Gambler s Ruin and. The Three State Markov Process. A Thesis Presented to the Faculty of. California Polytechnic University, Pomona

Transcription

1 Gamb s ad Th Th Sa Maov oss A Thss sd o h Fay of afoa oyh Uvsy, omoa I aa Ffm of h ms fo h Dg Mass of S Mahmas y a 5

2 Sga ag Thss: Gamb s ad h Th Sa Maov oss Aho: Da Sbmd: a As Dam of Mahmas ad Sass D. Aa K Thss omm ha Mahmas ad Sass D. ada J. Swf Mahmas ad Sass D. Jy Sws Mahmas ad Sass D. bs vo m Mahmas ad Sass

3 Aowdgms I wod o ha my advso D. Aa K. Who h oss hos of gda ad ddao ha h gav o h m, hs hss wod o hav b ossb. D. K s a sao o oy o m, b o vyo who osss hs ah. Thas o D. Jy Sws fo a of h owdg ad sgh. Aso o D. ady J. Swf, who aswd hosads of my sos hs as wo yas. Addoa has a owd o D. bs vo m, D. Sws ad D. Swf fo h obos as hss omm mmbs. Las b o as, I wod o ha my as, da ad ay, fo h osa ov ad oagm. Thas o my boh y fo aways movag m o sv o b h bs. Aso, has o Lassa fo g wh dss hos of sdyg ad mah babb. To a of my famy ad fds I am ay gaf o yo fo sadg bhd m dg my oy of bomg a mahmaa.

4 ASTAT obabs a dmd fo a vay of gamb s mods. Sfay, obabs a fod h assa gamb s mod agmd o d aasoh ad wdfa obabs. Ths obm s sovd boh a f m ad f m sg. I h f m as, a ah ombaos ays a y o o o sam ahs of h Maov ha. I h f m as, aos a sovd sg obaby gag fos ad h hoy of dff aos. Th as obaby fos a y dmd fo h ga h sa Maov oss. Ths soos a agozd o h ds ass ad fo foms; ams of ah y of as obaby fo a sd.

5 Tab of os ha - Gamb s F Tm Iodo 6 So - Gamb s F Tm 7 So - Gamb s wh aasohs F Tm 5 So - Gamb s wh aasohs ad Wdfas F Tm ha - Gamb s If Tm Iodo 7 So Th Gamb s obm 8 So - Gamb s wh aasohs If Tm 9 So - Gamb s wh aasohs ad Wdfas If Tm ha Th Ga Th Sa Maov oss Iodo Th Ga Th Sa Maov oss 5 oso 65 fs 67 5

6 ha Gamb s F Tm Iodo I hs ha, h gamb s obabs, f m, a dmd wo dff sgs. Th gamb s wh aasohs s sovd So ad h h gamb s obm wh aasohs ad wdfas s dmd So. W bg wh a y a ah og s fo sovg h ass gamb s f m. Ths ombao mhod s dd o sov h gamb s wh aasohs ad h gamb s wh aasohs ad wdfas, by soag h ass gamb s wh hs gazd mods. Th a wo ah og aoahs o fd h gamb s wh aasohs f m. Th fs mhod avods aasohs ad ods a soo ha s dd o h aasoh obaby oy my. Th sod aoah s mo d ad s y aasoh dd. Th sod h s h sd o sov h gamb s wh aasohs ad wdfas So. 6

7 So - Gamb s F Tm Th gamb s obm s ov h hdd ffy yas od. I a b ad ba o ovsao s ba ad foh bw as asa ad Fma, s []. asa osdd h obm so dff ha h dobd whh Fma wod b ab o sov. I asa s oo, s [], was mo dff ha a h oh obaby obms hy had dsssd. Th gamb s obm gos as foows: a ay sas wh a gv amo of moy, -doas, ad mas a ss of o doa bs. Assm hs ay h ws a doa o oss a doa o ah b. Th ay s fo may b vsazd as a Maov ha o h sa dagam show bow Fg - - Fg - Th sa dagam fo h gamb s obm wh <, <, ad. Eah sa ss h amo of moy h ay has a ay gv m. Sag a sa, h ay ws o oss a doa, movg o h f o h gh o Fg. Th gam ds wh h ay oss a of hs moy o wh h ahs hs goa of wg doas. Th obaby s h ha of ahg sa bfo sa. Th a wo vaas of h 7

