We cll ths gme the dscete shg gme Poposto By tetg ( d (, fte fte umbe of steps, we wll hve = = =, e evey peso eds up wth the sme umbe of dmes Ths poblem c be foud [] The poof follows esly fom two obsevtos Fst, the mxmum mout oud the tble cot cese, o c the mmum mout decese Secod, f the mmum mout s stctly smlle th the mxmum mout, the fte the ext step, ethe the mmum mout wll hve cesed, o else the umbe of plyes hvg the mmum mout wll hve decesed Togethe, these obsevtos mply tht fte ftely my steps, the mxmum d mmum mouts wll cocde We leve the detls of the poof to the ede Aothe teestg poblem s to deteme the umbe of tetos eeded to ech equdstbuto tems of the tl dt (see Fgues, d 3 A tetve shg gme tht teds towd the equl dstbuto of the plyes mouts s wht we cll

2 C D ANDREA, E GÓMEZ Fgue 3-D gph of the dscete shg gme wth plyes d wth tl dstbuto s show I ths exmple, equdstbuto s cheved t the 6-th teto Fgue -D gph of the gme gve Fgue Ths gph does ot show how the tl mouts e dstbuted oud the tble Fgue 3 -D gph of geelzed dscete shg gme wth plyes d k = 3

3 C D ANDREA, E GÓMEZ equdstbuto pocess You my thk tht the bove exmple, the equdstbuto s due to the fct tht the shg s doe by hlves Howeve, Poposto geelzes wthout much effot: suppose ll the s e tlly multples of fxed umbe k Replce ( wth k +, =,,, k k d ( wth sml step whee you dd dmes utl you ech the fst multple of k gete th o equl to fo ll =,, The g you wll ed up wth = = = fte fte umbe of steps Shg oud Tble wth Complex Numbes Suppose thee e pesos oud ccul tble, umbeed couteclockwse s befoe But ths tme ech of them stts wth tl mout,, tht c be y complex umbe Let be postve el umbe, < < It wll ply the ole of bove We wt to study the behvo of k +, =,,, s the umbe of tetos goes to fty Note tht the blcg step ( does ot mke sese hee ymoe, s ethe the s e ecessly teges o = fo some k We c gve elstc feelg to the k gme f we estct the s to el umbes The t ech step, ech peso s shg he welth ( > o debt ( < by gvg poto of he umbe to the peso o he ght But ths estcto s uecessy, so we wll wok the moe geel settg of complex umbes I cotst to the dscete veso of the lst secto, we cll ths the complex shg gme Fo =,,,, let be the mout tht peso hs fte the -th step The =, d we lso exted the defto by settg = wheeve mod, so tht mkes sese fo y tege The ule fo shg t ech step the yelds = +, whee the sub-dces e udestood modulo Note tht f =, the = Sce <, clely ths shg gme wth oly two people s equdstbuto pocess Ths s ptcully tvl f =, whch cse both pesos wll hve the sme umbe fte ust oe step Fom ow o, we ssume > Note lso tht the sum of the umbes oud the tble ems costt wth ech step Let us deote ths costt by S, tht s S = + +, The Shg Tsfomto d Its Egevlues We c model the shg pocess (3 wth the d of the -le tsfomto T : ( = ( + ( + ( + ( T z,, z z z, z z,, z z Wth the otto s bove, we see tht ( = T ( gve by,,,,, whee the expoet o T dctes the umbe of tmes tht ths tsfomto s composed wth tself: T T T I the cocl bss of, T hs mtx A =, whch hs etes log the m dgol, o the dgol mmedtely below d t the uppe ght (3

4 C D ANDREA, E GÓMEZ coe, d zeo elsewhee Ths s stochstc o Mkov mtx, s ts etes e ll o-egtve d the sum of the etes ech ow s equl to Fo moe fomto o these mtces d the teestg popetes, we efe the ede to [] We wll pesetly pove some of these popetes fo ou mtx A ode to mke ths ote moe self-coted, but the ede should be we tht both Popostos d Theoem below hold fo y stochstc mtx Poposto The mtx A hs dstct egevlues λ =, λ,, λ wth λ < fo =,, f t be the chctestc polyoml of T Computg t explctly, we obt Poof Let ( ( f t = det t A = t Let,, ω ω be the th dstct complex oots of The t s esy to see tht the (dstct oots of f e λ = ω +, =,, If we set ω =, the we hve tht λ = Fo =,,, we hve λ = ω + < ω + = The equlty s stct, s ω s ot postve el umbe f > A egevecto ssocted to the egevlue s v = (,,, I ou gme, ths egevecto coespods to the cse whee evey peso hs the sme mout ( = S, =,, Clely, ths stuto s stble, e does ot chge fte pplyg T Let v,, v deote egevectos fo the egevlues λ,, λ espectvely Thus T( v = λv fo ech Now we e edy fo the m esult of ths secto, whch s tht the complex shg gme lso teds towd equl dstbuto mog the plyes (See Fgues 4 d 5 Theoem The complex shg gme descbed by (3 s equdstbuto pocess Tht s, Poof Note tht { v,, v} (,, = = α v S S lm (,, =,, s bss fo cosstg of egevectos of wth α, =,, The Poposto ow mples tht ( (,, = T,, = T αv= αλ v = = ( = αv= ( αα α lm,,,,, T Wte Theefoe the gme s deed equdstbuto pocess Note tht, sce the sum of the umbes ems costt t ech step of the teto bove, we must hve + S α = = Thee s othe wy to obt the vlue of α Let C be the subspce of de fed s C = {( z,, z : z + + z = } Lemm Let { v,, v} be the bss of egevectos fxed bove The {,, v v} s bss of C Poof Wte v (,, = v v fo > Sce T( v = λv, we hve ( v + ( v v + ( v = λ ( v v,,,, Addg the coodtes o ech sde of ths detty, we get v v v v + = λ + + Sce λ fo >, we coclude tht v + + v = v,, v s coted C, d sce both of these spces hve dmeso, we must hve equlty, povg the clm We c use ths sttemet to compute the vlue of α We hve Recll tht we chose (,,, ( Theefoe, the subspce sped by { },, = α v = α v + α v = = v = Sce αv C =, the sum of the coodtes of ths vecto s equl

5 C D ANDREA, E GÓMEZ 3 Fgue 4 Evoluto of the complex shg gme wth plyes d = 3/4 The tl mouts e teges betwee d Fgue 5 Evoluto of the complex shg gme wth 5 plyes d = / The tl mouts e teges betwee d 5 to zeo, d hece + + = α Fom ths we deduce g tht + S α = = A Explct Fomul fo Recll tht we defed =, d ptcul =, wheeve mod Let us fd fomul, tems of the tl dt,,, fo the umbe tht ech peso wll hve fte y gve step of the complex shg gme: Poposto 3 Fo y d, we hve = (

6 4 C D ANDREA, E GÓMEZ Poof We use ducto o, the tl step beg obvous Next, ssume the fomul holds fo ll d fo ll wth, fo fxed The, ( = + = ( + ( ( = ( + ( = = (, ( + so the fomul holds fo + d the poof s complete I ptcul, whe ech peso shes oe hlf of he mout t ech step, we hve Coolly If, = the = fo ll d By goupg the boml coeffcets coguece clsses of modulo, we obt equvlet fomulto of ths esult (lwys fo = : = = + +, l=, l l l=, l l l=, l l whee ech coguece s udestood to be modulo If ll the tl mouts,, e equl, the shg pocess s stble fom the begg, d ths fomul educes to the well kow fct bout the sum of the etes the -th ow of the Pscl tgle, = Wht s moe teestg s tht, lght of S Theoem, lm = fo ll, egdless of the tl mouts So gve, f we goup the boml coeffcets by the coguece clsses of modulo d gve ech clss bty weght, the s we go futhe dow the ows of the Pscl tgle, the weghed sum of the etes dvded by the sum wth o weghts,, lwys teds to the vege of the weghts Moe geelly, egdless of the vlue of, Theoem S shows tht the sums gve the sttemet of Poposto 3 ll ted to the sme lmt s 3 Egevectos of T d Vdemode Mtces Let ω be y -th oot of uty, e ω = The t s esy to check tht Tht s, (, ωω,,, ω ( ωω ω = ( ω + ( ωω ω T,,,,,,,, s egevecto of T wth egevlue ω +, the ltte beg of couse oe of the λ bove Theefoe, f ω p s pmtve -th oot of uty, the the vectos v,, v gve by (,,,, v = ωp ω p ω p fom bss fo cosstg of egevectos of T Note tht we hve g v =,,, s bove, d

7 C D ANDREA, E GÓMEZ 5 ( ( tht the egevlue fo v s λ = ω p + Ude the (odeed bss v,,, v T hs dgol mtx λ D = λ The mtx whose colums e the coodtes of the vectos v,, v chges the bss fom the cocl to v,, v Ths s Vdemode mtx ωp ωp ωp V( ω p =, ( ( ω p ωp ω p d t s well kwo tht ts vese s othe (omlzed mtx of the sme type, V ( ω = V ( ω p p We c use ths lst mtx to expess the vecto of tl mouts (,, { v v } As the poof of Poposto, let (,, = = α v The,, V ( ω p α α = α tems of the bss d stghtfowd computto yelds + wy to obt α = = S ( ( p = α = ω fo =,, Ths gves us yet othe 3 Shg t Ifte Tble Suppose ow tht, sted of ccul tble, we hve fte umbe of pesos seted o oe sde of ftely log ectgul tble, the umbe beg ubouded both fom the left d the ght Ech peso hs tl (complex umbe mout to be shed the sme wy s the gme wth the ccul tble: t ech step, ech peso gves poto of he mout to the peso o he ght As befoe, s fxed postve el umbe, < < We cll ths veso the fte shg gme The tl dt ow s sequece ( of complex umbes We defe s (3 fo d Note tht ths tme thee s o coguece modulo, s thee e ftely my plyes, dexed by the teges Ideed, the ectgul tble c be egded s the lmt of ccul tbles hvg The ecuso fom shg s the sme s befoe: = +, (4 Poposto 3 lso holds hee, d the poof s the sme The oly dffeece s tht ow we do ot hve = wheeve mod, the codto tht effectvely mde the gme ccul (d wth plyes befoe So we stll get = ( Ths fomul shows somethg tht s obvous the ew cotext, mely tht the mout of the peso the -th plce wll deped oly o the tl mouts of those people seted to he left (d o he ow tl mout, of couse So thee s o eso to beleve tht ths ew gme should be equdstbuto pocess (5

8 6 C D ANDREA, E GÓMEZ Fo stce, f thee s cocetto of welth (by whch we me umbes wth lge bsolute vlue some secto of the tble, t wll eve sped to the left Howeve, s the followg esult shows, f oe of the sequeces ( coveges, the the gme wll ted towd equl dstbuto Theoem 3 If lm exsts fo some, the lm exsts fo ll, d ll of these lmts cocde Poof Let be such tht L = lm exsts The, usg (4, we hve lm = ( L ( L = L, e L exsts d t s equl to Now, usg (4 g, L By ducto, we hve the tht ( = L exsts d L L fo ll Sce L s, gve ε > thee exsts such tht But the we hve tht fo k, becuse thks to (6, d + k + k k k ( ( ( L ( k k L + L ε k + ( + L ε k + ( + L, k k k + ( + L + L ε + + = < k k L ( ( = L = fo ll (6 ε L < fo ll k ε Fo k, we c get ( + L < whch, togethe wth (7, shows tht L = lm + = L+, d fom hee we deduce (g by ducto tht L = lm = L fo ll > It would be teestg poblem to chcteze those sequeces ( of tl dt tht duce equdstbuto pocess the fte ectgul tble Note fo stce tht f f : s y peodc fucto d we set = f fo, the the gme wll be equdstbuto pocess (ths s essetlly the complex gme oud ccul tble, whee the umbe of plyes s the peod of f, lfted to the fte ectgul tble Oe tul questo to sk s ths: f the tl dt s bouded, must the fte gme be equdstbuto pocess? Suppose tht C fo ll The, sce ( = ( + ( =, we see fom (5 tht C fo ll d fo some (d hece ll? The swe s NO, s the followg exmple shows Exmple Suppose, d set Does ths mply tht the sequece ( f =, = k othewse k k k k f 3 3 fo some >, (7 (8 coveges

