Rent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental

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1 Rnt, Las or Buy: Randomizd Algorithms for Multislop Ski Rntal Zvi Lotkr Dpt. of Comm. Systms Enginring Bn Gurion Univrsity Br Shva Isral Boaz Patt-Shamir Dror Rawitz {boaz, School of Elctrical Enginring Tl Aviv Univrsity Tl Aviv Isral May 3, 2010 Abstract In th Multislop Ski Rntal problm, th usr nds a crtain rsourc for som unknown priod of tim. To us th rsourc, th usr must subscrib to on of svral options, ach of which consists of a on-tim stup cost ( buying pric ), and cost proportional to th duration of th usag ( rntal rat ). Th largr th pric, th smallr th rnt. Th actual usag tim is dtrmind by an advrsary, and th goal of an algorithm is to minimiz th cost by choosing th bst altrnativ at any point in tim. Multislop Ski Rntal is a natural gnralization of th classical Ski Rntal problm (whr thr ar only two availabl altrnativs, namly pur rnt and pur buy), which is on of th fundamntal problms of onlin computation. Th Multislop Ski Rntal problm is an abstraction of many problms, whr onlin choics cannot b modld by just two altrnativs,.g., powr managmnt in systms which can b shut down in parts. In this papr w study randomizd algorithms for Multislop Ski Rntal. Our rsults includ an algorithm that producs th bst possibl onlin randomizd stratgy for any additiv instanc, whr th cost of switching from on altrnativ to anothr is th diffrnc in thir buying prics; and an -comptitiv randomizd stratgy for any (not ncssarily additiv) instanc. Kywords: onlin algorithms; comptitiv analysis; ski rntal; randomizd algorithms. An xtndd abstract was prsntd at th 25th Intrnational Symposium on Thortical Aspcts of Computr Scinc (STACS), Supportd in part by th Isral Scinc Foundation (grant 664/05) and by Isral Ministry of Scinc and Tchnology.

2 1 Introduction Arguably, th rnt or buy dilmma is th fundamntal problm in onlin algorithms: intuitivly, thr is an ongoing gam which may nd at any momnt, and th qustion is to commit or not to commit. Choosing to commit (th buy option) implis paying larg cost immdiatly, but low ovrall cost if th gam lasts for a long tim. Choosing not to commit (th rnt option) mans high spnding rat, but lowr ovrall cost if th gam nds quickly. This problm was first abstractd in th Ski Rntal formulation [11] as follows. In th buy option, a on-tim cost is incurrd, and thraftr usag is fr of charg. In th rnt option, th cost is proportional to usag tim, and thr is no on-tim cost. Th dtrministic solution is straightforward (with comptitiv ratio 2, i.., th cost paid by th algorithm is at most twic th bst possibl cost for th givn usag tim). In th randomizd modl, th algorithm chooss a random tim to switch from th rnt to th buy option (th advrsary is assumd to know th algorithm, but not th actual outcoms of random xprimnts). As is wll known, th bst possibl randomizd onlin stratgy for classical ski rntal has comptitiv ratio of (in xpctation). In many ralistic cass, thr may b som intrmdiat options btwn th xtrm altrnativs of pur buy and pur rnt: in gnral, it may b possibl to pay only a part of th buying cost and thn pay only partial rnt. Th gnral problm, calld hr th Multislop Ski Rntal problm, can b dscribd as follows. Thr ar svral stats (or slops), whr ach stat i is charactrizd by two numbrs: a buying cost b i and a rntal rat r i (s Figur 1). Without loss of gnrality, w may assum that for all i, b i < b i+1 and r i > r i+1, namly that aftr ordring th stats in incrasing buying costs, th rntal rats ar dcrasing. Th basic smantics of th multislop problm is natural: to hold th rsourc undr stat i for t tim units, th usr is chargd b i + r i t cost units. An advrsary gts to choos how long th gam will last, and th task is to minimiz total cost until th gam is ovr. Th Multislop Ski Rntal problm introducs ntirly nw difficultis whn compard to th classical Ski Rntal problm. Intuitivly, whras th only qustion in th classical vrsion is whn to buy, in th multislop vrsion w nd also to answr th qustion of what to buy. Anothr way to s th difficulty is that th numbr of potntial transitions from on slop to anothr in a stratgy is on lss than th numbr of slops, and finding a singl point of transition is qualitativly asir than finding mor than on such point. In addition, th possibility of multipl transitions forcs us to dfin th rlation btwn multipl buys. Following [1], w distinguish btwn two natural cass. In th additiv cas, buying costs ar cumulativ, namly to mov from stat i to stat j w only nd to pay th diffrnc in buying prics b j b i. In th non-additiv cas, thr is an arbitrarily dfind transition cost b ij for ach pair of stats i and j. Our rsults. In this papr w analyz randomizd stratgis for Multislop Ski Rntal. (W us th trm stratgy to rfr to th procdur that maks onlin dcisions, and th trm algorithm to rfr to th procdur that computs stratgis.) Our main focus is th additiv cas, and our main rsult is an fficint algorithm that computs th bst possibl randomizd onlin stratgy for any givn instanc of additiv Multislop Ski Rntal problm. Our algorithm is basd on a simpl charactrization of 1

