Dynamic Programming for Estimating Acceptance Probability of Credit Card Products

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1 Joual of Copute ad Couicatios, 07, 5, ISSN Olie: 7-57 ISSN Pit: 7-59 Dyaic Pogaig fo Estiatig Acceptace Pobability of Cedit Cad Poducts Lai Soo Lee,, Ya Mei Tee, Hsi Vo Seow Depatet of Matheatics, Faculty of Sciece, Uivesiti Puta Malaysia, Sedag, Malaysia Laboatoy of Coputatioal Statistics ad Opeatios Reseach, Istitute fo Matheatical Reseach, Uivesiti Puta Malaysia, Sedag, Malaysia Nottigha Uivesity Busiess School, The Uivesity of Nottigha Malaysia Capus, Seeyih, Malaysia How to cite this pape: Lee, LS, Tee, YM ad Seow, HV (07) Dyaic Pogaig fo Estiatig Acceptace Pobability of Cedit Cad Poducts Joual of Copute ad Couicatios, 5, Received: Decebe, 06 Accepted: Decebe 6, 07 Published: Decebe 9, 07 Copyight 07 by authos ad Scietific Reseach Publishig Ic This wok is licesed ude the Ceative Coos Attibutio Iteatioal Licese (CC BY 0) Ope Access Abstact Baks have ay vaiats of a poduct which they ca offe to thei custoes Fo exaple, a cedit cad ca have diffeet iteest ates So deteiig which vaiats of a poduct to offe to the ew custoes ad havig soe idicatio o acceptace pobability will aid with the pofit optiisatio fo the baks I this pape, the authos look at a odel fo axiisatio of the pofit lookig at the past ifoatio via ipleetatio of the dyaic pogaig odel with eleets of Bayesia updatig Nueical esults ae peseted of ultiple vaiats of a cedit cad poduct with the odel povidig the best offe fo the axiu pofit ad acceptace pobability The poduct chose is a cedit cad with diffeet iteest ates Keywods Cedit Cad, Cedit Scoig, Dyaic Pogaig, Pofitability Itoductio Taditioally, cedit cad issues chaged a fixed iteest ate o thei cedit cads fo all thei custoes Accodig to [], sice 99 howeve, soe cedit cad fis have switched to vaiable iteest ate as a esult to the cedit cad ledig aket becoig oe copetitive [] As epoted by [], pofitability of cedit cad ledes cosequetly suffeed a substatial loss due to this copetitio Hece it is becoig iceasig ipotat to be able to secue the acceptace DOI: 06/jcc Dec 9, Joual of Copute ad Couicatios

2 L S Lee et al of a offe i ode to have pofit So, the ledes have to be able to pesuade the custoe to accept thei offe Whe a good custoe is willig to accept a offe, he o she will geeate pofit to a ogaisatio Oe way of doig so is to custoise the offe to the custoe The ledes could use ifoatio about the custoe s pefeeces so as a guide to ake a decisio o what type of offe the custoe ay be iteested i This ifoatio is aleady available fo iitial collectio fo cedit scoig puposes By lookig at which type of poduct accepted by diffeet custoes, the ledes ca lea about the pefeeces of thei custoes Hece, the decisio o what offe to ake ca be odeled Thee ae a ube of eseaches who have eseached acceptace pobability fo fiacial poducts to axiise pofitability; fo exaple [] [] [5] I this pape, the authos exteded a acceptace odel based o the wok doe by [] The lede s decisio poble has bee odeled as a Makov Decisio Pocess ude ucetaity The objective of this odel is the axiisatio of pofit usig a dyaic pogaig [6] odel with Bayesia updatig to icopoate the usage of past custoe ifoatio to optiise acceptace pobability The poble is discussed i the ext sectio The the optial solutios fo vaiats of poducts ae descibed Fially, the ueical esults ae tabled ad the coclusios ae daw i the last sectio The Poble Baks have ay vaiats of a pesoal fiacial poduct which they ca offe to thei custoes The attactiveess of the vaiats to the custoe ca be odeed i such a way that the likelihood of acceptig that vaiat by the custoe is ootoically deceasig while the lede s pofitability of the vaiat is ootoically iceasig Fo exaple, a cedit cad with diffeet iteest ates likes 5%, 0% ad so o The decisio o which offe to ake to the ext applicat is based o the give kowledge of the pevious offes ad whethe the offe accepted by pevious custoes The objective of odelig the acceptace pobability is to axiise the pofit to the bak I the odel hee, the authos follow the exaple of [] ad odel the poble as a cedit cad poduct with diffeet vaiat of iteest ates It is assued that the custoes ae fo hoogeous populatio ad the pobability of ay custoe choosig vaiat t is p t whee t =,,, ad p p p The pofit to the lede of t vaiat chose by custoes is P t whee P P P ad Pt 0 Thus, the bak s axiu pofit is deteied by ax { pp t t} Howeve, oe does ot kow the pobability of p t, so we defied p t as with the coditio of p p p Hee assue Offe is a cedit cad with 5% iteest ate aually, Offe is a cedit cad with 0% iteest ate aually, Offe is a cedit cad with 5% iteest ate aually ad Offe is a cedit cad with 0% iteest ate aually Also, assue the ube of potetial custoes has a geoetic distibutio with paaete β with the last custoe is β DOI: 06/jcc Joual of Copute ad Couicatios

3 L S Lee et al At the begiig, two vaiats of the cedit cads ae cosideed i the odel Thee ae a few assuptios ade i this odel Fist, assue that if a custoe ejected vaiat t, eaig that he/she would also eject all wose vaiats u, whee u > t Siilaly, if a custoe accepted vaiat t, he/she would accepted all bette vaiats v, whee v< t With so u > t > v To illustate, let u = 0% iteest ate o a cedit cad, t = 0% ad v = 5% If oe ejects a offe of 0% iteest ate (t), the oe is also likely to eject a cedit cad of 5% iteest ate (u) Ad if oe accepts the offe of 0% iteest ate, oe is likely to accept a bette offe of 5% (v) iteest ate We esue this by defiig a set of coditioal pobabilities whee p is the pobability of acceptig Offe ad q is the Beoulli ado vaiable q = p; q = Pobability (custoe would accept Offe /custoe would accept Offe ) Sice q = p, hece p = qp = q( q) This coditio esues that p p Fo thee vaiats of iteest ates fo the cedit cad, the coditioal pobability is as follows: Sice p= qq, hece p= qp = q( qq ) ad this esues that p p p Ad so fo the fou vaiats of iteest ates fo the cedit cad, the coditioal pobability is as follows: Sice p= qqq, hece p= qp= q( qqq ) ad this esues that p p p p Fo ay vaiats of iteest ates fo the cedit cad, the coditioal pobability is defied as: u= u p = q p = q, whee p = Pobability of acceptig offe, q = Pobability (custoe would accept Offe /custoe would accept Offe ) This coditio esues that p p p p Give that q t ae all Beoulli ado vaiables so i a Bayesia settig, oe could descibe the bak s kowledge of the ifoatio as a Beta distibutio The pio fo q t is by B( t, t t) whose pobability desity fuctio is t t t qt ( qt) ad expectatio is t whee t = the ube of custoes t that have accepted the offe t ad t = the ube of custoes who wee exteded offe t At ay poit, the bak s belief about the acceptace pobabilities p p p p is give by the paaetes (,,,,,, ) Let the expected axiu total futue pofit to the bak as V (,,,,,, ),,,,,, give that the cuet belief is DOI: 06/jcc Joual of Copute ad Couicatios

4 L S Lee et al, ae the paaetes of the Beta distibutio descibig oe s belief of p So if Offe is accepted, the paaetes will get updated to +, + If it is ejected, they get updated to, + Thus, oe could eitepet these as: = ube of custoe who aleady accepted Offe (Offe 5% i this odel); ad = ube of custoe who have bee offeed Offe (Offe 5%) Hece, ae the paaetes of the Beta distibutio descibig oe s belief of p Note the assuptio that the offe of Offe will have to be accepted fist befoe Offe ca be cosideed If Offe is accepted, the paaetes get updated to +, + So whe it is ejected, they get updated to, + Thus, = ube of custoe who aleady accepted Offe (Offe 0% i this odel); ad = ube of custoe who have bee offeed Offe (Offe 0%) Note that, ae the paaetes of the Beta distibutio descibig oe s belief of p If Offe is accepted, the paaetes get updated to +, + Whe Offe is ejected, ad the custoe is assued to would have accepted Offe but could eject Offe ; o accepted Offe ad Offe Hece they get updated to, + ad the, ad, is updated depedig o the coditios of Offe ad Offe Thus, = ube of custoe who aleady accepted Offe (Offe 5% i this odel); ad = ube of custoe who have bee offeed Offe (Offe 5%), ae the paaetes of the Beta distibutio descibig oe s belief of p If Offe is accepted, the paaetes get updated to +, + If it is ejected, the thee ae thee possibilities: ) The custoe would have accepted Offe but ejected Offe ad Offe ; ) The custoe would have accepted Offe ad Offe but ejected Offe ; ) The custoe would have accepted Offe, Offe ad Offe Thus, = ube of custoe who aleady accepted Offe (Offe 0% i this odel); ad = ube of custoe who have bee offeed Offe (Offe 0%) I the above fou cases, t t fo t =,,, By icludig the ifoatio obtaied fo the past acceptace ad ejectio of each vaiats of the poduct, the odel becoes a leaig odel to suppot akig decisios o which poduct to offe to the ext custoe With such a belief distibutio, the expected pobability of Offe beig accepted is, Offe is, Offe is ad Offe is k u Fo k offes, this is defied as u= u Let V (,,, ) = expected axiu futue pofit fo the ext custoe Coside a two vaiat case, give that oe has to choose which of the two DOI: 06/jcc Joual of Copute ad Couicatios

5 L S Lee et al vaiats of the poduct to offe to the ext custoe, fuctio (,,, ) to satisfy the optial equatio (see [7]): (,,, ) V P+ β V( +, +,, ) + V(, +,, ) ; = ax P + β V ( +, +, +, + ) + V( +, +,, + ) + V(, +,, ) V has Fo the vaiats of the cedit cad poduct, fuctio V (,,,,, ) satisfies the optial equatio of: (,,,,, ) V () P+ β V ( +, +,,,, ) + V (, +,,,, ) ; P + β V (,,,,, ) + V ( +, +,, +,, ) = ax + V (, +,,,, ) ; P + β V ( +, +, +, +, +, + ) + V ,,,,, + V +, +,, +,, + V +,,, ( ),, () Fo the vaiats of the cedit cad poduct, fuctio V,,,,,,, satisfies the optial equatio of: DOI: 06/jcc Joual of Copute ad Couicatios

6 (,,,,,,, ) V L S Lee et al P+ β V ( +, +,,,,,, + V (, +,,,,,, ; P + β V ( +, +, +, +,,,, + V ( +, +,, +,,,, + V (, +,,,,,, ; P + β V +, +, +, +, +, +,, + V ( +, +, +, +,, +,, = ax + V ( +, +,, +,,,, + V (, +,,,,,, ; P + β V +, +, +, +, +, +, +, + + V ( +, +, +, +, +, +,, + ) + V ( +, +, +, +,, +,, + V ( +, +,, +,,,, + V (, +,,,,,, Fo vaiats of poducts, fuctio V (,,,,,, ) optial equatio: (,,,,,, ) V k k u u Pk β V u u k: k u= u u= u () satisfies the ( ) = ax + +, +,, +, + (,,,, ) k k u l + V + + l l + l = u= u l The fist te i each offe is the pobability that a custoe will accept the () DOI: 06/jcc Joual of Copute ad Couicatios

7 L S Lee et al vaiat offeed ultiplied by the pofit to the bak The eaiig tes depeds o the chace β that thee will be aothe custoe I the β equatio, the fist te coespods to the cuet offe beig accepted The eaiig tes coespod to the offe beig efused ad it looks at the diffeet ways it ca happe Fo exaple, the te V ( +, +,, + ) coespods to the efusal of the Offe While V (, +,, ) eas oe believes Offe has bee efused thus thee is o updatig of Offe The te V +, +,,, + coespods to the efusal of the l-th offe l l Optial Solutio fo May Vaiats of the Poduct Coside a vaiatio of the poble i () whee the lede has a cost of β V (, +,, ) if a offe is ade to a custoe whee the state is (,,, ) iespectively of which offe is ade Sice the cost is idepedet of the offe ade, it caot affect the optial actio Let V (,,, ) be the optial expected pofit fo the odified poble The, we kow the optial V,,, V,,,, with policy whe solvig fo ( ) is the sae as fo B + V ( +, +,, ); V,,, ax B + V = β + V ( ) (,,, ),,, Pi whee Bi =, Pi 0, ad i =, β Fo the vaiats case, the optial expected pofit is defied as: V (,,,,, ) B+ V ( +, +,,,, ) ; B + V ( +, +, +, +,, ) + V ( +, +,, +,, ) ; = β ax B + V +, +, +, +, +, + + V ( +, +, +, +,, + ) + V ( +, +,, +,, ) ; Pi whee Bi =, i =,, β Fo the vaiats case, the optial expected pofit is defied as: (5) (6) DOI: 06/jcc Joual of Copute ad Couicatios

8 (,,,,,,, ) V = β L S Lee et al B+ V ( +, +,,,,,, ; B + V ( +, +, +, +,,,, + V ( +, +,, +,,,, ; B + V ( +, +, +, +, +, +,, + V ( +, +, +, +,, +,, ax + V ( +, +,, +,,,, ; B + V ( +, +, +, +, +, +, +, + ) + V +, +, +, +, +, +,, + + V ( +, +, +, +,, +,, + V ( +, +,, +,,,, ; Pi whee Bi =, i =,,, β Recall that Equatio () is the optial equatio fo vaiats of the poduct which is the extesio of Equatios ()-() i the, ad -vaiats cases espectively We subtact a cost of β V (, +,,,,, ) fo all the actios i state of (,,,, ) of Equatios (5)-(7) We kow that this caot affect the decisios ade but allows us to siplify Equatios (5), (6) ad (7) to a geeal equatio of: V,,,, k k j j = β ax Bk + V ( +, +,, k, k, k+, k+ ) k: k j= j j= j k k j l + V ( +, +,, l, l +,,, ) l= j= j l The poof of the theoe ca be efeed i Seow ad Thoas [6] If we have vaiats,,, oe chooses Offe ) (7) (8) DOI: 06/jcc Joual of Copute ad Couicatios

9 L S Lee et al >, oe chooses Offe ) (, ) If we have vaiats, ) (,,, ), oe chooses Offe ) (, ), oe chooses Offe >,, oe chooses Offe ) If we have vaiats, ) (,,,,,, oe chooses Offe ) (,,,, oe chooses Offe ) (,, oe chooses Offe >,, oe chooses Offe ) So, if we have vaiats, fo the theoe as i [] it is foud that: At ay state (,,,, ), thee exists fuctios i ( i+, i+,,, ), i =,,, so that: ) Oe chooses Offe to all futue custoes if (,,,, ) ; ) Oe chooses Offe to all futue custoes if (,,,, ) ;,,,, ; ad ) Oe chooses Offe t to all futue custoes if t+ t t+ t+ ) Oe chooses Offe t + to all futue custoes if (,,,, ) We have poved that thee is exists at ost oe,,,,, t,,,, i the followig Lea t t+ t+ = Lea : At ay state (,,,, ) > t+ t t+ t+, thee is exists at ost oe,,,,, t,,,, fo the choice of t vaiats of all futue t t+ t+ = custoes Poof To pove the Lea above, we eed to coside two cases Case whee thee is exactly oe switch: ) Let P() be the stateet that oe chooses Offe to all futue custoes if t+ t ( t+, t+,,, ) ad oe chooses Offe t + to all futue custoes >,,,, if t+ t t+ t+ ) Let P() which is base offe be the default Hece fo =, oe chooses Offe ) Fo =, assue P() is coect, that is: a) (, ), oe chooses Offe b) > (, ), oe chooses Offe Note that thee is oe (, ) Fo the followig stateets,,,, is used to diffeetiate,,, sice,,, ae ot the poit to switch the offe ) Suppose P(K) is tue, fo = K, whee P(K) is the stateet that oe t+ t t+, t+,, K, K ; ad oe >,,,, P(K) chooses Offe t to all futue custoes if chooses Offe t + to all futue custoes if also eas that: a) Oe chooses Offe if (, ) b) Oe chooses Offe if (,,, ) t+ t t+ t+ K K ad oe chooses Offe if DOI: 06/jcc Joual of Copute ad Couicatios

10 L S Lee et al (, ) c) Oe chooses Offe if (,,,,, Offe if (,,, ) ; ad oe chooses Offe if (, ) ; oe chooses d) Oe chooses Offe if t (,,,,,,, K, K) ; oe chooses Offe if t (,,,,, K, K) ; oe chooses Offe if t (,,, K, K) ; oe chooses Offe if t 6 5 ( 5, 5, 6, 6,, K, K) ; ad so o, the oe chooses Offe t if t+ t ( K, K) e) Oe chooses Offe t + if t+ > t ( K, K) ad thee is oe t ( t+, t+,, K, K) o switch of offes 5) It ca be show that P( K + ) is tue whee P( K + ) is the stateet,,,,, that oe chooses Offe to all futue custoes if t+ t t+ t+ K+ K+ othewise oe chooses Offe t + to all futue custoes if t+ > t ( t+, t+,, K+, K+ ) P( K + ) also eas that: a) Oe chooses Offe if (, ) b) Oe chooses Offe if (,,, ) ad oe chooses Offe if (, ) c) Oe chooses Offe if (,,,,, ; oe chooses,,,, Offe if ; ad oe chooses Offe if d) Oe chooses Offe if,,,,,,,, ; oe chooses Offe t+ ( K+ K+ ) if t+ (,,,,, K+, K+ ) t+ (,,, K+, K+ ) t+ 6 5 ( 5, 5, 6, 6,, K+, K+ ) Offe t if t+ t ( K+, K+ ) e) Oe chooses Offe t + if > (, ) ; oe chooses Offe if ; oe chooses Offe if ; ad so o, the oe chooses Sice P(K) is tue, that is: t+ t K+ K+ ) Oe chooses Offe if (, ) ) Oe chooses Offe if (,,, ) (, ) ) Oe chooses Offe if (,,,,, Offe if (,,, ) ; ad oe chooses Offe if (, ) ) Oe chooses Offe if t (,,,,,,, K, K) t (,,,,, K, K) t (,,, K, K) t 6 5 ( 5, 5, 6, 6,, K, K) if ( ) ad oe chooses Offe if ; oe chooses ; oe chooses Offe if ; oe chooses Offe if ; oe chooses Offe if ; ad so o, the oe chooses Offe t t t K, K The itoductio of a additioal optio of choice; the te t+ t ( K+, K+ ) i (5, d); ca also be expessed as t+ t ( K, K, K+, K+ ) Sice (,,, ) has a additioal optio choice of (, ) t+ t K K K+ K+ K+ K+, the the ext choice is oe chooses vaiat t to all futue custoes if DOI: 06/jcc Joual of Copute ad Couicatios

11 L S Lee et al t+ t ( K+, K+ ) whee oe chooses vaiat t + if (, ) befoe it coes to the last coditio of choice which is tue, that is: > Hece, P(K + ) is also t+ t K+ K+ ) Oe chooses Offe if (, ) ) Oe chooses Offe if (,,, ) (, ) ) Oe chooses Offe if (,,,,, Offe if (,,, ) ; ad oe chooses Offe if ad oe chooses Offe if ; oe chooses, ) Oe chooses Offe if,,,,,,,, ; oe chooses Offe t+ ( K+ K+ ) if t+ (,,,,, K+, K+ ) t+ (,,, K+, K+ ) t+ 6 5 ( 5, 5, 6, 6,, K+, K+ ) Offe t if t+ t ( K+, K+ ) 5) Oe chooses Offe t + if t+ t ( K+, K+ ) (,,,, ) ; oe chooses Offe if ; oe chooses Offe if ; ad so o, the oe chooses t t+ t+ K+ K+ > ad thee is oe Case whee thee is o chage i the decisio of offe If thee is o chage i the decisio of the offe, fo above poof of case eas that thee is oly oe vaiat at ay state ad ( ),,,,, t,,,, does ot exist tivially t t+ t+ = Epiical Results ad Aalysis I this sectio, the data eeded to get ifoatio fo leaig the switch of offes has bee geeated usig the dyaic pogaig odel This is based o expected pofit geeated (i ) Soe esults geeated by the odel ae show i the followig tables We fist defied β = 05 fo ad vaiats i the odel The defied β = 0999 fo vaiats We have subtacted the fee fo the odel, hece the values show ae ot the full pofits Please ote the choice of β = 05 ad 0999 was based o the pupose to illustate the pofit geeated at 50% discoutig facto ad alost 00% discoutig facto Two Vaiats Case If thee ae vaiat of poducts (5% ad 0% iteest ates), the vaiat 5% will be chose if P > P ad othewise, vaiat 0% will be chose if P > P Table ad Table peset soe of the esults geeated by the odel The bold i the ow is the poit whe the switch of offes occus We choose = ad = 0 to epeset a case whee oe s belief of the acceptace of vaiat 5% is p = 0 DOI: 06/jcc Joual of Copute ad Couicatios

12 L S Lee et al Table Pat of esults geeated by the acceptace odel whe P = 0000, P = 5000, β = 05, = 5, p = 0 Pofit ( ) Offe % % % % % Table Pat of esults geeated by the acceptace odel whe P = 0000, P = 5000, β = 05, = 5, p = 0 Pofit ( ) Offe % % % % % % % % % % % % % % % Table ad Table peset a case whee oe s belief of the acceptace of vaiat 5% with the atio of p = ad soe of the belief poits at which the offe decisio chages Table 5 ad Table 6 peset a case whee oe s belief of the acceptace of 5 vaiat 5% with the atio of p = ad soe of the belief poits at which the 6 offe decisio chages Table 7 ad Table 8 show a exaple whee thee is o ay poit of the switch of offes occus That is t ( t+, t+,,, ) does ot exists i ay state We choose = ad = 5 to epeset a case whee oe s belief of the acceptace of vaiat 0% is p = 5 DOI: 06/jcc Joual of Copute ad Couicatios

13 L S Lee et al Table Chagig of offes whe =, =, P = 0000, P = 5000, β = 05, = 5, p = Table Chagig of offes whe = 8, = 6, P = 0000, P = 5000, β = 05, = 5, p = Table 5 Chagig of offes whe = 5, = 6, P = 0000, P = 5000, β = 05, = 5, p = Table 6 Chagig of offes whe = 5, = 8, P = 0000, P = 5000, β = 05, = 5, p = Table 7 Offe 5% to all futue custoes whe P = 0000, P = 50000, β = 0999, = 5, p = 0 Pofit ( ) Offe % % % Table 8 Offe 0% to all futue custoes whe P = 0000, P = 50000, β = 0999, = 5, p = 0 Pofit ( ) Offe % % % % % % Thee Vaiats Case If thee ae vaiat of poducts (5%, 0% ad 5% iteest ates), the vaiat 5% will be chose if P > P ad P > P Vaiat 0% will DOI: 06/jcc Joual of Copute ad Couicatios

14 L S Lee et al be chose if P > P ad P > P Othewise, vaiat 5% will be chose if P > P ad P > P Table 9 ad Table 0 peseted soe of the esults geeated by the odel fo thee vaiat of poducts The bold i the ow is the poit whe the switch of offes occus We choose = ad = 6 to epeset a case whee oe s belief of the acceptace of vaiat 5% ad vaiat 0% ae p = p = The 8 chages of offe show ae fo vaiat 5% to vaiat 5% Table pesets soe of the esults geeated by the odel fo thee vaiats of the poduct The bold ow is the poit whe the switch of offes occus We choose =, = ad =, = 6 to epeset a case whee oe s belief of the acceptace of vaiat 5% ad 5% ae p = ad p = espectively The chages of offe ae fo vaiat 5% to vaiat 0% Table 9 Pat of esults geeated by the acceptace odel whe P = 0000, P = 5000, P = 5000, β = 05, = 6, p = 6 Pofit ( ) Offe % % % Table 0 Pat of esults geeated by the acceptace odel whe P = 0000, P = 5000, P = 5000, β = 05, = 6, p = 6 Pofit ( ) Offe % % % % % Table Pat of esults geeated by the acceptace odel whe P = 0000, P = 5000, P = 5000, β = 05, = 6, p = 6 Pofit ( ) Offe % % % % % DOI: 06/jcc Joual of Copute ad Couicatios

