On Markov chain-gambler's ruin problem with ties allowed

Transcription

1 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 On Markov chan-gabler's run proble wth tes allowed M. A. El-Shehawey * M. E. El-Tantawey ** and Gh. A. Al-Shreef *** Abstract. A Markov chan- gabler's run proble n the case that the probabltes of wnnng/ losng a partcular gae depend on the aount of the current fortune wth tes allowed s consdered. Closed-for forulas for the soluton of the MC-GR proble and the expected duraton of the gae n the presence of partally absorbng- partally segregatng- partally reflectng boundares are deduced and soe very sple closed fors whch edately lead to exact and explct forulas for soe specal cases are gven. Key words: Gabler's run; Markov chan-rando walk; probablty of run partally; the expected duraton of the gae.. Introducton Markov chan-gabler's run (MC-GR) proble s typcally represented by Markov chan-rando walk (MC-RW) on the set of ntegers. It has been long consdered. It can be used n varous dscplnes: n statstcs to analyze sequental test procedures n coputer scence to estate the sze of the World Wde Web usng randozed algorths n ecology to descrbe ndvdual anal oveents and populaton dynacs n physcs as a splfed odel of Brownan oton n econocs to odel share prces and ther dervatves n edcne and bology absorbng barrers gve a natural odel for a wde varety of phenoena (cf. Yaaoto (22) and Ue (23)). There are nuerous applcatons of MC-GR proble and has attracted soe attenton n the seventes n the context of cancer growth odelng (cf. Bell (976) and Beyer and Wateran (977)). In partcular there was sgnfcant nterest n estatng the expected te for a cancerous clone of cells to reach tuor sze fro a sngle way-ward cell rather than dyng off before reachng tuor sze. The analyss can help to deterne the appearance of the frst wayward cancerous cell and perhaps to dentfy reasons for the cancerous growth (cf. El-Shehawey (994) and references theren). We now gve a bref hstorcal revew of the use of barrers n a MC-RW. Weesakul (96) dscussed MC-RW restrcted between a reflectng and an absorbng barrer. Usng generatng functons he obtans explct expressons for the probablty of absorpton. Lehner (963) studes MC-RW wth a partally reflectng barrer usng cobnatoral ethods. Dua Khadlkar and Sen (976) fnd the bvarate generatng functons of the probabltes of a partcle reachng a certan state under dfferent condtons. Percus (985) consders an asyetrc MC-RW wth one or two boundares. Usng generatng functons the Departent of Matheatcs Daetta Unversty Faculty of Scence P. O. Box 6 ew Daetta Egypt. E-als: (*) el_shehawy@du.edu.eg (**) tantaw@yahoo.co (***) gh_alshreef@du.edu.eg

2 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 probablty dstrbuton of beng at poston after n steps s obtaned. El-Shehawey (2) obtans absorpton probabltes at the boundares for a MC-RW between one or two partally absorbng boundares as well as the condtonal ean for the nuber of steps before stoppng gven the absorpton at a specfed barrer usng condtonal probabltes. The MC-GR proble was essentally forulated as follows: Two players play a seres of sngle gaes n whch one unt changes hands untl one player goes bankrupt. Suppose that a one player (refer as gabler) starts wth an ntal captal of unts aganst the other (refer as the opponent) has unts. In each sngle gae the gabler ether wns one unt wth probablty p fro hs adversary or loses t wth probablty q pto hs adversary. The gaes go on for a whle untl ether the gabler loses all hs unts or wns all hs opponent's unts. The actual captal possessed by the gabler s thus represented by a MC-RW that s X... n n whose state space s S... wth absorbng barrers at and and transton probabltes p p p for Absorpton beng nterpreted as the run of one or the other of the gablers.. Stateent of the general MC-GR Proble We consder a generalzaton of the MC-GR proble by ntroducng the followng assuptons: - The ntal captal of the gabler s unts. 2- The ultate fortune whch the gabler wants to accuulate s unts. 3- We allow tes n the sngle gaes (the possblty of stayng wth the sae fortune when no unt s changed). 4- The probabltes of wnnng and losng depend on the current fortune. 5- The probabltes of wnnng losng and nether wnnng nor losng one unt respectvely are p q and r p q nstead of p and q pas above. 6- If the gabler fortune drops down to unt; at the next gae the gabler goes broke (runed) wth probablty wn the rght to spn agan wth probablty or starts agan wth one unt captal wth probablty. 7- At the boundary the gabler reaches hs obectve; at the next gae hs adversary ay go broke wth probablty wn the rght to spn agan wth probablty or start agan wth captal of unts wth probablty. One ay thnk of ths generalzaton as a general MC-RW wth transton probabltes p p p q and p ( p q ) 2... ; wth the followng odfcatons at the boundares. When the process reaches the boundary state b ( b or ) t s absorbed wth probabltyb b reflects (to the state b or b respectvely) wth probablty b or reans at the boundary state b wth probablty b. 2