3 cost b 4 b 3 b 2 b 1 s 1 s 2 s 3 s 4 gam duration tim Figur 1: A multislop ski rntal instanc with 5 slops: Th thick lin indicats th optimal cost as a function of th gam duration tim. optimal randomizd stratgis which w dvlop. W also giv a simplr stratgy which dcomposs a (k+1)-slop instanc into k two-slop instancs, whos comptitiv ratio is boundd by 1. W not that whil w ar not abl to giv a closd-form xprssion for th comptitiv ratio of an optimal algorithm, th bound for th simplr algorithm implis that th optimal comptitiv ratio is nvr wors than 1. For th non-additiv modl, w giv a simpl -comptitiv randomizd stratgy, improving on th bst known bound of about 2.88 [5]. Rlatd work. Variants of ski rntal (somtims calld th lasing problm [4]) ar implicit in many onlin problms. Th classical ski rntal problm has two slops, whr th buying cost of th first slop and th rntal rat of th scond slop ar both 0. Th problm was introducd in [11] with an optimal dtrministic stratgy achiving a comptitiv factor of 2. An optimal randomizd 1 -comptitiv stratgy was givn in [10]. Karlin t al. [9] apply th randomizd stratgy to TCP acknowldgmnt batching and othr problms. A strict gnralization of th two-slop cas, calld th bahncard problm was studid by Flischr [6]. In this variant, tim is discrt, and th rsourc nds or nds not to b usd in ach stp; th dilmma of th algorithm is whthr or not to buy a bahncard: having a valid bahncard rducs th usag cost whn th rsourc nds to b usd. In addition, a bahncard has an absolut xpiration tim. Flischr prsntd two optimal dtrministic stratgis for this problm. H also gav a randomizd stratgy that is optimal for th cas whr th bahncard has no xpiration tim. Not that this spcial cas of th bahncard problm is quivalnt to two-slop ski rntal without pur buy. Anothr solution for th continuous vrsion of two-slop ski rntal without pur buy is givn in [12]. Azar t al. [2] considr a problm that can b viwd as non-additiv multislop ski rntal whr slops bcom availabl ovr tim, and obtain an onlin stratgy whos comptitiv ratio is Bjrano t al. [3], motivatd by rrouting in ATM ntworks, study th non-additiv multislop problm. Thy giv a dtrministic 4-comptitiv stratgy, and show that th factor of 4 holds assuming only that th slops ar concav, i.., whn th rnt in a slop may dcras with tim. Damaschk [5] considrs a static vrsion of th problm from [2], namly non-additiv multislop ski 2