15 L S Lee et al Table pesets soe of the esults geeated by the odel fo thee vaiats of the poduct The bold ow is the poit whe the switch of offes occus We choose =, = ad =, = 7 to epeset a case whee oe s belief of the acceptace of vaiat 5% ad 0% ae p = ad p = 7 espectively Table ad Table peset a case whee oe s belief of the acceptace of vaiat 5% with the atio of p = p = ad acceptace of vaiat 0% with 5 the atio of p = p = espectively fo soe of the belief poits at which the 8 offe decisio chages Tables 5-7, peseted hee show that thee is o ay poit of the switch of offes occus That is ( ) does ot exists i ay state of (,,,, ) I Table 5, we choose = 5, = ad =, = to epeset a case 5 whee oe s belief of the acceptace of vaiat 0% is p = ad acceptace of vaiat 5% is p = I Table 6, we choose = 6, = 5 ad 7 =, = to epeset a case whee oe s belief of the acceptace of vaiat 0% is p = ad acceptace of vaiat 5% is p = I Table 7, we 5 choose = 6, = ad =, = to epeset a case whee oe s belief Table Pat of esults geeated by the acceptace odel whe P = 0000, P = 5000, P = 5000, β = 05, = 6, p = 6 Pofit ( ) Offe % % % % % Table Chagig of offes whe 5 =, 5 = 5, 0 =, 0 = 5, P = 0000, P = 5000, P = 5000, β = 05, = 6, p = Table Chagig of offes whe 5 = 6, 5 = 6, 0 = 6, 0 = 6, P = 0000, P = 5000, P = 5000, β = 05, = 6, p = DOI: 06/jcc Joual of Copute ad Couicatios

16 L S Lee et al Table 5 Offe 5% to all futue custoes whe P = 0000, P = 5000, P = 5000, β = 05, = 6, p = 6 Pofit ( ) Offe % % % % % % Table 6 Offe 0% to all futue custoes whe P = 0000, P = 5000, P = 5000, β = 05, = 6, p = 6 Pofit ( ) Offe % % % % % % % % % % Table 7 Offe 5% to all futue custoes whe P = 0000, P = 5000, P = 5000, β = 05, = 6, p = 6 Pofit ( ) Offe % % % % % 6 of the acceptace of vaiat 0% is p = ad acceptace of vaiat 5% is p = Table 8 shows that as iceases but the ate ad ae fixed, the cucial value whee oe chages offes, (, ), is ootoically o iceasig We also give esults fo the effect of oe ifoatio (iceet of DOI: 06/jcc Joual of Copute ad Couicatios

17 L S Lee et al Table 8 Effect of oe ifoatio o the switch of offes,, Chages of Offe 6, 6, Reai 5% 6, 6, % to 5% 6, 6 8, % to 5% (, ) 6, 6, % to 5% 6, 6 5, % to 5% 6, 6 6, % to 5% ) i the table The (, ) fo vayig fo to 6 Note that the hyphe ( ) i the table eas that thee is o chages of the offe occus whee Offe 5% is the oly offe Fou Vaiats Case If thee ae vaiats of poducts, the vaiat 5% will be chose if P > P, P > P, ad P > P Vaiat 0% will be chose if P > P, P > P, ad P > P Vaiat 5% will be chose if P > P, ad will be chose if P > P P > P, P > P Othewise, vaiat 0% P > P, P > P, ad Table 9 pesets soe of the esults geeated by the odel fo fou vaiats of the poduct The bold ow is the poit whe the switch of offes occus We choose =, =, =, = 7, ad =, = 7 to epeset a case whee oe s belief of the acceptace of vaiat 5%, 5% ad 0% ae p = ad p = p = 0 espectively The chages of offe ae fo vaiat 5% to vaiat 0% Table 0 pesets soe of the esults geeated by the odel fo fou vaiats of the poducts The bold ow is the poit whe the switch of offes occus We choose =, = 6, =, = 6 ad =, = to epeset a case whee oe s belief of the acceptace of vaiat 5%, 0% ad 0% ae p = p = ad p = espectively The chages of offe show is fo vaiat 5% to vaiat 5% Table pesets soe of the esults geeated by the odel fo fou vaiats of the poducts The bold ow is the poit whe the switch of offes occus We DOI: 06/jcc Joual of Copute ad Couicatios

18 L S Lee et al Table 9 Pat of esults geeated by the acceptace odel whe P = 0000, P = 50000, P = 80000, P = 00000, β = 0999, =, p = Pofit ( ) Offe % % % % % Table 0 Pat of esults geeated by the acceptace odel whe P = 0000, P = 50000, P = 80000, P = 00000, β = 0999, =, p = Pofit ( ) Offe % % Table Pat of esults geeated by the acceptace odel whe P = 0000, P = 50000, P = 80000, P = 00000, β = 0999, =, p = Pofit ( ) Offe % % % % % choose = 5, = 6, = 6, = 7, ad = 6, = 7 to epeset a case whee 5 oe s belief of the acceptace of vaiat 5%, 0% ad 0% ae p = ad 6 6 p = p = espectively The chages of offe show is fo vaiat 5% to 7 vaiat 0% Tables - show that the chages of offe is fo vaiat 0% to vaiat 5%, vaiat 0% to vaiat 0% ad vaiat 5% to vaiat 0% espectively Tables 5-8 peseted hee show o poit fo the switch of offes That is,,,,,,,, ( ) does ot exist i ay state of t t+ t+ Table Pat of esults geeated by the acceptace odel whe P = 0000, P = 50000, P = 80000, P = 00000, β = 0999, =, p = Pofit ( ) Offe % % % % DOI: 06/jcc Joual of Copute ad Couicatios

19 L S Lee et al Table Pat of esults geeated by the acceptace odel whe P = 0000, P = 50000, P = 80000, P = 00000, β = 0999, =, p = Pofit ( ) Offe % % % % % Table Pat of esults geeated by the acceptace odel whe P = 0000, P = 50000, P = 80000, P = 00000, β = 0999, =, p = Pofit ( ) Offe % % % Table 5 Offe 5% to all futue custoes whe P = 0000, P = 50000, P = 80000, P = 00000, β = 0999, =, p = Pofit ( ) Offe % % % Table 6 Offe 0% to all futue custoes whe P = 0000, P = 50000, P = 80000, P = 00000, β = 0999, =, p = Pofit ( ) Offe % % % Table 7 Offe 5% to all futue custoes whe P = 0000, P = 50000, P = 80000, P = 00000, β = 0999, =, p = Pofit ( ) Offe % % % % DOI: 06/jcc Joual of Copute ad Couicatios

20 L S Lee et al Table 8 Offe 0% to all futue custoes whe P = 0000, P = 50000, P = 80000, P = 00000, β = 0999, =, p = Pofit ( ) Offe % % % % % 5 Coclusio Fo the esults, we ca clealy see that thee is at ost oe poit of switch offes No atte how ay vaiats of the poduct offeed, the switchig offe will ot oe tha oe Based o this obsevatio, the odel ca tell the best offe to exted to the ext custoe i a efficiet ae ad axiise the pofit eaed Hece the odel is able to idetify the best offe fo vaiats of cedit cads Futhe eseach would be to test this o diffeet fiacial poducts like otgages Ackowledgeets This eseach is suppoted by the Reseach Uivesity Gat Schee (RUGS) RU Disclosue Policy The authos declae that thee is o coflict of iteest egadig the publicatio of this pape Refeeces [] Stago, V (000) Copetitio ad Picig i the Cedit Cad Maket The Review of Ecooics ad Statistics, 8, [] Godzicki, D (0) The Evolutio of Copetitio i the Cedit Cad Maket Wokig pape Stafod Uivesity, Palo Alto, CA [] Keeey, RL ad Olive, RM (005) Desigig Wi-Wi Fiacial Loa Poducts fo Cosues ad Busiess Joual of the Opeatioal Reseach Society, 56, [] Seow, H-V ad Thoas, LC (006) Usig Adaptive Leaig i Cedit Scoig to Estiate Take-Up Pobability Distibutio Euopea Joual of Opeatioal Reseach, 7, [5] Ma, P, Cook, J ad Asell, J (00) Modellig Take-Up ad Pofitability Joual of the Opeatioal Reseach Society, 6, 0- [6] Bella, R (957) Dyaic Pogaig Piceto Uivesity Pess, Piceto, NJ [7] Putea, ML (99) Makov Decisio Pocess: Discete Stochastic Dyaic Pogaig Joh Wiley & Sos, Ic, Hoboke, NJ DOI: 06/jcc Joual of Copute ad Couicatios