9 C D ANDREA, E GÓMEZ 7 Ths sequece c be defed ecusvely by fst settg ll ts vlues equl to zeo fo, the =, d fom thee o we move to the left two zeoes, the thee oes Evey tme we ed wth lst of oes, we put sequece of zeoes doublg the umbe of the sequece ledy bult so f (sttg fom Ad evey tme we ed wth lst of zeoes, we put sequece of oes equl to the umbe of the sequece ledy bult so f (sttg fom, lwys movg to the left So, the sequece looks lke ths (sttg fom o the ght:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Wth ths sequece s the tl dt d (5, oe c show tht fo y postve tege k, so the sequece (,, 3 k k k k 3 3 does ot covege d, lght of Theoem 3, ( does ot covege fo y A mmedte cosequece of Theoem 3, s tht f thee exst d such tht = fo ll <, the the fte gme s equdstbuto pocess Ths s, fct, ptcul cse of moe geel fct: t tus out tht equdstbuto follows fom the exstece of lm (d ot, s we ust sw, fom the boudedess of ( Theoem 3 Suppose tht lm = The lm = fo evey Poof Cosde the sequece ( b defed s b = fo ll Fo fxed, due to (4, t s esy to see tht lm = f d oly f lm b =, so we c ssume wthout loss of geelty tht lm = Fo fxed, sce the sequece ( coveges, thee exsts C > such tht ε < C fo ll Let ε > be gve Sce, thee exsts such tht < f > Fo >, we use (5 to wte As ( ( ( = + + (becuse of (8, we get esly + As fo the fst summd of (9, we hve: Sce <, fo ε ( < + ( ( ( C C + = ( ( C ( ( we get C ε <, d hece we c boud (9 s follows: ε ε + =ε,, whch poves tht lm = Whe the tl mouts e ll el, (, oe c smlly show tht f lm = + (esp, the + (esp s, fo evey Note tht the exstece of lm s suffcet but ot ecessy fo the fte gme to be equdstbuto pocess Ths c be see fom the bove exmple whee the tl dt tke o the vlues of peodc fucto f : As metoed bove, t would be teestg to fd moe explct ecessy d (9

10 8 C D ANDREA, E GÓMEZ suffcet codtos th the exstece of lm ( tlly the chctezto we get fom Theoem 3 d (5 fo some, whch s esse- Ackowledgemets We e gteful to Adá Pez fom whom we let bout the dscete shg gme d the efeece [] We would lso lke to thk José Igco Bugos fo teestg dscussos, Gevso Gómez fo pogmmg sevel smultos d poducg the fgues, d Ju Sb fo suggestg the sttemet of Theoem 3 s t ppes the text, whch s stoge d moe geel th wht we hd oglly wtte The fst utho ws ptlly suppoted by the esech poect MTM79 fom the Msteo de Cec e Iovcóo (Sp REFERENCES [] G Z Chg d T W Sedebeg, Ove d Ove g, New Mthemtcl Lby, 39 Mthemtcl Assocto of Amec, Wshgto DC, 997 xv+39 pp ISBN: [] G Ltouche d V Rmswm, Itoducto to Mtx Alytc Methods Stochstc Modelg, st Edto, PH Dstbutos, ASA SIAM, 999