8 gamb s obm: o assms f m o a md mb of bs ad h oh assms a f m o a md mb of bs. Th soo o boh gamb s obms das ba a as o h a 6 s [,]. I addo o asa ad Fma, soos w obad by. ygs, J. o, A. d Mov,. d Momo ad N. o, s [, ]. I hs ha, w aayz h f m gamb s obm sg a ah ombaos. W sos h s oy m fo wags. Th obaby of havg doas af mag bs gv yo sad wh doas s dod by y hoy of Maov has, [] f, fo,,,,,. O... O h Ρ has s. No: ad a h obabs of bg absobd by,,, m. Tha s, h obaby ha by h h b h ay has o moy f s, ad, s h obaby of mg hs goa of doas by h h b. A h absobg sas ad h s o a o go b ma h. L ob bg sa af -ss ay a sa, as aady dfd ad, ob absoo a sa zo -ss ay a sa. W dsss a mhod o dm a aay sso fo., 8

9 Fo smy assm,. Tha s, w assm, a boh sd h bo Fg. - - Fg L L, s h oo a a ahs gog fom o -ss, bodd by hozoa s y ad y ad sd o o h hs bodas. Fg Tya La ah To dm,, h obm oms dow o fdg h mb of a ahs gog fom o -ss o hg zo o. Th mb of a ahs L, s sd by L,. 9

10 Lmma -. Fo,, L wh h sbs of mas < > o f f Ths s s dvd sg ombaos, h fo, ad h mhod of so/so. Th oof may b fod [9, ]. wg h sm of boma offs h dg mma as h ows w oba a s dsovd by a oy omoa ofsso Da Mas, s [7]. Afwads, was fod o ha hs s gos ba may yas, f. []. Lmma -. Fo m a wh a, m a who mbs m a m a g w w m g mod ad wh w m, a oo of y. h m ombg Lmmas -. ad -. ad smfyg gvs h foowg moa ooso.

11 ooso -. Fo, L os s s, No: s o obvos h dg sso s a g, ao ha os h mb of a ahs. W a h of h w-ow Fboa foma. S [6] fo h om das of h oof of hs ooso. Fom hs ooso a ooay foows. ooay -. as, os s s, as os s s os, oof. Fo as, o ha,, L d

12 wh s h mb of ss ad d h mb of dow ss. As bfo, s h obaby of movg fom sa o sa, wg a doa ad s h obaby of movg fom sa o sa -, osg a doa. Th mb of sam ahs, Fg, ha sa a sa ad d a sa -ss who hg h wo absobg sas s L,. Moov d d d wh s h mb of wad ss ad d s h mb of dowwad ss. Th oa mb of ss mad,, s a o h mb of wad ss ad dowwad ss. Th dff wad ss ad dowwad ss s va o h dff dg ad sag osos. Ths gvs as, sg h foma fo L, fod ooso -.. Fo as, h obaby of sag a sa ad gog o sa -ss s va o h obaby of sag a sa, gog o sa o - ss, h gog o zo o h h -s, o sag a sa, gog o sa o - ss, h gog o zo o h - h -s, o Ths s a o h sm,ov, of obabs gog fom o -ss ms, h obaby of movg fom o sa zo o s.,,, s dmd by as sg. Ths,

13 os s s, os s s os s s No ha os s a aa gom ss wh os Ths, os os os So,, os s s, os os s s Ths oms h oof of as.

14 o Eam A o ay sas wh a gv amo of moy,,, o doas, ad mas a ss of o doa bs. Assm h ay h ws a doa o oss a doa o ah b of d o ba. 8/8 8/8 8/8 8/8 5 /8 /8 /8 /8 Fg 5 mamm va, 8/8 obaby of a w, ag a b o ba o d, /8 obaby of a oss, sag amo hav b aad h foowg ab sd as,., 5 5 5,,,,,,