4 rntal problm whr ach slop is bought from scratch. 1 For dtrministic stratgis, [5] givs an uppr bound of 4 and a lowr bound of ; [5] also prsnts a randomizd stratgy whos comptitiv factor is 2/ln As far as w know, Damaschk s stratgy is th only randomizd stratgy for multislop ski rntal (additiv or not) to appar in th litratur. Irani t al. [8] prsnt a dtrministic 2-comptitiv stratgy for th additiv modl that gnralizs th stratgy for th two slops cas. Thy motivat thir work by nrgy saving: ach slop corrsponds to som partial slp mod of th systm. Augustin t al. [1] prsnt a dynamic program that computs th bst dtrministic stratgy for non-additiv multislop instancs. Th cas whr th lngth of th gam is a stochastic variabl with known distribution is also considrd in both [8, 1]. Myrson [14] dfins th smingly rlatd parking prmit problm, whr thr ar k typs of prmits of diffrnt costs, such that ach prmit allows usag for som duration of tim. Myrson s rsults indicat that th parking prmit problm is not vry closly rlatd to Multislop Ski Rntal, at last from th comptitiv analysis point of viw: it is shown in [14] that th comptitiv ratio of th parking prmit problm is Θ(k) and Θ(log k) for dtrministic and randomizd stratgis, rspctivly. Organization. Th rmaindr of this papr is organizd as follows. In Sction 2 w dfin th basic additiv modl and mak a fw prliminary obsrvations. In Sction 3 w giv a simpl algorithm to solv th multislop problm, and in Sction 4 w prsnt our main rsult: a charactrization of th optimal stratgis, from which w driv fficint algorithms computing thm. W also prsnt som xprimntal rsults for th thr slops cas in Sction 5. An -comptitiv algorithm for th non-additiv cas is prsntd in Sction 6. W conclud in Sction 7 with som opn problms. Th appndix contains som additional proofs. 2 Th Additiv Modl: Problm Statmnt and Prliminaris In this sction w formaliz th additiv vrsion of th multislop ski rntal problm (th non-additiv modl is addrssd in Sction 6). A k-ski rntal instanc is dfind by a st of k + 1 stats, and for ach stat i thr is a buying cost b i and a rnting cost r i. A stat can b rprsntd by a lin: th ith stat corrsponds to th lin y = b i + r i x. Figur 1 givs a gomtrical intrprtation of a multislop ski rntal instanc with fiv stats. W us th trms stat and slop intrchangably. Th rquirmnt of th problm is to spcify, for all tims t, which slop is chosn at tim t. W assum that stat transitions can b only forward, and that stats cannot b skippd, i.., th only transitions allowd ar of th typ i (i + 1). W strss that this assumption dos not rstrict gnrality in th additiv modl, whr a transition from stat i j for j > i + 1 is quivalnt to a squnc of instantanous transitions i (i+1)... j. It follows that a dtrministic stratgy for th additiv multislop ski rntal problm is a monoton non-dcrasing squnc (t 1,...,t k ), whr t i [0, ) corrsponds to th transition (i 1) i. Notic that t i is undfind if th transition (i 1) i is nvr prformd by th stratgy. A randomizd stratgy can b dscribd using a probability distribution ovr th family of dtrministic stratgis. Howvr, for our purposs it is mor convnint to dscrib randomizd stratgis 1 It can b shown that stratgis that work for this cas also work for th gnral non-additiv cas (s Sction 6). 3

5 using a mor gnral formalism. W spcify, for all tims t, a probability distribution ovr th st of k+1 slops. Th intuition is that this distribution dtrmins th actual cost paid by any onlin stratgy. Formally, a randomizd profil (or simply a profil) is spcifid by a vctor p(t) = (p 0 (t),...,p k (t)) of k+1 functions, whr p i (t) is th probability to b in stat i at tim t. Th corrctnss rquirmnt of a profil is k i=0 p i(t) = 1 for all t 0. Clarly, any stratgy is rlatd to som profil. In th squl w considr a spcific typ of profils for which a randomizd stratgy can b asily obtaind. Th prformanc of a profil is dfind by its total accrud cost, which consists of two parts as follows. Givn a randomizd profil p, th xpctd rntal rat of p at tim t is R p (t) df = i p i (t) r i, and th xpctd total rntal cost up to tim t is t 0 R p (z)dz. Th scond part of th cost is th buying cost. In this cas it is asir to dfin th cumulativ buying cost. Spcifically, th xpctd total buying cost up to tim t is B p (t) df = i p i (t) b i. Th xpctd total cost for p up to tim t is X p (t) df = B p (t) + t 0 R p (z)dz. Th goal of th algorithm is to minimiz total cost up to tim t for any givn t 0, with rspct to th bst possibl. Intuitivly, w think of a gam that may nd at any tim. For any possibl nding tim, w compar th total cost of th algorithm with th bst possibl (offlin) cost. To this nd, considr th optimal solution of a givn instanc. If th gam nds at tim t, th optimal solution is to slct th slop with th last cost at tim t (th thick lin in Figur 1 dnots th optimal cost for any givn t). Mor formally, th optimal offlin cost at tim t is opt(t) = min i (b i + r i t). For i > 0, dnot by s i th tim t satisfying b i 1 + r i 1 t = b i + r i t (i.., s i is th tim whr slops i 1 and i intrsct), and dfin s 0 = 0. It follows that th optimal slop for a gam nding at tim t is th slop i for which t [s i,s i+1 ] (if t = s i for som i thn both slops i 1 and i ar optimal). As customary, w shall b intrstd in th comptitiv ratio of a profil p, dfind to b sup {X p (t)/opt(t) t 0}. A profil (or stratgy) whos comptitiv ratio is c is said to b c- comptitiv. Finally, lt us rul out a fw trivial cass. First, not that if thr ar two slops such that b i b j and r i r j thn th cost incurrd by slop j is nvr lss than th cost incurrd slop i, and w may thrfor just rmov slop j from th instanc. Consquntly, w assum hncforth, without loss of gnrality, that th stats ar ordrd such that r i 1 > r i and b i 1 < b i for 1 i k. 4

6 Scond, using similar rasoning, w may considr only stratgis that ar monoton ovr tim with rspct to majorization [13], i.., stratgis such that for any two tims t t w hav j p i (t) i=0 j p i (t ). (1) i=0 Intuitivly, Eq. (1) mans that thr is no point is rolling back from an advancd slop to a prvious on: if at a givn tim w hav a crtain composition of th slops, thn at any latr tim th composition of slops may only improv. Not that Eq. (1) implis that B p is monoton non-dcrasing and R p is monoton non-incrasing, i.., ovr tim, a stratgy invsts non-ngativ amounts in buying, rsulting in non-incrasing rntal rats. 3 An 1-Comptitiv Stratgy In this sction, as a warm-up, w dscrib a simpl solution to th multislop problm via a rduction to th classical two-slop vrsion. Th advantag of this solution is that it is asy to prov that th rsulting randomizd stratgy has comptitiv factor at most 1. An algorithm finding th optimal stratgy is dscribd in Sction 4, but w do not know how to analyz its comptitiv factor dirctly. Th cas of pur buy option (r k = 0). Our stratgy is basd on th wll known 1 -comptitiv randomizd stratgy for th classical two slop ski rntal problm: Fact 1 ([10]). Th optimal randomizd stratgy in th classical two slop ski rntal problm is to switch from slop 0 to slop 1 at a tim t [0,b 1 /r 0 ] chosn by th probability dnsity function Th comptitiv ratio of this stratgy is 1. d dt p 1(t) = d t r 0/b 1 1 = r 0 t r0/b1 dt 1 b 1 1. Now, suppos w ar givn an instanc (b,r) with k + 1 slops, whr r k = 0 and k > 1. W dfin th following k instancs of th classical two-slops ski rntal problm: in instanc i for i {1,...,k}, w st instanc i: b i 0 = 0 and ri 0 = r i 1 r i ; b i 1 = b i b i 1 and r i 1 = 0. (2) An xampl is givn in Figur 2. Obsrv that b i 1 = ri 0 s i, i.., th two slops of th ith instanc intrsct xactly at s i, thir intrsction point at th original multislop instanc. Now, lt opt(t) dnot th optimal offlin solution to th original multislop instanc, and lt opt i (t) dnot th optimal solution of instanc i at tim t. Explicitly, opt i (t) = min{b i 1,ri 0 t}. W hav th following dcomposition lmma. Lmma 1. For all t 0, opt(t) = k i=1 opti (t). 5

7 cost t t cost cost t 0 + t 2 + 0t t t s 1 = 2 s 2 = 3 tim s 1 1 = 2 tim s 2 1 = 3 tim (a) 3-slop instanc. (b) Instanc 1: r 1 0 = 1, b 1 1 = 2. (c) Instanc 2: r 1 0 = 0.5, b 1 1 = 1.5. Figur 2: A multislop ski rntal instanc with thr slops, and th rsulting two classical ski rntal instancs. Proof. Considr a tim t and lt i(t) b th optimal multislop stat at tim t. Thn, opt i (t) = b i 1 + r0 i t i=1 i:s i t i:s i >t = (b i b i 1 ) + (r i 1 r i ) t i:s i t i:s i >t = b i(t) + r i(t) t = opt(t), and th lmma follows. Givn th dcomposition (2), it is asy to obtain a stratgy for any multislop instanc by pasting togthr th stratgis for k 2-slop instancs as follows. Lt p i dnot an 1-comptitiv profil for (th 2-slop) instanc i, as obtaind from Fact 1. Dfin th profil ˆp for th multislop instanc by ˆp 0 (t) = p 1 0 (t), ˆp i(t) = p i 1 (t) pi+1 1 (t) for 1 i < k, and ˆp k (t) = p k 1 (t). Th profil is wll dfind, as th following lmma shows. Lmma 2. For all t 0: (1) p i 1 (t) pi 1 1 (t) for 1 i k, and (2) k i=0 ˆp i(t) = 1. Proof. By Fact 1 w hav that th stratgy for th ith instanc is p i 1 (t) = (t ri 0 /bi 1 1)/( 1). Claim (1) of th lmma now follows from that fact that b i 1 /ri 0 = s i > s i 1 = b i 1 1 /r0 i 1 for vry i {1,...,k}. Claim (2) follows from th tlscopic sum k 1 ˆp i (t) = p 1 0 (t) + ( p i 1 (t) p i+1 1 (t) ) + p k 1 (t) = p1 0 (t) + p1 1 (t) = 1. i=0 i=1 Nxt, givn a profil ˆp, w convrt it into a stratgy as follows. Dfin ˆP i (t) df = j i ˆp j(t) and lt U b a uniform random variabl ovr [0,1]. Th stratgy is as follows: for vry stat i > 0, w mov 6

8 from stat i 1 to stat i whn U = ˆP i (t). Namly, th ith transition tim t i is th tim t such that U = ˆP i (t). Not that it is sufficint to us a singl random xprimnt in constructing, sinc our aim is th xpctd cost, and arbitrary dpndncis btwn th diffrnt p i ar allowd. Thus w obtain th following: Thorm 1. For vry tim t, th xpctd cost of th stratgy dfind by ˆp is at most 1 optimal cost. tims th Proof. W first show that by linarity, th xpctd cost of th combind profil is th sum of th costs of th two-slop profils, i.., that Xˆp (t) = k i=1 X pi(t). Th buying cost of th combind profil is: Bˆp (t) = k 1 ˆp i (t) b i = i=0 i=0 (p i 1 (t) pi+1 1 (t)) b i + p k 1 (t) b k = p i 1 (t) (b i b i 1 ) = B p i(t). i=1 i=1 Similarly, Rˆp (t) = k i=1 R pi(t) by linarity, and thrfor, Xˆp (t) = Bˆp (t) + t 0 Rˆp (z)dz = B p i(t) + i=1 t 0 ( ) R p i(z) dz = i=1 X p i(t). i=1 Finally, by Lmma 1 and th fact that th profils p 1,...,p k ar 1-comptitiv w conclud that Xˆp (t) = X p i(t) i=1 which mans that ˆp is 1 -comptitiv. i=1 1 opti (t) = 1 opt(t) Th cas of r k > 0. W not that if th smallst rntal rat r k is positiv, thn th comptitiv ratio is strictly lss that 1 : this can b sn by considring a nw instanc whr r k is subtractd from all rntal rats, i.., b i = b i and r i = r i r k for all 0 i k. Suppos p is 1-comptitiv with rspct to (r,k ) (not that r k = 0). Thn th comptitiv ratio of p at tim t with rspct to th original instanc is c(t) = X p(t) opt(t) = X p (t) + r k t opt (t) + r k t 1 opt (t) + r k t opt = (t) + r k t d Clarly, dt opt (t) = r i r k for t [s i 1,s i ). Hnc, th ratio opt (t) r k t c(t) is monoton dcrasing as wll. It follows that c r 0 r k r k + 1 = r k/r opt (t) r k t + 1. is monoton dcrasing, and thus Obsrv that c = 1 whn r k = 0, and that c = 1 whn r k = r 0 (i.., whn k = 0). W rmark that it can b shown [6, 12] that th xact comptitiv ratio of th gnral two-slop cas is in fact 1+r 1 /r 0. 7

9 4 An Optimal Onlin Algorithm In this sction w dvlop an optimal onlin stratgy for any givn additiv multislop ski rntal instanc. W rduc th st of all possibl stratgis to a subst of much simplr stratgis, which on on hand contains an optimal stratgy, and on th othr hand is asir to analyz, and in particular, allows us to ffctivly find such an optimal stratgy. Considr an arbitrary profil. Our first simplification is to confin ourslvs to continuous profils only. W justify this simplification in two stps: First, w rul out profils whr som p i s hav infinitly many discontinuitis. This allows us to avoid masur-thortic pathologis without xcluding any rasonabl solution within th Church-Turing computational modl. As w show in th appndix, this in fact mans that w may considr only continuous profils. So considr a continuous profil p = (p 0,...,p k ). W show that it can b transformd into a profil of a crtain structur without incrasing th comptitiv factor. Our chain of transformations is as follows. First, w show that it suffics to considr only simpl profils w call prudnt (Dfinition 1). Prudnt stratgis buy slops in ordr, on by on, without skipping and without buying mor than on slop at a tim. W thn dfin th concpt of tight profils (Dfinition 3), which ar prudnt profils that spnd mony at a fixd rat rlativ to th optimal offlin stratgy. As w prov, thr xists a tight optimal profil. This fact is th ky to dvloping an optimal stratgy, bcaus th bst tight profil can b ffctivly computd: Givn a constant c, w show how to dcid whthr thr xists a tight c-comptitiv stratgy, and this way, using binary sarch on c, w can find th bst tight stratgy. