Online Resource Allocation under Arbitrary Arrivals: Optimal Algorithms and Tight Competitive Ratios

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1 Onlne Resource Allocaton under Arbtrary Arrvals: Optmal Algorthms and Tght Compettve Ratos Wll Ma Operatons Research Center, Massachusetts Insttute of Technology, Cambrdge, MA 02139, Davd Smch-Lev Insttute for Data, Systems, and Socety, Department of Cvl and Envronmental Engneerng, and Operatons Research Center, Massachusetts Insttute of Technology, Cambrdge, MA 02139, We consder the problem of allocatng fxed resources to heterogeneous customers arrvng sequentally. We study ths problem under the framework of compettve analyss, whch does not assume any predctablty n the sequence of customer arrvals. Prevous work has culmnated n optmal algorthms under two scenaros: () there are multple resources, each of whch yelds reward at a constant rate when allocated; or () there s a sngle resource, whch yelds reward at dfferent rates when allocated to dfferent customers. In ths paper, we derve optmal allocaton algorthms when there are multple resources, each wth multple reward rates. Our algorthms are smple, ntutve, and robust aganst forecast error. Ther tght compettve rato cannot be acheved by combnng exstng algorthms, whch consder the tradeoffs between multple resources and multple reward rates separately. By showng how to ntegrate these tradeoffs whle makng allocaton decsons, we expand the applcablty of compettve analyss n many areas, such as onlne advertsng, matchng markets, and personalzed e- commerce. We test our methodologcal contrbuton on the hotel data set of Bodea et al. (2009), where there are multple resources (hotel rooms), each wth multple reward rates (fares at whch the room could be sold). We fnd that applyng our algorthms, n conjuncton wth algorthms whch attempt to forecast and learn the future transactons, results n the best performance. 1. Introducton In ths paper we study a general onlne resource allocaton problem, stated n revenue management termnology. A frm has multple tems, each wth an unreplenshable startng nventory, and a 1

2 2 set of feasble prces at whch ts unts of nventory could be sold. Heterogeneous customers arrve sequentally over tme. Upon a customer s arrval, the probablty that she would buy each tem at each prce s revealed; these probabltes can be 0 for tems she s not nterested n, or prces that are too hgh. The frm then chooses an avalable tem and feasble prce to offer her, after whch her purchase decson s mmedately realzed accordng to the probablty gven. The frm s goal s to maxmze ts expected revenue before the nventores run out, or there are no more customers. A specal case of our problem s the determnstc case, where all purchase probabltes are 0 or 1. In ths case, the frm knows the maxmum a customer s wllng to pay for each tem, possbly 0. Therefore, the frm s decson can be reduced to choosng an tem to assgn to the customer (chargng her maxmum wllngness-to-pay for that tem), or rejectng the customer f her wllngness-to-pay s low for every tem. We study these problems under the framework of compettve analyss. In compettve analyss, no nformaton s gven about the sequence of customers, nor are they assumed to follow any observable pattern. The algorthm s performance s expressed as a fracton of an optmum whch knows the complete customer sequence n advance. For c 1, f an algorthm can guarantee that ths fracton s at least c for every problem nstance (and customer sequence), then t s sad to acheve a compettve rato of c. The goal s to develop robust algorthms whch acheve the optmal compettve rato,.e. a rato c such that no algorthm, wthout knowng the customer sequence n advance, can do better Prevous Work n Compettve Analyss Our model nvolves multple tems, as well as multple feasble prces for each tem. Ths combnes two challenges n compettve analyss, whch have prevously been studed separately. 1. Multple Items: The challenge of how to prortze between multple tems, when a customer can only be offered (or assgned) one of them, has been consdered n the onlne b- matchng problem (Kalyanasundaram and Pruhs 2000), Adwords problem (Mehta et al. 2007, Buchbnder et al. 2007), and onlne assortment problem (Golrezae et al. 2014). The optmal

3 3 algorthms for these problems all perform some knd of nventory balancng, placng lower prorty on sellng tems wth lower remanng nventory. Inventory balancng algorthms are also related to the randomzed rankng algorthms used n the onlne bpartte matchng problem (Karp et al. 1990, Aggarwal et al. 2011). 2. Multple Prces: The challenge of when to reject a customer only wllng to pay a low prce, to preserve nventory for customers wllng to pay hgher prces, has been consdered n the sngle-tem, determnstc case of our problem (Ball and Queyranne 2009, Lan et al. 2008). The optmal algorthm employs bookng lmts, rejectng customers wth low wllngness-to-pay once a threshold amount of the tem has been sold. Our model studes the challenges ntroduced when multple prces are ncorporated nto the aforementoned problems wth multple tems. In Secton 6, we explan how our technques can be extended to allow for fractonal nventory consumpton, lke n the Adwords problem; or multple tems to be offered to each customer, lke n the onlne assortment problem. We now dscuss two addtonal ways to vew our model, whch emphasze the ncrease n modelng power from allowng for multple prces: Frst, one can thnk of each of our (tem, prce)-combnatons as an ndependent product. By allowng for multple prces, we have allowed the multple products correspondng to each tem to draw from the same nventory, or resource. The dfferent products can also consume dfferent amounts of that resource, under the extenson wth fractonal nventory consumpton. Second, n some applcatons, the customers are classfed under a fnte number of types, and nstead of a prcng decson, there s a dfferent reward (correspondng to match qualty ) for allocatng each tem to each customer type. Ths can be reduced to our problem, wth the feasble prces for an tem beng that tem s match qualtes over all types. By allowng for multple prces, we have allowed each tem to yeld dfferent rewards when allocated to dfferent types, as opposed to yeldng the same reward for all types.

4 Integratng the Challenges We ntroduce a bd-prce control polcy whch acheves the optmal compettve rato under both multple tems and multple prces. Our algorthm mantans for each tem a bd prce, whch s the value placed on one unt of ts nventory. The pseudorevenue assocated wth an (tem, prce)- combnaton s then the prce mnus the value of the tem. The algorthm offers to each customer the (tem, prce)-combnaton wth the hghest expected pseudorevenue, never offerng combnatons wth non-postve pseudorevenue. Bd-prce control s a classcal dea n revenue management (see Tallur and Van Ryzn (2006), Lu and Van Ryzn (2008)), where the bd prces are computed usng an LP, based on the remanng nventory and forecasted dstrbuton of remanng customers. However, snce we make no assumptons about future customers, our bd prces are based on only the remanng nventory. Our bd prces are very smple they are computed separately for each tem, lke the multplcatve penaltes n Golrezae et al. (2014). Let w be the fracton of the startng nventory of whch has already been sold. At each pont n tme, the bd prce of tem s set to Φ (w ), where Φ s a value functon dependent on the set of feasble prces for tem. To llustrate our algorthm, we dsplay the form of Φ for an example tem whch could be sold at fares $150 or $450, n Fgure 1. Note the followng: As the fracton of tem sold ncreases over tme, the value of one unt of nventory ncreases, hence the pseudorevenues assocated wth the feasble prces of tem decrease, and the bdprce algorthm places lower prorty on offerng/assgnng tem. Ths captures the nventory balancng used to address the challenge of multple tems. Let α be the value at whch Φ (α ) = 150. The algorthm stops sellng tem at the lower prce of 150 once ts fracton sold reaches α, because the pseudorevenue assocated wth the lower prce s 150 Φ (w ), whch s non-postve for w α. Therefore, our algorthm captures the bookng lmts used to address the challenge of multple prces. The specfc value of Φ (w ) also tells the algorthm how to choose between a lower prce whch may have hgher expected revenue, versus a hgher prce whch has lower expected nventory consumpton.

5 5 Fgure 1 The value functon for an tem wth feasble prce set {150, 450}. $450 Value of 1 unt of Inventory, Φ $150 α 0 1 Fracton Sold, w Φ ncreases from 0 to the maxmum prce of 450 over [0,1], and s pecewse-convex. In general, each value functon Φ s desgned to maxmze the compettve rato CR assocated wth t. As we wll explan, the exact functon Φ s defned as the soluton to a dfferental equaton arsng from a prmal-dual analyss. The bookng lmts mpled by such a Φ are dfferent than the bookng lmts derved by Ball and Queyranne (2009) whch are optmal when s the sngle tem beng sold. For example, f tem has two prces, wth r beng the rato of hgh to low prce, then the value of α s α(r ), where α(r) = ln 2(r 1) 1 + 4r(r 1)/e 1. (1) Meanwhle, the optmal bookng lmt from the sngle-tem case s r 2r 1. α s greater than r 2r 1, wth the ntuton beng that wth multple tems, there s less upsde to reservng nventory for hgher prces, because the reserved unts may have to compete wth other tems to be sold. Indeed, when there are both multple tems and multple prces, the optmal algorthm must ntegrate nventory balancng when settng bookng lmts, nstead of usng the sngle-tem bookng lmts Compettve Rato Results The overall compettve rato assocated wth our algorthm s mn CR, beng lmted by the tem wth the smallest value of CR. Whle ths compettve rato s not acheved by the exact bd-prce algorthm specfed n the prevous subsecton, we prove the followng results n ths paper: 1. A varant of the bd-prce algorthm, whch we call Mult-prce Balance, acheves a compettve rato of mn CR n the asymptotc regme, where all startng nventores go to.

6 6 2. A dfferent varant of the bd-prce algorthm, whch we call Mult-prce Rankng, acheves a compettve rato of mn CR n the determnstc case of our problem. 3. A counterexample, whch can be made to fall under both the asymptotc regme and the determnstc case, shows that the compettve rato of any algorthm cannot exceed mn CR. When there s a sngle feasble prce for an tem, CR = 1 1. Our statements 1 3 are gen- e eralzatons of results that exst when every tem has only one prce. Statement 1 corresponds to the nventory balancng algorthm of Golrezae et al. (2014) achevng a compettve rato of 1 1 e. Statement 2 corresponds to the rankng-based algorthm of Aggarwal et al. (2011) achevng the same compettve rato. Statement 3 shows that both of these results are tght. These results may not be tght n the non-asymptotc, non-determnstc settng, whch s an mportant open problem (Devanur et al. 2013) n the sngle-prce case as well. Nonetheless, we establsh lower bounds on the compettve rato acheved whch hold n the non-determnstc settng, and are parametrzed by k, the mnmum startng nventory of an tem. As k ncreases, these bounds sharply approach the tght guarantee of mn CR from the asymptotc regme. In the sngleprce case, our bounds show that the multplcatve gap from 1 1 e s at most (1 + k)(1 e 1/k ), whch mproves the prevously-best-known gap from Golrezae et al. (2014). We llustrate our bounds on the case where every tem has two feasble prces, n Fgure 2. The compettve rato CR assocated wth an tem s 1 e α(r ), where r s ts rato of hgh to low prce, and α s defned n (1). Thus the overall compettve rato mn CR can be wrtten as 1 e α(r), (2) where r = max r. (2) s decreasng n r. As r 1, α(r) 1 and (2) approaches the known value of The smallest compettve rato occurs as r, wth (2) approachng 1 1 e e.393. The formal statements of our theorems, whch allow each tem to have an arbtrary set of feasble prces, are deferred to Secton 2. We analyze Mult-prce Balance n Secton 3 and Mult-prce Rankng n Secton 4. Descrptons of our technques are also deferred to these sectons.

7 7 Fgure 2 Compettve ratos acheved n the two-prce case, where r denotes the maxmum rato of an tem s hgh to low prce, and k denotes the mnmum startng nventory of an tem. The guarantees mprove from bottom to top (as r decreases), and from left to rght (as k ncreases). 1 1 [GNR14] 2 1 1/e (1+k)(1 e 1/k ) [Thm. 1()] 1 1 e.632 [GNR14] 1 1 e [AGKM11] 1 r r [Thm. 1()] 2(2r 1) 1 e α(r) [Cor. 1,Thm. 3] 1 e α(r) [Thm. 2,Thm. 3] r 2r 1 [BQ09] tght bounds non-tght bounds 1 1 e α(r) [Thm. 1()] (1+k)(e 1/k 1) 1 4 [CMSLX16] k Non-determnstc Settng 1 1 e e Determnstc Case 1 2 Determnstc, Sngle-tem Case The smallest guarantee of 1 4 n ths dagram s also mpled by the results of Chen et al. (2016). In general, the tght compettve rato of CR can approach 0 s the feasble prce set for tem contans both a large number of prces and a large rato from hghest to lowest prce, whch s a known negatve result (Aggarwal et al. 2011). Nonetheless, n many applcatons, one can enumerate the prce ponts (e.g., an tem whch could only be sold at $19.99 or $24.99), or bound the rato between the hghest and lowest prces (e.g., an advertser who bds between.1 and.2) Applcaton on Hotel Data Set of Bodea et al. (2009) We frst summarze the general benefts of applyng compettve analyss, and the compettve algorthms derved from ths research. In contrast to tradtonal algorthms, whch optmze based on a forecast of future demand, or attempt to learn the demand, compettve algorthms hedge aganst some worst case, and operate wthout any demand nformaton. Most mmedately, they are useful for products wth hghly unpredctable demand (Ball and Queyranne 2009, Lan et al. 2008), or for ntalzng new products wth no hstorcal sales data (Van Ryzn and McGll 2000). Second, by eschewng stochastc processes for generatng demand, compettve algorthms are usually smple

8 8 and flexble, leadng to clean nsghts about the problem (Borodn and El-Yanv 2005). Thrd, past research has reported on cases where compettve algorthms perform well n practce (Feldman et al. 2010), or on average n numercal experments (Golrezae et al. 2014, Chen et al. 2016). In Secton 7, we run smulatons on the publcly-accessble hotel data set of Bodea et al. (2009). We use the product avalablty nformaton to estmate customer choce models, and the transactonal data as the sequence of arrvals. Ths leads to an onlne assortment problem lke n Golrezae et al. (2014), wth multple prces (advance-purchase rate, rack rate, etc.) for each tem (Kng room, Two-double room, etc.). We compare the performance of our Mult-prce Balance algorthm, usng the extenson dscussed n Secton 6 whch can offer assortments, to varous benchmarks and forecastng algorthms. The man concluson from our smulatons s that the best performance s acheved by hybrd algorthms (see Golrezae et al. (2014)). These are forecastng-based algorthms whch contnuously reference our forecast-ndependent value functons Φ 1,..., Φ n, and adjust ther decsons accordngly. Although ths only changes a small fracton ( 5%) of decsons, these tend to be the decsons where the forecast s beng most overconfdent. Therefore, not only does ths boost average performance, t drastcally reduces the varance n performance caused when the forecast s wrong Other Related Work We brefly dscuss some related papers whch has not been mentoned untl now. Alternate Approaches to Onlne Matchng. Our problem captures the onlne edge-weghted bpartte matchng problem, whch has been studed under varous settngs desgned to get around a basc mpossblty result (see Aggarwal et al. (2011)). One such settng s free dsposal (Feldman et al. 2009). Alternatvely, one could assume that the arrvals appear n a random order, whch allows for some form of learnng (Kesselhem et al. 2013); ths approach s very general and has been extended to onlne lnear programmng (Agrawal et al. 2014). However, to our knowledge, we are the frst to focus on the weght-dependent compettve rato for the onlne edge-weghted

9 bpartte matchng problem, nstead of makng assumptons such as free dsposal or randomlyordered arrvals. For a survey of onlne matchng, we refer to Mehta (2013). Known Stochastc Processes. When the stochastc process generatng the arrvals n our problem s gven, the resultng optmzaton problem s stll computatonally ntractable. Nonetheless, many effectve heurstcs have been proposed, under dfferent varatons of the model (Zhang and Cooper 2005, Jasn and Kumar 2012, Cocan and Faras 2012, Chen and Faras 2013). These heurstcs can earn 1 2 of the LP optmum n general settngs (Chan and Faras 2009, Wang et al. 2015, Gallego et al. 2015). Manshad et al. (2012) derve an mproved performance rato when the gven stochastc process s IID. From a modelng perspectve, our problem wth multple tems and multple prces s smlar to the mult-fare, parallel flghts problem of Zhang and Cooper (2005), and the appontment schedulng wth customer preferences problem of Wang et al. (2015). Alternate Metrcs. Compettve/approxmaton rato both consder the algorthm s expected reward as a fracton of an LP optmum. Our problem has been analyzed under other metrcs as well. When the arrval process s unknown but assumed to be IID, one popular metrc s regret, whch measures the addtve loss from optmum (see Ferrera et al. (2016)). When the arrval process s known, the flud and dffuson analyss approaches have also been used (see Reman and Wang (2008)). However, unlke compettve rato, these metrcs all tend to focus on asymptotc performance as the number of customers grows to nfnty. Fnally, a recent metrc whch has been studed s regret rato (Zhang et al. 2016). For a comprehensve revew of dfferent metrcs to use under dfferent models of demand (for a sngle tem), we refer the reader to Araman and Caldentey (2011). 2. Problem Defnton, Algorthm Sketch, and Theorem Statements A frm s sellng n N dfferent tems. Each tem [n] 1 starts wth a fxed nventory of k N unts, and could be offered at one of m N feasble prces, wth correspondng fares r (1),..., r (m ) R 9 1 For a general postve nteger b, let [b] denote the set {1,..., b}.

10 10 satsfyng 0 < r (1) <... < r (m ). For convenence, we let r (0) = 0 for each. In Appendx E.1, we allow for a contnuum of feasble prces n some range [r mn, r max ]. There are T N customers arrvng sequentally. Upon the arrval of customer t [T ], the frm observes p (j) t,, the probablty that customer t would buy tem at prce j, for all [n] and j [m ]. 2 The frm chooses up to one of the tems wth nventory remanng, and offers t to customer t, at any prce j. The customer accepts the offer wth probablty p (j) t,, n whch case the frm earns revenue r (j), and the nventory of tem s decremented by 1. In Secton 6, we dscuss models where multple tems can be offered or multple unts of nventory can be consumed at a tme. We defne an nstance I of the problem to consst of all of the followng: 1. Intal nformaton n, {k, m, r (1),..., r (m ) : [n]}; 2. Arrval nformaton T, {p (j) t, : t [T ], [n], j [m ]}. An onlne algorthm prescrbes, based on the ntal nformaton, how to make the offerng decson at each tme t, wthout knowng {p (j) t, : [n], j [m ]} for future customers t > t nor the length of the tme horzon T. For an onlne algorthm, let ALG(I) denote the revenue earned on a run on nstance I, whch s a random varable wth respect to the customers purchase decsons as well as any con flps n the algorthm. Meanwhle, we can wrte the followng LP based on nstance I: max T m n t=1 =1 j=1 p (j) t, r (j) m T t=1 j=1 m x (j) t, (3a) p (j) t, x (j) t, k [n] (3b) n =1 j=1 x (j) t, 1 t [T ] (3c) x (j) t, 0 t [T ], [n], j [m ] (3d) LP (3) encapsulates the executon of any algorthm, whch could make full use of the arrval nformaton at the start, on nstance I. x (j) t, represents the uncondtonal probablty of the algorthm 2 These probabltes can be 0 for tems the customer s not nterested n, or prces that are too hgh. A ratonal customer would have p (1) t,... p(m ) t,, although we do not need ths assumpton.

11 offerng tem at prce j to customer t. (3b) enforces that startng nventores are respected, whle (3c) enforces that at most one combnaton of tem and prce s offered to each customer. Objectve functon (3a) represents the expected revenue earned by the algorthm. Let OPT(I) denote the optmal objectve value. The compettve rato of the onlne algorthm s then defned to be nf I 11 E[ALG(I)] OPT(I). (4) We say that an algorthm acheves a compettve rato of c f (4) s lower-bounded by c. Gven any fxed onlne algorthm, (4) consders the worst-case nstance, ncludng the worst-case arrval sequence. The goal for the algorthm s to hedge aganst the worst-case arrval sequence, possbly by usng randomness. Defnton (4) provdes a guarantee on E[ALG(I)] relatve to any algorthm whch could have been possble, due to the followng result. Lemma 1. OPT(I) s an upper bound on the expected revenue of any algorthm, whch could make full use of the arrval nformaton at the start, on nstance I. The proof of Lemma 1 s deferred to Appendx A. The defnton of OPT based on the LP s standard n problems wth stochastc purchase realzatons and arbtrary customer arrvals we leave ts justfcaton to Mehta and Pangrah (2012), Golrezae et al. (2014). In the determnstc case of our problem, every p (j) t, s 0 or 1. The problem can be smplfed by lettng j t, = max{j [m ] : p (j) t, = 1}, wth j t, = 0 f the set s empty, for all t [T ] and [n]. We say that tem s assgned to customer t to ndcate that s offered to customer t at prce j t,, whch results n a sale; there s no reason to offer any other prce. Customer t can also be rejected, e.g. f j t, s low for every. In the determnstc case, the LP (3) s ntegral, so OPT(I) s equal to the revenue of the best algorthm knowng the arrval sequence at the start The Mult-prce Value Functon Φ For an arbtrary tem, we specfy ts value functon Φ, whch s dependent on ts feasble prces r (1),..., r (m ). Recall that Φ s a functon of w, the fracton of tem sold. For w [0, 1], Φ (w ) s the value the algorthm currently places on one unt of nventory of.

12 12 Frst we defne bookng lmts α (1),..., α (m ), whch are the fractons of startng nventory reserved for the respectve fares r (1),..., r (m ), va the followng proposton. Proposton 1. Consder any tem, wth an arbtrary number of dscrete prces satsfyng 0 < r (1) <... < r (m ). There are unque postve values α (1),..., α (m ) whch sum to 1 and satsfy 1 e α(1) = 1 1 r (1) /r (2) (1 e α(2) ) =... = 1 1 r (m 1) /r (m) There are also unque postve values σ (1),..., σ (m ) whch sum to 1 and satsfy σ (1) = 1 1 r (1) /r (2) σ (2) =... = 1 1 r (m 1) /r (m) (1 e α(m) ). (5) σ (m). (6) The proof of Proposton 1 s deferred to Appendx A. Whle fndng the exact soluton to (5) requres fndng the roots of a degree-m polynomal, a numercal soluton can easly be found va bsecton search. Proposton 1 contrasts α (1),..., α (m ) wth the bookng lmts σ (1),..., σ (m ) derved by Ball and Queyranne (2009), whch are optmal when s the sngle tem beng sold. α (1),..., α (m ) and Φ are a by-product of our analyss, and maxmze the compettve rato. Our method for dervng them s deferred to Appendx E. For now, we complete the defnton of Φ : Defnton 1. For each tem, defne the followng based on α (1),..., α (m ) : L (j) : the sum j ) j =1 α(j, defned for all j = 0,..., m (note that L (0) = 0 and L (m ) = 1); l ( ): a functon on [0, 1], where l (w) s the unque j [m ] for whch w [L (j 1) that l (L (j) ) = j + 1 for j = 0,..., m 1; we defne l (L (m ) ) to be m ). The value functon Φ s then defned over w [0, 1] by Φ (w ) = r (l (w ) 1) + (r (l (w )) r (l (w ) 1), L (j) ) (note ) exp(w L (l (w ) 1) ) 1. (7) exp(α (l (w)) ) 1 An example of a value functon Φ wth 2 prces was plotted n Fgure 1. In general, Φ s contnuously ncreasng and pecewse-convex over m segments of lengths α (1),..., α (m ), separated by segment borders L (0),..., L (m ). For each j, Φ reaches the value of r (j) at L (j), hence prce j stops beng offered once the fracton sold w reaches L (j).

13 13 We wll see that the compettve rato CR assocated wth Φ s 1 e α(1), whch s optmal. When m = 1, t can be seen that α (1) = 1, Φ (w ) = r (1) ew 1, and CR e 1 = 1 1, whch correspond e to known results. The functons Φ 1,..., Φ n facltate the tradeoff between mmedate reward and future nventory. We develop two algorthms, whch use them n dfferent ways Sketch of our MULTI-PRICE BALANCE and MULTI-PRICE RANKING Algorthms We frst sketch Mult-prce Rankng, whch s smpler. It assumes that k = 1 for all, whch does not lose generalty snce an tem whch starts wth multple unts of nventory can be transformed nto multple dsparate tems. At the start, the algorthm fxes for each tem a random seed W, drawn ndependently and unformly from [0, 1]. It then treats Φ (W ) as the bd prce for the sngle unt of tem : t offers to each customer t the avalable tem and prce j maxmzng the expected ( pseudorevenue, p (j) (j) t, r Φ (W ) ). Mult-prce Rankng hedges aganst the ambguty n customer arrvals by usng randomness, whch s standard n compettve analyss. The random seed W determnes the random mnmum prce at whch the algorthm s wllng to sell tem, as well as a random prorty for sellng when the algorthm s choosng between multple tems. We now sketch Mult-prce Balance, whch updates the bd prce of each tem based on the fracton w of ts k unts whch has been sold. However, the algorthm does not drectly use Φ (w ) as the bd prce of tem that would ncur a systematc roundng error, as w would always be a multple of 1 k, whle Φ s based on the segment borders L (0),..., L (m ) whch may not be multples of 1 k. Instead, at the start, the algorthm fxes for each tem a random value functon Φ, based on random segment borders L (0),..., L (m ). (0) L,..., L (m ) have been randomly rounded to multples 1 (0) of k satsfyng 0 = L... L (m ) = 1; they correspond to a random value functon Φ defned on dscrete ponts {0, 1 k,..., 1} satsfyng 0 = Φ (0) Φ ( 1 k )... Φ (1). The algorthm treats Φ (w ) as the bd prce for tem : t offers to each customer t the tem and prce j maxmzng the expected pseudorevenue, p (j) t, ( Φ ( L (j) ) Φ (w ) ).

14 14 The defnton of pseudorevenue at prce j s Φ ( L (j) s exactly 0 when w = L (j). As a result, the random segment borders ) Φ (w ), nstead of r (j) Φ (w ), so that t (0) L,..., L (m ) mply random bookng lmts for tem. In general, the realzed Φ wll be close to Φ, wth Φ ( L (j) ) r (j). In the asymptotc regme wth k, Φ = Φ determnstcally. However, for small k, optmzng a randomzed procedure for ntalzng Φ, based on r (1),..., r (m ) as well as k, nstead of havng a determnstc Φ, based on only r (1),..., r (m ), allows us to acheve a greater compettve rato Statements of Our Results Theorem 1. Mult-prce Balance acheves a compettve rato of mn CR, where for all, CR s lower-bounded by all of: () 1 e α(1) (1) ; () σ ; and () 1 e 1, f m (1+k )(e 1/k 1) 2 (1+k )(1 e 1/k ) = 1. Corollary 1. Mult-prce Balance acheves a compettve rato approachng 1 exp( mn α (1) ) as each startng nventory k approaches. Theorem 1 s our general result, where for each, CR s the compettve rato assocated wth the optmal randomzed procedure for ntalzng Φ, based on r (1),..., r (m ) and k. Lower bound () on CR s attaned by a randomzed procedure whch defnes Φ based on the fxed functon Φ. The numerator n () s a functon of the feasble prces r (1),..., r (m ), whle the denomnator s a functon of the startng nventory k. Note that the denomnator (1 + k )(e 1/k 1) decreases toward 1 as k, resultng n Corollary 1. Lower bound () s attaned from solvng an optmzaton problem for the best randomzed procedure, whch s tractable when k = 1. Interestngly, the bound turns out to be based on the sngle-tem bookng lmts σ (1),..., σ (m ), nstead of α (1),..., α (m ). Lower bound () s the mprovement of () n the sngle-prce case, where we have ganed a factor of e 1/k. It smplfes and mproves the bound from Golrezae et al. (2014). Mult-prce Balance s formalzed and Theorem 1 s proven n Secton 3. We explan the deas behnd our prmal-dual analyss, why we need random value functons, and how to overcome the resultng analytcal challenges. Theorem 2. Mult-prce Rankng acheves a compettve rato of 1 exp( mn α (1) ) n the determnstc case of our problem.

15 15 Mult-prce Rankng s formalzed and Theorem 2 s proven n Secton 4. Whle we descrbed Mult-prce Rankng as an algorthm for our general problem n Subsecton 2.2, t s most amenable to analyss n the determnstc case. Ths s also true n the sngle-prce case, as our analyss uses the framework of Devanur et al. (2013) and extends t to handle to multple prces. Theorem 3. Consder a set of m prces satsfyng 0 < r (1) <... < r (m), from whch α (1) and σ (1) are defned accordng to Proposton 1. Then there exsts a dstrbuton over nstances I (a randomzed nstance ) wth m = m and r (1) = r (1),..., r (m) = r (m) for each tem, on whch no onlne algorthm can have expected revenue greater than (1 e α(1) )E[OPT(I)]. Furthermore, for every nstance I n the support of the dstrbuton: 1. the startng nventores k can be made arbtrarly large; 2. I falls under the determnstc case of our problem. Theorem 3 s proven n Secton 5. It mples that no onlne algorthm can acheve a compettve rato greater than 1 e α(1), va Yao s mnmax prncple (Yao 1977). The counterexample can be made to satsfy the condtons of both Corollary 1 and Theorem 2, showng that these results are tght. In our counterexample, a large number of customers arrve accordng to a random permutaton, lke n Karp et al. (1990), Mehta et al. (2007), Golrezae et al. (2014). In our case, the customers are further splt nto m phases, where the customers n phase j are wllng to pay r (j) for any of the tems they are nterested n. The lengths of the phases are optmzed by an adversary to mnmze the compettve rato. Interestngly, on the related counterexamples from the lterature (Karp et al. 1990, Mehta et al. 2007, Ball and Queyranne 2009, Golrezae et al. 2014), all (reasonable) algorthms have the same performance. On our counterexample, wth the adversarally-optmzed phase lengths, the unque optmal algorthm turns out to be our two algorthms. We say unque because Mult-prce Balance and Mult-prce Rankng converge toward the same algorthm as the startng nventores go to ; ths phenomenon has also been noted n the sngle-prce case by Aggarwal et al. (2011).

16 16 Proposton 2. For m 2 prces satsfyng 0 < r (1) <... < r (m), from whch α (1) and σ (1) are defned accordng to Proposton 1, the followng nequaltes hold: (1 1 e ) σ(1) < 1 e σ(1) < 1 e α(1) ; (8) ln r(m) r (1) < σ (1) ; (9) 1 e α < 1 e α(1), where α s the unque soluton to 1 e α = 1 α ln r(m) r (1). (10) Fnally, Proposton 2, whch s proven n Appendx A, puts our tght compettve rato of 1 e α(1) nto perspectve. σ (1) s the exstng tght compettve rato for a sngle tem, whle 1 1 e s the exstng tght compettve rato for multple tems wth one prce each. (8) shows that our compettve rato for multple tems wth multple prces s not a nave combnaton of the exstng compettve ratos, and hence our algorthms cannot be obtaned by combnng exstng algorthms. Wth a sngle tem whose prce can take any value n the contnuum [r (1), r (m) ], the tght compettve rato s 1 1+ln(r (m) /r (1) ) (Ball and Queyranne 2009). 1 e α, wth α as defned n (10), s our correspondng compettve rato when there are multple tems (see Appendx E.1). (9) and (10) say that when the prces vary wthn a dscrete subset of [r (1), r (m) ], the compettve ratos can only be greater. (9) combned wth (8) shows that our compettve rato of 1 e α(1) s Ω( 1 log(r (m) /r (1) ) ). 3. MULTI-PRICE BALANCE and the Proof of Theorem 1 Mult-prce Balance, as sketched n Subsecton 2.2, s formalzed n Algorthm 1. For now, we consder a generc randomzed procedure for ntalzng L (0),..., L (m ) and Φ n Step 1, whch determnstcally satsfes the followng monotoncty condtons: L (0),..., L (m ) {0, 1 (0),..., 1}, 0 = L... k L (m ) = 1; (11) Φ (0), Φ ( 1 k ),..., Φ (1) R, 0 = Φ (0) Φ ( 1 k )... Φ (1). (12) Snce Φ s non-decreasng, the expresson Φ ( L (j) ) Φ ( N k ) n (13) s non-postve once the number sold N reaches k. Therefore, Algorthm 1 never tres to offer an tem whch has stocked out.

17 Algorthm 1 Mult-prce Balance L (0) 1: Intalze,..., L (m ), Φ randomly and ndependently for each [n] 2: N 0 for all [n] (N tracks the total number of copes of tem sold, at any prce) 3: for t = 1, 2,... do 4: Compute max p (j) [n],j [m ] t, ( Φ ( L (j) 17 ) Φ ( N k )) (13) 5: f the value of (13) s strctly postve then 6: Offer any tem t and prce jt maxmzng (13) to customer t 7: f customer t accepts (occurrng wth probablty p (j t ) t, ) then t 8: Z t Φ t ( L (j t ) ) Φ t t (N t /k t ) (ths s the pseudorevenue earned) 9: N t N t : end f 11: end f 12: end for Theorem 4. Suppose n Lne 1 of Algorthm 1, for each [n], the segment borders (1) L,..., L (m ) and value functon Φ are randomly ntalzed n a way such that k ( Φ ( N + 1 k ) Φ ( N k )) + Φ ( L (j) ) Φ ( N ) r(j) k F, j [m ], N {0,..., L (j) k 1}; (14) E[ Φ ( L (j) )] r (j), j [m ]. (15) Then Algorthm 1 acheves a compettve rato of F. Theorem 4 dentfes condtons whch, when satsfed by the randomzed procedure for each, yelds a compettve rato of F. Note that (14) needs to hold for every potental nstantaton of Φ, whle (15) only needs to hold n expectaton over the nstantatons. We prove Theorem 4 n Appendx B, but outlne ts proof here and provde some ntuton. Frst, we take the dual of the LP (3): n mn k y + T =1 t=1 z t (16a) p (j) t, y + z t p (j) t, r (j) t [T ], [n], j [m ] (16b) y, z t 0 [n], t [T ] (16c) By weak dualty, OPT(I) s bounded from above by the objectve value of any feasble dual soluton.

18 18 Durng the (random) executon of Algorthm 1, t mantans a dual varable y = Φ ( N k ) for each. At each tme t, only f a sale s realzed, does the algorthm set z t to a non-zero value Z t (Lne 8) and ncrement the y -varables by ncrementng N t (Lne 9). We prove three clams: 1. Durng each tme t [T ], the gan n the dual objectve s at most some multple 1 F of the revenue earned by the algorthm; 2. Durng each tme t [T ], the condtonal expectaton of Z t over the random purchase decson of customer t, combned wth the current value of y, make the LHS of (16b) at least p (j) t, Φ ( L (j) ), for all [n] and j [m ]; 3. The expectaton of Φ ( L (j) ), over the random segment borders and value functon ntally chosen by the algorthm, s at least r (j), for all [n] and j [m ]. Clam 1 follows from condton (14), whle Clam 3 follows from condton (15). Clams 2 and 3 can be combned to show that the dual varables y and z t mantaned by the algorthm are feasble, after takng an expectaton over all sample paths. We explan the ntuton behnd our dea of a random value functon, and the resultng analyss. Even for a sngle tem, wth a small startng nventory and a large rato r from ts hghest to lowest prce, n order to acheve a constant compettve rato whch does not scale wth r, one must use random bookng lmts (Ball and Queyranne 2009). Wth multple tems, our equvalent s to have the confguraton of segment borders (0) L,..., L (m ) be random, and defne an arbtrary value functon Φ correspondng to each one. In order to average over these confguratons n the analyss, we relax dual feasblty to only hold n expectaton. The dea of feasblty n expectaton has been prevously seen, but n dfferent contexts: n Devanur et al. (2013), over a random seed, and n Golrezae et al. (2014), over a random purchase decson (smlar to our Clam 2) Optmzng the Randomzed Procedures Theorem 4 reduces the problem of dervng a compettve algorthm to that of fndng a randomzed procedure for ntalzng Φ 1,..., Φ n satsfyng (14) (15). We can consder ths problem separately for each, based on r (1),..., r (m ) and k, and omt the subscrpt.

19 19 A randomzed procedure conssts of a dstrbuton over the all of the confguratons satsfyng (11), and for each confguraton, values for Φ( 1 ), Φ( 2 ),..., Φ(1) satsfyng (12). We would lke k k to fnd a randomzed procedure whch satsfes (14) (15) wth a maxmal value of F. Whle ths optmzaton problem s ntractable n general, we can use the ntuton behnd the defntons of L (0),..., L (m) and Φ from Subsecton 2.1 to specfy a near-optmal randomzed procedure. Defnton 2. Defne the followng randomzed procedure for ntalzng Φ: 1. Draw a random seed W unformly from [0, 1]; 2. For each j, set L (j) = L(j) k +1 k f W < L (j) k L (j) k, and L (j) = L(j) k k otherwse; 3. For q {0, 1 k,..., 1}, let l(q) be the unque j [m] such that L (j 1) q < L (j) (note that l( L (j) ) = j + 1 for j = 0,..., m 1; we defne l( L (m) ) to be m). The value functon Φ s then defned over q {0, 1,..., 1} by k Φ(q) = l(q) 1 j=1 (r (j) r (j 1) ) exp( L (j) L (j 1) ) 1 exp(α (j) ) 1 + (r ( l(q)) r ( l(q) 1) ) exp(q L ( l(q) 1) ) 1. (17) exp(α ( l(q)) ) 1 It s mportant that the random segment borders L (0),..., L (m) are rounded comonotoncally (n a perfectly postvely correlated fashon) usng a sngle seed, both to ensure that they satsfy the monotoncty condton n (11), and to reduce the number of potental confguratons on whch (14) needs to hold. Φ ncreases over the m (possbly empty) segments of ts doman {0, 1 k,..., 1}, whch are bordered by L (0),..., L (m). (17) s smlar to defnton (7) for Φ, except the sum n (17) does not telescope, snce L (j) L (j 1) equals α (j) only n expectaton. Theorem 5. The randomzed procedure for ntalzng Φ from Defnton 2 satsfes (14) (15) wth F = 1 e α(1) 1 e. Furthermore, f m = 1, then the value of F can be mproved to (1+k)(e 1/k 1) α (1). (1+k)(1 e 1/k ) Theorem 5 s proven n Appendx B. It, n conjuncton wth Theorem 4, establshes bounds () and () from our man result for Mult-prce Balance, Theorem 1. In Appendx B, we state the complete proof of Theorem 1, ncludng bound (), whch nvolves explctly formulatng the optmzaton problem over randomzed procedures and solvng t when k = 1.

20 20 Algorthm 2 Mult-prce Rankng n the Determnstc Case 1: Intalze W unformly at random from [0, 1], ndependently for each [n] 2: avalable true for all [n] 3: for t = 1, 2,... do 4: Compute max (r (j t,) Φ (W )) (18) [n],j [m ]:avalable =true 5: f the value of (18) s strctly postve then 6: Offer any tem t maxmzng (18) to customer t, at prce j t, t 7: avalable t false 8: end f 9: end for 4. MULTI-PRICE RANKING and the Proof of Theorem 2 In Subsecton 2.2, we sketched Mult-prce Rankng, for our general problem. In Algorthm 2, we formalze t n the determnstc case, whch s the case analyzed n Theorem 2. Recall that we have assumed, wthout loss of generalty, that k = 1 for each tem. Our analyss extends the framework of Devanur et al. (2013) to ncorporate multple prces. It uses the dual LP defned n (16), where every p (j) t, s 0 or 1. If Algorthm 2 assgns tem to customer t (chargng prce j t, ), then we set dual varables Z t = r (j t,) Φ (W ) and Y = Φ (W ), where Φ s the fxed functon defned n Subsecton 2.1 (we gnore the measure-zero set where Φ s undefned). All dual varables not set durng a tme perod are defned to be zero. The followng lemmas are proven n Appendx C: Lemma 2. If Algorthm 2 assgns tem to customer t, then (1 e α(1) )(Y + Z t ) r (j t,) w.p.1. Lemma 3. Settng y = E[Y ], z t = E[Z t ] for all, t forms a feasble soluton to the dual LP (16). The proof of Theorem 2 s straght-forward gven these lemmas: Proof of Theorem 2. Lemma 3 mples OPT(I) n =1 E[Y ] + T t=1 E[Z t], va weak dualty. However, by Lemma 2, the revenue earned by Algorthm 2, or ALG, s at least mn [n] {1 e α(1) } ( n Y =1 + T Z ) t=1 t, wth probablty 1. Thus, E[ALG] (1 exp( mn [n] α (1) )) OPT(I). 5. Randomzed Instance and the Proof of Theorem 3 We formalze the randomzed nstance descrbed n Subsecton 2.3 and use t to prove Theorem 3.

21 The n N tems all have m = m, r (j) = r (j) for all j, and k = k for some k N. We thnk of n as gong to, whle k s arbtrary. Throughout ths example, we often express quanttes as portons τ of n. We abuse notaton and wrte τn to refer to an nteger, even f τ s rratonal, snce the error from roundng τn to the nearest nteger s neglgble as n. The arrval sequence s randomzed followng the classcal constructon of Karp et al. (1990). There are T = nk customers, splt nto n groups of k dentcal customers each. Unformly draw a random permutaton π = (π 1,..., π n ) of (1,..., n) from the n! possbltes. For [n], all k customers n group would determnstcally buy any tem n {π,..., π n }. Our constructon dffers 21 from exstng ones n that the n groups of customers are further splt nto m phases. Let β 1,..., β m be postve numbers summng to 1, correspondng to the fracton of groups n each phase, whose values we specfy later. For all j [m], the customers n groups (β β j 1 )n + 1,..., (β β j )n are wllng to pay r (j) for any of the tems n ther nterest set. Defnton 3. Defne the followng shorthand notaton for all j = 1,..., m + 1: A j := m l=j α(l) (note that A 1 = 1 and A m+1 = 0); B j := m l=j β l (note that B 1 = 1 and B m+1 = 0). Proposton 3. Gven m N, 0 < r (1) <... < r (m), and α (1),..., α (m) as defned n Proposton 1, there exsts a unque soluton to the followng system of equatons n varables B 2,..., B m : B m r (m) e α(m) =... = B 2 r (2) e α(2) = r (1) e α(1), (19) wth 0 < B m <... < B 2 < B 1 = 1. We defne B 2,..., B m accordng to Proposton 3. Ths mples defntons for β 1,..., β m, whch are strctly postve and sum to 1. Now, regardless of the permutaton π, the optmal algorthm allocates the k copes of tem π to the customers n group, for each [n], successfully servng all T = nk customers and earnng revenue m j=1 r(j) (β j n)k. Ths s also the optmal objectve value of the LP (3). Therefore, OPT(I) = m j=1 r(j) (β j n)k determnstcally, whch we can rewrte as m (r (j) r (j 1) )B j nk. (20) j=1

22 Upper Bound on Performance of Onlne Algorthms Lemma 4. The expected revenue of an onlne algorthm on ths randomzed nstance s upperbounded by the maxmum value of subject to 0 λ j ln B j B j+1 m r (j) B j n(1 e λ j )k (21) j=1 for j [m 1], 0 λ m, and m j=1 λ j 1. Lemma 4 drastcally smplfes the analyss of the onlne algorthm, because t restrcts to algorthms whch are ndfferent to the realzed permutaton π, allowng for a determnstc analyss. However, our analyss dffers from exstng ones (e.g. (Golrezae et al. 2014, Lem. 6)) n that despte the tem symmetry, the onlne algorthm has a decson how many customers n each phase to serve, as opposed to reservng nventory for customers n future phases. Ths s controlled by the λ-varables, where λ j denotes the expected fracton of tem π n s nventory sold to phase-j customers. The expected number of groups served durng phase j s then at most B j n(1 e λ j), resultng n the upper bound (21). Constrant λ j ln B j B j+1 B j n(1 e λ j) must not exceed the total number of groups n phase j, β j n. comes from the fact that Lemma 5. Let j [m] and τ [0, 1]. The maxmum value of subject to λ l 0 for all l = j,..., m as well as m l=j λ l τ s nk m l=j m r (l) B l n(1 e λ l )k (22) l=j ( r (l) B l 1 exp ( α (l) + A j τ ) ). (23) m j + 1 Lemma 5 establshes the optmal objectve value of the optmzaton problem from Lemma 4. The upper bound of ln B j B j+1 the proof of Theorem 3 s easy. on λ j for j [m 1] turns out to not be bndng. Wth both lemmas, Proof of Theorem 3. The value of (23) wth j = 1 and τ = 1 s m m nk r (l) B l (1 e α(l) ) = (1 e α(1) ) (r (l) r (l 1) )B l nk, (24) l=1 l=1

23 23 where we have used (5) to derve the equalty. Combnng Lemmas 4 5, we get that the RHS of (24) s an upper bound on E[ALG(I)], for any onlne algorthm. Meanwhle, OPT(I) on ths randomzed nstance s always equal to (20), whch s exactly the RHS of (24) dvded by (1 e α(1) ). We have establshed that E[ALG(I)] (1 e α(1) )E[OPT(I)], whch s the desred result. Fnally, the second condton of Theorem 3 s clearly satsfed; the frst condton s also satsfed because our analyss holds for any value of k N, hence k can be made arbtrarly large. Remark 1. Suppose k. It can be seen that our algorthm (ether Mult-prce Balance or Mult-prce Rankng, whch behave dentcally on ths nstance see Aggarwal et al. (2011)), wth bookng lmts α (1),..., α (m), s the unque optmal algorthm on ths nstance. The proof of Lemma 4 shows that gven λ 1,..., λ m, the domnant strategy for the onlne algorthm s to deplete the nventores of tems evenly (whch s possble snce k ), n whch case upper bound (21) s attaned. The proof of Lemma 5 shows that the unque optmal values for λ 1,..., λ m are α (1),..., α (m). It only remans to show that λ j = α (j) s feasble, namely α (j) ln B j B j+1 for j < m. Applyng (19), ths s equvalent to showng e α(j) 1 e α(1) Extendng our Technques r(j) e α(j) r (j+1) e α(j+1), or e α(j+1) r(j) r (j+1), whch follows from (5) snce We explan how our technques can be extended to allow for fractonal nventory consumpton lke n the Adwords problem (Mehta et al. 2007), or offerng multple tems lke n the onlne assortment problem (Golrezae et al. 2014). The extenson to contnuous prce sets n deferred to Appendx E.1. Consder the followng modfcaton of our problem from Secton 2: when customer t s offered tem at prce j, she determnstcally pays p (j) t, r (j) and consumes a fractonal amount p (j) t, 1 of tem s nventory, nstead of payng r (j) and consumng 1 unt wth probablty p (j) t,. We assume that mn k. Ths generalzes the Adwords problem under the small bds assumpton, by allowng each budget to be depleted at m dfferent rates r (1),..., r (m ).

24 24 For ths problem, we use Mult-prce Balance, except snce we are takng mn k, we can determnstcally set each Φ = Φ. The three clams used to establsh Theorem 4 are smpler: Clam 2 now holds determnstcally nstead of requrng a condtonal expectaton over Z t, whle Clam 3 also holds determnstcally snce Φ s always Φ. In Theorem 5, condton (14) s now only satsfed under an addtonal error term ε, snce N s no longer a dscrete nteger. Nonetheless, the roundng error ε approaches 0 as k, so the optmal compettve rato s stll acheved. For onlne assortment, we use the term product to refer to an (tem, prce)-combnaton (, j). Consder the followng modfcaton of our problem from Secton 2: upon the arrval of customer t, for any subset (assortment) S of products and (, j) S, we are gven p (j) t, (S), the probablty that customer t would pck product (, j) when offered the choce from S. After beng gven these probabltes, we must offer an assortment S to customer t. Ths generalzes the orgnal onlne assortment problem, by allowng each tem to have multple feasble prces. The executon of an algorthm can be encapsulated by the followng modfcaton of the LP (3): max T x t (S) t=1 S T x t (S) t=1 S (,j) S j:(,j) S r (j) p (j) t, (S) (25a) p (j) t, (S) k [n] (25b) x t (S) = 1 t [T ] (25c) S x t (S) 0 t [T ], S {(, j) : [n], j [m ]} (25d) Mult-prce Balance can be drectly appled to ths problem, wth the change that t offers the assortment S maxmzng expected pseudorevenue, (,j) S p(j) t, (S) ( Φ ( L (j) ) Φ (w ) ), to each customer t. In the analyss, dual constrants (16b) now requre z t (,j) S p(j) t, (S)(r (j) y ) for all t and S, whch s stll mpled by the condtons of Theorem 4 so long as the choce probabltes for customers satsfy a mld ratonalty assumpton (see Golrezae et al. (2014) for detals).

25 25 7. Smulatons on Hotel Data Set of Bodea et al. (2009) We test our algorthms on the publcly-accessble hotel data set collected by Bodea et al. (2009). Based on the data, we consder a mult-prce onlne assortment problem, as defned n Secton 6. In general, we am to follow the expermental setup of Golrezae et al. (2014) Expermental Setup We consder Hotel 1 from the data set, whch has more transactons than the other four hotels. For each transacton, we use bookng to refer to the date the transacton occurred, and occupancy to refer to the dates the customer wll stay n the hotel. We consder occupances spannng the 5-week perod from Sunday, March 11th, 2007 to Sunday, Aprl 15th, Although the data contans occupances for a couple of weeks outsde ths range, such transactons are sparse. We merge the dfferent rooms nto 4 categores: Kng rooms, Queen rooms, Sutes, and Twodouble rooms. Rooms under the same category draw from the same nventory. We merge the dfferent fare classes nto two: dscounted advance-purchase fares and regular rack rates. We use product to refer to any of the 8 combnatons formed by the 4 room categores and 2 fares. We estmate a Multnomal Logt (MNL) choce model on these 8 products, for each of 8 customer types. The customer types are based on the bookng channel, party sze, and VIP status (f any) assocated wth a transacton. These types capture preference heterogenety (for example, party szes greater than 1 tend to prefer Sutes and Two-double rooms). The detals of our choce estmaton are deferred to Appendx F. We should pont out that more sophstcated segmentaton and estmaton technques have been employed on ths data set (van Ryzn and Vulcano 2014, Newman et al. 2014). Nonetheless, MNL has been reported to perform relatvely well (van Ryzn and Vulcano 2014, sec. 5.2). The MNL choce model s convenent for our purposes because under t, both the assortment optmzaton problem, as well as the choce-based LP (25) wth exponentally many varables, can be solved effcently (Tallur and Van Ryzn 2004, Lu and Van Ryzn 2008, Cheung and Smch-Lev 2016).

26 26 We treat each occupancy date as a separate nstance of the problem, for whch we defne a sequence of arrvals, wth one arrval for each transacton whch occupes that date. The choce probabltes for each arrval are determned by the customer type assocated wth the correspondng transacton. 3 The number of days n advance of occupancy that each arrval occurred s also recorded, but ths nformaton s only relevant for algorthms whch attempt to forecast the remanng number of arrvals based on the remanng length of tme. Before we proceed, we dscuss the lmtatons of our analyss and the data set: 1. In the data set, 55% of the transactons occupy multple, consecutve days. However, we treat such a transacton as a separate arrval n the nstances for each of those occupancy dates. Whle ths s a smplfyng assumpton, the focus of our paper s on the basc allocaton problem wthout complementarty effects across consecutve days, and our goal n usng the data set s to extract an arrval pattern over tme. 2. It s not possble to deduce from the data the fxed capacty for each category of room. Nonetheless, we consder a wde range of startng capactes n our tests. 3. Estmatng the number of customers who do not make a purchase s a standard challenge n choce modelng, whch s exacerbated n ths data set by the fact that the arrvals are rather non-statonary. We test varous assumptons on the weght of the no-purchase opton n the MNL model for each customer type. In general, we assume that ths weght s large, whch causes the revenue-maxmzng assortments to be large, allowng for tenson between offerng large assortments whch maxmze mmedate revenue, and offerng small assortments whch regulate nventory consumpton (detals n Appendx F) Instance Defnton An nstance conssts of a fxed capacty for each room category, correspondng to a specfc occupancy date. Each customer nterested n that occupancy date arrves n sequence, after whch her characterstcs (channel, party sze, VIP status) are revealed. The problem s to show a personalzed assortment of (room, fare)-optons to each customer. The nstances we test are defned below. 3 The choce realzed n that transacton was used for choce model estmaton, but s not used n defnng the arrval.