3 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 Physcally ths corresponds to the stuaton upon reachng the boundary state b ( b or ) the partcle s ether reans at the boundary state b wth probablty b or lost fro the syste wth probablty b b or turned back to the state b or b of the syste wth probablty b and reduces to the classcal MC-RW proble wth tes allowed. We also generalze the case of spatal hoogenety by assung that q p q p 2 2 and p q r 2 for S \ (.) s non-negatve constant. otce that the spatal hoogenety s the partcular case when. We wll concern wth gvng closed for forulas for the solutons of the general MC-GR proble.e. we copute the probablty of run proble and the expected duraton wth "wnnng partally". Ths proble s a generalzaton to prevous works (cf. Feller (968) p. 345 Srnvasan and Mehata (978) p. 9 Percus (985) El- Shehawey (2) and El-Shehawey and El-Shreef (29)) t s apparently not covered by the lterature. 2. Closed for soluton of the general MC-GR proble Let P ; p q ; P P p q P " ; ; " S be the probablty of run partally "wnnng partally" for a gabler startng wth unts S and playng aganst an adversary wth ntal captal unts. The probabltes of wnnng losng and nether wnnng nor losng one unt respectvely are p q and r p q n the presence of partally segregatng- partally reflectng- partally absorbng barrer wth probabltes and at ; and at respectvely. The probablty P " P " satsfes the followng dfference equaton \ P p P r P q P S (2.) wth the boundary condtons and P subect to P (2.2) P P (2.3) \ P p P r P q P S (2.4) P P P P. (2.5). (2.6) h h h h 3

4 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 otce that contrary to the classcal MC-GR proble here allow tes n the sngle gaes (the possblty of stayng wth the sae fortune when no unt s changed) the probabltes of wnnng losng and nether wnnng nor losng depend on the current fortune also at the boundares and t s possble to start to stay to leave or to reach. 2.. Man result. The an purpose of ths contrbuton s to fnd closed for forula for the soluton for the general MC-GR proble. Solvng the dfference equaton (2.) wth (2.2) and (2.3) (or (2.4) wth (2.5) and (2.6)) systeatcally the result s forulated n the next theore: Theore 2.. The probablty P ; p q ; P P The copleentary probablty of run partally s P P ; p q ; for h 4 takes the for S (2.7) S (2.8) the su n the nuerator s to be nterpreted as and q p. otce that the copleentary probablty P h h h ay be obtaned fro (2.7) by nterchangng p and q respectvely and replacng by S. Clearly P P ; p q ; P. 3. Soe specal consequences Many nterestng specal cases can be derved fro theore 2. through an approprate choce of the ntal fortune/ wnnng and/or losng probabltes/ boundary probabltes soe cases are lsted below 3.. MC-GR wth varous ntal fortune In ths subsecton the probabltes of wnnng losng and nether wnnng nor losng one unt respectvely are p q and r p q and at the boundares and there are partally segregatng-partally reflectng-partally absorbng barrers wth respectve probabltes and at ; and at.

5 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 () If the gabler starts fro the orgn (.e. wthout ntal captal) the probablty of wnnng partally the gae s gven by P P ; p q ; and q p. (3.) () If the gabler starts fro the boundary (.e. wth ntal captal s aganst an opponent wthout ntal captal) then the probablty of run partally s P P ; p q ; and q p. () If the gabler and hs opponent's start at the ddle (.e. then the probablty of run partally s P s P s; p q ; s s 2 2 s 2s 2s 2s 2s 2s and q p. (3.2) s and 2s ) (3.3) We note that the case wth s and 2s corresponds to the popular wnnng-aseres-by- s -gaes schee n sports (often cobned wth a requreent on the nu nuber of gaes to be won) (cf. Sauels (975) Segrst (989) and Lengyel (29)) MC-GR wth varous boundary probabltes In ths subsecton the ntal captal of the gabler s unts and the probabltes of wnnng losng and nether wnnng nor losng one unt respectvely are p q and r p q. 5