15 So - Gamb s wh aasohs F Tm I hs so, a gazao of h oga gamb s obm s osdd. A addd obaby,, of a aasoh s assmd. Tha s, a ha ha a ay o dg h bg a ay a os a of hs moy s ow ossb. A om sysm may b modd a sma ma. A had dv may b hogh of as addg bys of fomao wh obaby ad osg o ossg fomao wh obaby, b a ay o m h had dv mgh ash osg a of s fomao. Th a ah og s fod ooso -., som agba, ad bas os of obaby, ovd h oos d o sov h gamb s obm wh aasohs. I hs so, w s wo dff mhods of soo. Th sod mhod aas mo ga, aowg s o osd mo ga ad obms s So. owv, hs s so, o may goa s o oba a ga aay sso fo h obaby of bg d f m wh h gamb s obm s agmd o d aasoh obabs. Movad by g ad oao mods, osd ha a ay s,,..- h s a obaby of a aasoh ag yo o sa zo as show Fg 5. 5

16 - Fg 5 - Gamb s wh aasohs wh,,, <, ad. Wh aso ma M O O... O O M h Ρ has s., No: ad a h obabs of bg absobd by m. As bfo, w wsh,, o fd a sao fo. Assm, as show Fg 6., Aga,, ob bg sa -ss ay a sa., ob absoo a sa zo -ss ay a sa. 6

17 - Fg 6 - Gamb s wh aasohs h bo, Th foowg ooay foows fom ooay -. ooay -. as, os s s, as os s s os, oof. I as, h mb of sam ahs sd h bo of Fg 6 s h sam as h mb of sam ahs sd h bo of Fg. Th oy dff s ha ow 7

18 < s. Thfo h sam og foma ooso -. hods ad as foows h as dd ooay -.. Fo as, fdg, s va o h obm of fdg, wh h s ad s vsd Fg. I So, h obaby of movg fom sa o sa zo -ss was show ooay -. o b, os s s os Η Swhg h s ad h s ad ag h a sa wh - gvs, os s s os, whh sabshs as ooay -.. To dm h obabs o Fg 5 w s wo dff mhods. Thom -. Fo os s s os os s s, oof. S,,,, hods, ad h obaby foows fom ooay -.. 8

19 No: h dg hom dds o h obaby a m way s. Th foowg hom gvs a mo ga aav sso fo h obabs o Fg 5. h ddy o s. Thom -. Fo, os s s os s s oof Th hom w b show assmg. Th agm ods by oog a wh yo w a o s ba fom, assmg yo a d ss. Ths mas ha w h sad a sa ad w o sa - ss ad h o h as s w o sa wh obaby, a oss; o sag a w oyd o sa - ss ad h o h as s w o sa wh obaby, a aasoh. If w add ov a ossb ahs havg ss ad ov a ossb as osos, w oba,,, Thom -. ow foows fom sbso of as of ooay -.. No: Ths ah og agm as o h mo ga sao of. 9

20 Eam wh aasohs A ay, ayg a a ga aso, sas wh a gv amo of moy,,, o doas, ad mas a ss of o doa bs ag a b. Assm h ay h ws a doa o oss a doa o ah b. A ay m dg h ay s bg h aso has obaby.5 of bg osd dow, asg h ay o os a of hs moy Fg 7 5, / obaby of a w, /8 obaby of a oss, /8 obaby of aasoh, sag amo hav b aad h foowg ab, sd as,. Fo am h, obaby fo bg sa h af bs gv yo sad wh o doa s fod h ab, by g off h foh ow,, om, y..7,

21 5 5,5,,,,, Th mbs o hs ab a od o fo sgfa dgs.

22 Smmazg h as ooay ad h wo as homs, h -s aso obaby fos of h gamb s obm wh aasohs Fg 5 sag a sa wh a: os s s, os s s os, os s s os os s s, Aavy,, os s s os s s whv.

23 So - Gamb s wh aasohs ad Wdfas F Tm I hs so, w gaz h mhods of So o dm h obabs of a gamb s sysm havg boh aasohs ad wdfas. Th mhod dsssd Thom -. as dy o gv a, aay sso fo h obabs of a sysm havg aasohs ad wdfas addo o h sa w/os asos. As bfo, o aoah s b o h ah og s of ooso -. So. Th ma da s ha a va aasoh o wdfa, wh os, w oy ha a h d of a oy. Ths aows s o osd was o Fg 8 bow o b ssay a ahs h d of h wa. ooso -. s s how o o sh ahs. osd s s - Fg 8 - Gamb s wh aasohs ad Wdfas

24 wh aso ma M O O... O O s s M s s h Ρ has s. No: ad a h obabs of bg absobd by,,, m. osd ha a ay s,,,- h s a obaby of a aasoh ag yo o sa zo ad ha h s obaby s of a wdfa ag yo o sa wh s,,,, ad. As bfo w wsh o fd a sao fo., s s - Fg 9 Gamb s wh aasohs h bo,

25 Th aso obaby fos a dmd fo wo ds ass by h foowg ooay. ooay -. as Fo, os s s, as Fo, os s s os s s oof. Fo as, h oof foows as bfo fom ooso -.. as foows fom h vos ah og s dsssd Thom -.. I a sma way, hs ah og h aows a d aoah fo fdg h obaby of ahg yo goa,, ss. Ths m,,,, s whh gvs 5

26 , os s s os s s s. I smmay, h ga -s aso obaby fos of h gamb s wh aasohs ad wdfas a, sag fom sa,, o,, os s s,. Th obaby, sag fom sa,, s, os s s os s s Th goa obaby, sag fom sa,, s, os s s os s s s As fa as w ow, hs foma a w h aasoh ad wdfa sg. 6

27 ha Gamb s If Tm Iodo I hs ha, w aga osd h assa gamb s obm b wh a f amo of m. Mo gay, w osd ah of h h mods sd ha d h assmo of havg a df amo of m absoo. Ths m, o hs do o y s ah og. Th obm s addssd as sovg a sysm of aos d aoa boday odos. Two hs a moyd: dff aos ad obaby gag fos. I aa, h f m gamb s wh aasohs obm s sovd So sg h hoy of sod od, osa off aos ad som as ad obaby hoy o hod a ogh. Ths s ad h mhod a h gazd So o sov h gamb s wh boh aasohs ad wdfas f m sg dff ao hs. A sg obaby gag fo aoah may aso b sd o dm h obabs h aasoh ad wdfa as. Ths soo mhod s aso sd So. 7

28 So - Th Gamb s obm W osd h assa gamb s obm b ow wh a f amo of m, - - Fg - Th sa dagam fo h gamb s obm wh <, <, ad. Th obaby s h ha of vay ahg sa bfo sa. Th assa gamb s obm f m has h foowg ga soo, s [, ]. T < T fo fo Th obm has b sovd may ways []. 8

29 So - Gamb s wh aasohs If Tm Ths so oos a h sam s- as So - b wh f m. W a oog fo h obabs, ha s, obabs of va absoo a sa zo. I hs so, w dv ssos fo boh h obabs ad h obaby of ahg o goa,, gv ha yo sa a sa. Ths a obad sg h hoy of a, osa off aos. Fo f m, a hs Maov ha was fs dsbd So of ha. - Fg - Gamb s wh aasohs L obva absoo a sa zo ay a sa ad obva absoo a sa ay a sa. 9

30 Th ma s of hs so s gv h foowg hom whh s odd h fom som o wo aag [5]. Thom -. Fo oof. odog o h fs s,,,, fo M K M If yo sa sa h ha of bg d a b fod by sg wh yo a af o s. Af o s yo od b a sa wh obaby h g d, a - wh obaby h g d o a sa zo wh obaby h g d. Th obaby af o s of bg absobd a gv ha yo bga a sa zo s zo s zo s a absobg sa. Th aos fom a sysm of sod od, a, osa-off dff aos.

31 Ths fo,,,,..., wh haas ao,. Th oos of hs ao a ±. No ha hs oos a ds s s osv. h ga soo of h s of h fom s [8] wh ad a osas o b dmd by h wo boday odos ad. Ths gvs wh soo Ths,

32 So,, omg h oof.

33 So - Gamb s wh aasohs ad Wdfas If Tm I hs so, w osd a sysm of a osa off aos fo h obabs of Fg gv ha yo sa a sa. W s wo mhods of soo. Th fs aoah ss obaby gag fos. O sod soo mhod whh w gafy aowdg was sggsd by hss omm mmb, D. ada J. Swf s a dff ao aoah whh ssay gazs h agms sd h vos so. osd h Maov ha sad Fg. s s - Fg - Gamb s wh aasohs ad Wdfas

34 W s h obaby of vay bg d, obva absoo a sa zo ay a sa. Thom -. Fo h Maov ha dsbd by Fg, h a gv bow. fo b fo a a a b a,,,..., ad,,,..., wh β α β α β α β α wh ad β α

35 oof. odog o h fs s gvs, M M,,..., wh s. So fo,,..., wh a odos, ad. Th obabs,,a b fod sg h h of gag fos s ag of []. L ad D gg wh ad myg hogh by gvs, 5

36 [ ] [ ] [ ] [ ] Myg hogh by, [ ] [ ] Sovg fo, [ ] [ ] Dvdg hogh by, Faog o a, 6

37 Th oos of h ada o h f had sd of h dg ao a,, ± ± ± ad may ow b fod. No ha assmg > o s>. W hav a h foma fo h f sm of a gom ss... So, [ ]... Dvdg hogh by gvs [ ]... y aa faos, wh A A Now A ad a b sd as ow ss. 7

38 A A A A Th [ ]... A ag A wh h ow ss gvs [ ] A [ ] A A A No ha ga f b b b a a a h 8

39 b a b a b a b a b a b a b a K M fo a a a,...,, ad,,,..., fo A b wh A A A ad A 9

40 Ta A β ad α o oba h sso Thom -.. Th obabs,, may ow b d off as h offs of h gag fo. Sovg fo h obabs mas ossb o fd h goa obabs, s, sg h fa ha as m s off o fy, fo a s h ha yo a a h of h wo absobg sas s. Tha s h s obaby of bg d o mg yo goa f yo aow f m.,,,,...,. fo a A aav way o sov h ao s sg a dff ao h, []. a h ao fod a, Th ga soo o h dg ao may b fod by h sm of h homogos soo ad a aa soo, []. To fd a aa soo, assm, a osa. Th

41 so s, s whh gvs h aa soo, s Now has h assoad homogos ao Th assoad haas ao s wh oos ad Th ga soo may ow b w as [] s

42 Th osas a dmd by h a odos,, ad : s s whh gvs s s s s s No ha o o odd wh s h gas g ss ha o a o. So a b w as,

43 s s ad s s s Ths h obabs s a s wh s s s s s

44 ha Th Ga Th Sa Maov oss Iodo I Maov osss, h sms bas am s h wo sa Maov oss whos as obaby fos a sovd om da mos boos o h sb, f. [, ]. Th s omy s h h sa Maov oss. owv, o o owdg, h as obaby fos of hs oss hav o b ad f gay. If h h sa Maov oss s addssd a a, oy a sa ass a sd s, fo am, h obm so of ha []. I s o ha h h sa Maov oss has h mahmaa ds foms ha h as obaby fos may a ad ha hs soos d goom fos ha dd o aa h wo sa Maov oss as. Thfo, fo h dg dagogay ad mahmaay sg asos, a dad soo of h as obaby fos fo h ga h sa Maov oss s dvod h.

45 So - Th Ga Th Sa Maov oss I hs so h as obaby fos,,, of h ga h sa Maov oss a y dmd. Ths as obaby fos a fod o hav h ds foms of soo whh a fy dsbd. Th so ods by sag ams of ah soo fom. osd h ga h sa Maov oss d bow, β γ Fg - Ga h sa aso a dagam havg aso a ma β β Q γ γ 5

46 W aso do ah y of Q by,. Aodg o h hoy of Maov osss,, h aso a of movg fom sa o sa fo,,,,. Th ma Q w aso b w as Q,,,,,,,,, Movg fom sa zo o sa o, fo am os a a a,, ad movg fom sa zo o sa wo os a a β,. S o oh asos o of sa zo o h a of sayg sa zo s h gav sm of h dag as, [], ha s, β,. Th ows of Q add o zo. Dagoa s a oosv. Off dagoa ms a o-gav. W a sd fdg h aso obaby fos,, wh,,,. s fod by sovg h,, Komogoov fowad o bawads aos, wh I, wh Q Q,,,,,,,,, s h ma of aso obaby fos ad Q s as abov. Th soo of h Komogoov bawads ao a b w [] as 6

47 Q ad h sady sa dsbo s gv by h foowg dsbo. Lmma -. Th sady sa dsbo of h ga h sa Maov oss d Fg s γ γ γ β β β wh β γ oof Th saoay ad sady sa dsbo may b fod s [] o [] by sovg o Q [ ] 7

48 [ ] [ γ γ β β ] Ths gvs h foowg sysm of aos, γ β γ β whh may b sovd fo,, ovdd w o b a dsbo, ha s,. ah ha odg a h agba das o sov hs sysm, s h sady sa dsbo s s ogh o vfy ha [ ],, Q wh γ β β β γ β β γ γ β γ γ o β β β γ γ γ 8

49 wh γ β Ths sagh fowad h s f o h ad ad oms h oof. S h soos o Q Q, a ow, [], o hav h fom b a wh a ds gvas of Q wh osa offs ad soo fom wh, fo am, s a dob oo, w ow o ao o fdg h gvas of Q. To dm h gvas of Q, a ha h haas ao of Q s,, b a [ ] d d I Q γ γ β β No ha vo s a gvo of Q s. Ths foows s ah of h ows of Q add o zo. Ths s a gva of Q. Ths mas h haas ao of Q may b w as Q wh 9

50 β γ β γ Ths s h sam ha aas Lmma -.. If, a h oos of, h ad, ± a h gvas of Q. Th a h ds ass o osd:... > < as > S > foows ha, ad s >. I aa,,, a h o-osv gvas of Q, whh a ds f zo s o a m oo ha s, f >. If zo s a m oo h h Maov oss ffvy boms a oss havg asos bw wo o fw of s h sas. Thfo, w hfoh assm,, a h o-osv, ds gvas of Q. Th, by [] fo am, a b. wh s h sady sa dsbo fo sa gv Lmma -. o ha as, ad a ad b a osas whh w ow dm. 5

51 Sbsg o ao. ods, δ a b o δ a b wh δ fo fo s h Ko da fo Fo ay sf,,,, a b o a b Th a b ad δ a b a b sd o dm a ad b : δ b b b δ a δ a δ I smmay, a b boms, fo as, > δ δ 5

52 wh β β β γ γ γ γ β γ β, ± wh h δ ad a a ow. as, dob oo I hs as, ± ± a dob oos of h haas ao, wh < ad γ β. Th as obaby fos fo dob oos a s [], d. 5

53 wh s aga h sady sa dsbo fo sa gv Lmma -.., ad d a o aga osas whh may ow b dmd. Sbsg o. gvs, δ δ wh δ fo fo Dffag. ad vaag a gvs [ ] d d Usg δ w fd o b d d δ Ths h as obaby fos, d fo,,, bom, fo as, [ δ ] δ wh 5

54 β β β γ γ γ γ β γ β wh δ ad a a ow. as <,, ± ± wh γ β γ β L ad b a b a wh 5

55 a b, f a b f ab a b f a b θ Usg E s ao, osθ sθ whh may b w as a os b s b f os b s b a Sg ao. gvs a a g osb h s b. δ g g δ S s a ad δ s a, ha mas g a. Dffag. wh s o ad sg, a g h b 55

56 w ow s a, g s a, b s a, ad s h magay so ha mas h magay, s ohws w wod g a om mb fo h aso a whh a ha. So, h b a g h b a g h h a g b a δ b Ths, g δ A δ a osb h a a a sb a osb a osb δ δ b b a a sb sb δ a osb a δ b a sb δ os δ s 56

57 Ths h as obaby fo, fo as, < s [ ] s os δ δ wh β β β γ γ γ γ β γ β wh δ ad a a ow. I smmay, h as obaby fos fo h ga h sa Maov oss wh aso a ma Q β β γ γ a gv by 57

58 as Fo > δ δ as Fo [ ] δ δ as Fo < [ ] s os δ δ wh β β β γ γ γ γ β γ β, ± wh δ ad a a ow. 58

59 Eams Eam as > β γ Fg, Q ad [ ] So h oos a, 8±,, 5 Th sady sa dsbo s 59

60 Ad 5 5 δ δ δ δ Fo ad, δ ad 5 6 Fo ad, δ ad Fo ad, δ ad

61 Eam as β γ Fg 5 Q Th oos of h haas ao a, ± So, - s a dob. Th sady sa dsbo s,, ad 6

62 wh ad ad [ δ ] δ [ δ ] [ δ ] Fo ad, δ ad Fo ad, δ ad Fo ad, δ ad 6

63 Eam as < γ Fg 6, Q Th oos of h haas ao a, ± ± Th sady sa dsbo s wh ad 6

64 [ ] s os s os δ δ δ δ Fo ad, δ ad os Fo ad, δ ad s os Fo ad, δ ad s os 6

65 oso I hs hss, obabs a dmd fo h gamb s wh aasohs ad wdfas. Fo h f m obm, a ah ombaos dsbd ha ay a y o dmg h obabs by ovdg s a foma fo og sam ahs of h Maov ha. I h f m as ha, h obaby aos hav b sovd sg obaby gag fos ad h hoy of dff aos. Ths obms gaz ad d h w-ow, assa gamb s obm as a sa as of h soos dvod wh hs hss. A may a sah das ha h soos dmd hs hss aa w. owv, h gamb s obm das ba so may yas ov h s ha a mo sv a sah s ds o b da. A h sam m, fh gazaos of h gamb s obms hav b ogssg a a oy omoa, s, fo am [5] wh obabs havg sa dd obabs a bg sdd. Th gamb s may aso b dd aog h s of a bah g sysm, ha s, havg a aso dagam wh a ay movs dow by wo ss sad of o s. I ha, h as obaby fos of h ga h sa Maov oss s dmd. O mhod of soo vovs obag ga, foma fo h gvas of Q ad h sady sa dsbo of h h sa Maov oss. Af hs, a agozao of h ds soo fo foms s gv ha. 65

66 Ev hogh hs aoah o dmg as obaby fos s w ow, so fa, w hav o s h soo of h h sa Maov oss ad o sh om gay. O ga soo ass som sg sos ha a o y omy dsood. Fo am, why dos h ay β γ aa boh h sady sa dsbo of h sysm ad h gvas of Q? Wha s h sgfa of? Th ho s ha o ga fom of soo vas w sghs o h as of how as obaby fos oo ga. Th s of f sah mgh b o fom a sma aayss fo h fo sa Maov oss. Th gvas of Q o aga may b dmd y ms of h s of Q -hs m by sg h foma fo h oos of a b ao. Th sady sa dsbo s aga sovab b o obvos. S h fo sa Maov oss oas h h ad wo sa Maov oss as sa ass, aga o may ho o a mo abo how h ga N-sa Maov oss as obaby oos. I sho, h ag of f wo o hs o s dss. 66

67 fs [] Edwads, A.W.F, asa s obm: Th Gamb s, Iaoa Sasa vw, 7-79, 98. [] Godbg, S., Iodo o Dff Eaos, wh Isav Eams fom Eooms, syhoogy, ad Sooogy, Dov baos, I., Nw Yo, 986. [] Goss, D ad as,.m., Fdamas of Qg Thoy, Sod Edo, Joh Wy ad Sos, Nw Yo, 985. [] o,., o, S., ad So., Iodo o Sohas osss, Wavad ss I., 987. [5],., K, A., Ngy,., Sws, J., vo m,., Aoahs o Gamb s wh aasoh,. [6] Kasfy,., Da osss o Dm Tas obaby Fos, Mass Thss, a oy omoa,. [7] K, A., bo, G., Mas, D., Swf,., Kasfy,., Lam,., Da osss o Sov Sg Sv Sysms, Joa of Sasa ag ad If, 5, Sa Iss o La ah ombaos ad Ds Dsbos, -7, 5. 67

68 [8] Mas, D., ombaos: A obm Od Aoah, Th Mahmaa Assoao of Ama, Washgo, 998. [9] Mohay, S. G., La ah og ad Aaos, Aadm ss, 979. [] Naayaa, T.V., La ah ombaos Wh Sasa Aaos, Uvsy of Tooo ss, Tooo, 979. [] Taas, L., O Th assa obms, JASA, 6, , 969. [] Wf,., Gagfooogy, Aadm ss, I., 99. [] Z, D., A Fs os Dffa Eaos wh Modg Aaos, 6 h Edo, oos/o bshg,