6 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 () For the case and the two barrers at and are partally absorbng- partally reflectng that s when the gabler s reduced to penury at the next gae the gabler goes broke or start agan wth one unt captal wth respectve probabltes and. When the gabler reaches hs obectve at the next gae hs adversary ay go broke or start agan wth a captal of unts wth respectve probabltes and. Thus the probablty of run partally s gven by P P ; p q ; S and q p. (3.4) The copleentary probablty of run partally s P P ; p q ; P S (3.5) P s gven by (3.4). a) If and the barrers at s perfectly absorbng and at s partally absorbng- partally reflectng that s when the gabler s reduced to penury he ay be runed and the gae ends. Hence when the gabler reaches hs obectve at the next gae hs adversary ay go broke or start agan wth a captal of unts wth respectve probabltes and. Thus the probablty of run partally s gven by P P ; p q ; S and q p. (3.6) The copleentary probablty of run partally s P P ; p q ; P S (3.7) P s gven by (3.6). If the gabler starts fro the boundary or then 6

7 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 ; ; P P p q and q p. (3.8) b) If and the barrers at s perfectly absorbng and at s partally absorbng- partally reflectng that s when the gabler reaches hs obectve hs adversary ay be runed and the gae ends. When the gabler s reduced to penury at the next gae the gabler goes broke or start agan wth one unt captal wth respectve probabltes and. Thus the probablty of run partally s gven by P P ; p q ; S and q p. (3.9) The copleentary probablty of run partally s P P ; p q ; P S (3.) P s gven by (3.9). If the gabler starts fro the boundary or then ; ; P P p q and q p. (3.) c) If and the syetrc boundares at and are partally-absorbng partally-reflectng wth respectve probabltes and. Then the probablty of run partally s ; ; ; ; P P p q P p q S (3.2) 7

8 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 and q p. The copleentary probablty of run partally s P P ; p q ; P S (3.3) P s gven by (3.2). If the gabler starts fro the boundary or then P P ; p q ; and q p. (3.4) q We see that wth the approprate change of notaton n the specal case p..e. tes are not allowed n the sngle gaes (stayng wth the sae fortune when no unt s changed possble) forulas (3.4) and (3.5) agree wth that of El-Shehawey (2). Forulas (3.6) and (3.7) wth (also (3.9) (3.) wth and (3.2) (3.3) wth ) agree wth the well known results for a MC-GR proble between two perfectly absorbng barrers (see Parzen (962) p. 233 and El-Shehawey (2)). ) If and the boundary probabltes at the two boundares and are syetry.e. there are partally segregatngpartally reflectng-partally absorbng barrers at and wth respectve probabltes and. Thus the probablty of run partally s gven by ; ; ; ; P P p q P p q S and q p. (3.5) The copleentary probablty of run partally s P P ; p q ; P S (3.6) P s gven by (3.5). a) If and the syetrc boundares at and are partally-absorbng partally- segregatng wth respectve probabltes 8

9 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 and.e. If the gabler fortune drops down to unt (reaches hs obectve to ) at the next gae the gabler goes broke (adversary ay go broke) wth probablty wn the rght to spn agan wth probablty. Then the probablty of run partally s P P ; p q ; P ; p q ; S. (3.7) The copleentary probablty of run partally s ; ; S P P p q and q p. 9 (3.8) otce that forulas (3.7) and (3.8) are - ndependent on the segregatng probablty 2- well known n the specal case (cf. Parzen (962) p.233 and El- Shehawey (2)). b) If and the probabltes at the two boundares and are syetry.e. there s a partally segregatng-partally reflectng barrer at both and wth respectve probabltes and. Thus the probablty of run partally s gven by P P ; p q ; P ; p q ; S (3.9) The copleentary probablty of run partally s P P ; p q ; S (3.2) and q p. otce that forulas (3.9) and (3.2) are - ndependent on both the segregatng probablty and the ntal captal of the gabler S 2- also obtaned n the presence of two perfectly - reflectng barrers at both and (.e. ) - segregatng barrer at both and (.e. ). () If the gabler starts fro the ntal captal S the probablty of wnnng partally the gae n the presence of partally segregatng-partally absorbng barrer at wth respectve probabltes and and partally segregatng- partally reflectng- partally absorbng barrer at wth respectve probabltes and s

10 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 ; ; P P p q and q p. S (3.2) The copleentary probablty of run partally s P P ; p q ; P S (3.22) P s gven by (3.2). otce that forulas (3.2) and (3.22) are - ndependent on the segregatng probablty. 2- also gven n the presence of - totally absorbng barrer at and partally-absorbng partally-segregatng partally-reflectng at ( and ). -totally segregatng at and partally-absorbng partally-segregatng partally-reflectng at ( and ). (v) If the ntal captal of the gabler s S the probablty of wnnng partally the gae n the presence of partally segregatng-partally absorbng barrer at wth respectve probabltes and and partally segregatng- partally reflectng- partally absorbng barrer at wth respectve probabltes and s gven by ; ; P P p q and q p. S (3.23) The copleentary probablty of run partally s P P ; p q ; P S (3.24) P s gven by (3.23). otce that forulas (3.23) and (3.24) are - ndependent on the segregatng probablty. 2- also gven n the presence of - totally absorbng barrer at and partally-absorbng partally-segregatng partally-reflectng at ( and ). - totally segregatng at and partally-absorbng partally-segregatng

11 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 partally-reflectng at ( and ). a) If and the boundares at and are partallyabsorbng partally- segregatng wth respectve probabltes and at and at.e. If the gabler fortune drops down to unt (reaches hs obectve to ) at the next gae the gabler goes broke (adversary ay go broke) wth probablty at at wn the rght to spn agan wth probablty at at. Then the probablty of run partally s ; ; P P p q S. (3.25) The copleentary probablty of run partally s ; ; S P P p q and q p. (3.26) otce that forulas (3.25) and (3.26) (see also (3.7) and (3.8)) are - ndependent of the segregatng probabltes and ( ) 2- well known n the specal case (see Parzen (962) p.233 and El-Shehawey (2)) MC-GR wth varous wnnng and losng probabltes In ths secton the ntal captal of the gabler s S and at the boundares there are two partally segregatng-partally reflectng-partally absorbng barrers wth probabltes and at and and at respectvely. () If the probabltes of wnnng losng and nether wnnng nor losng one unt respectvely are q p r p the and probablty of run partally s gven by P P ; p p ; The copleentary probablty of run partally s P P ; p p ; S (3.27)

12 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 S (3.28) a) For the syetrc gae ( S ) the probabltes of wnnng losng and nether wnnng nor losng one unt respectvely are p p and r 2p the probablty of run partally s gven by ; ; P P p p S (3.29) The copleentary probablty of run partally s P P ; p p ; P S (3.3) P s gven by (3.29). otce that - forulas (3.28) and (3.3) can be obtaned fro (3.27) and (3.29) respectvely by nterchangng and and replacng by. 2- forulas (3.27) and (3.28) are dependent of S and ndependent of the probablty of wnnng p. 3- forulas (3.29) and (3.3) are ndependent of the probablty of wnnng (or losng) p q r. 4- forulas (3.29) and (3.3) are also gven for the followng syetrc two cases - p q 2(.e. r ) and - p q r 3. () In the case the transton probabltes are gven as (.) generalzes the spatal hoogenety q q p p r r S \ and p q r. The probablty of run partally s gven by p q P P ; ; 2 2 q p q q q p p p q p q f p q p f p q (3.3) 2

13 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 The copleentary probablty of run partally s p q P P ; ; P 2 2 P s gven by (3.3). The case p q p. S (3.32) q edately follows fro the case p q after takng the lt as a) The one-boundary case wth the assupton that there s one partally segregatng-partally reflectng-partally absorbng barrers wth probabltes and at and the other at can be obtaned fro (3.3) and (3.32) by takng the lt as : p q P P ; ; 2 2 q f q p q p p (3.33) f q p and ; p q P P ; 2 2 q f q p q p p (3.34) f q p otce that the solutons of MC-GR proble n the case gven n (.) s the sae as for the spatal hoogenety " p p q q r r p q ". Therefore the constant has no effect at all on the probabltes U ( x ) as could have been expected. It s straghtforward to show that the present results are generalzed soe well known results see for exaple wth the approprate change of notaton forulas (3.33) and (3.34) n the specal cases agree wth that of El-Shehawey (22); and agree wth that of Kac (954 pp 295) n the specal cases and for q p( see also Percus (985)); expressons (3.3) and (3.32) n the specal cases and r agree wth that of El-Shehawey ((2) (28) a (28) b ) (see also Percus (985) n the cases and ) and n the specal case and agree wth those of El- Shehawey and El-Shreef (29) and n the specal cases and p q 2 agree wth those of Lefebvre (28) (see also Feller (968 pp ) Cox and Mller (965) Beyer and Wateran (977) Vannucc (99) El- Shehawey (994) Palacos (999) and Lengyel (29) aong others). 3

14 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec The Expected Duraton of the gae D p q ; ; D be the expected duraton of the gae untl the Let gae ends. By condtonng on the outcoe of the frst bet we have D p D r D q D S \ (4.) and the quttng polcy ples the followng boundary condtons D D (4.2) D D (4.3) Theore 4.. The expected nuber of bets D untl the gabler ether reaches a fortune of dollars or goes broke s gven by D D U V (4.4) s s ps 2 q k D U V s k pk s ps q k V U p k p k and U q q V s k k s k pk s2 k s pk (4.6) Corollary 4.. In the hoogeneous case p p q q r and partally absorbng partally reflectng syetrc boundares and the expected duraton of the gae s gven by (4.5) x x x 2 2q D (4.7) p q x x x q x. p Ths result agrees wth Percus (985). 5. Concluson. In ths paper we have consdered the so-called Markov chan- gabler's run proble wth varable wn/loss probabltes and tes allowed n the present of two partally absorbng- partally segregatng- partally reflectng boundares that generalzes the classcal MC-GR proble. The probabltes of wnnng losng and nether wnnng nor losng depend on the aount of the current fortune fro probablty boundary condtons vewpont are takng nto account. We derved exact and explct expressons for the solutons of the ntroduced general MC-GR proble and the expected duraton of the gae these forulas enable us to fnd any nterestng partcular cases ( soe results are known and the others are "new" apparently not covered by the lterature) through as approprate choce of the boundary probabltes or the transton probabltes. 4

15 The 48rd Annual Conference on Statstcs Coputer Scences and Operatons Research Dec 23 References Bell G.I Models of carcnogeness as an escape fro totc nhbtors Scence Beyer W. A. and Wateran M. S Syetres for condtonal run proble. Math. Mag Cox D. R. and Mller H. D The Theory of Stochastc Processes. Methuen London. Dua S Khadlkar S. and Sen K A odfed rando walk n the presence of partally reflectng barrers J. Appl. Prob El-Shehawey M. A.994. The frequency count for a rando walk wth absorbng boundares: a carcnogeness exaple. I J. Phys. A: Math. Gen. Vol El-Shehawey M. A. 2. Absorpton probabltes for a rando walk between two partally absorbng boundares: I. J Phys. A:Math. Gen El-Shehawey M. A. 22. A se-nfnte rando walk assocated wth the gae of roulette J.Phys.A: Math. Gen. Vol El-Shehawey M. A. 28a. On nverses of trdagonal atrces arsng fro Markov chanrando walk I. Sa J. Matrx Anal. Appl El-Shehawey M. A. 28b. Trdagonal rando walk wth one or two perfect absorbng barrers. Math. Slovaca El-Shehawey M. A. and Al-Shreef Gh. A. 29. On Markov chan roulette type gae. J Phys. A: Math. Theor (pp). Feller W An Introducton to Probablty Theory and Its Applcatons 3rd. vol. I. Wley ew York. Kac M Rando walk and the theory of Brownan oton. In Selected Papers on ose and Stochastc Processes (. Wax ed.) Dover ew York. Lefebvre M. 28. The gabler's run proble for a Markov chan related to the Bessel process. Statstcs and Probablty Letters Lehner G One-densonal rando walk wth a partally reflectng barrer Ann. Math. Stat Lengyel T. 29. The condtonal gabler's run proble wth tes allowed. Appled Matheatcs Letters Palacos J.L The run proble va electrc networks. Aercan. Stat. Assoc Parzen E Stochastc Processes. Holden- Day Inc. London. Percus O. E Phase transton n one- densonal rando walk wth partally reflectng boundares. Adv. Appl. Prob Sauels S.M The classcal run proble wth equal ntal fortunes Math. Mag. 48 (5) Segrst K n-pont wn-by-k gaes J. Appl. Probab Srnvasan S. K. and Mehata K. M Stochastc Processes Mc Graw Hll ew Delh. Ue van T. 23. Maxu and nu of odfed gabler's run proble. arxv: Vannucc L. 99. Gabler's run proble wth Markovan turns. Rv. Mat. Sc. Econo. Socal. 3 o Weesakul B. 96. The rando walk between a reflectng and an absorbng barrer Ann. Math. Statst Yaaoto K. 22. Exact soluton to a gabler's run proble wth a nonzero haltng probablty.. arxv: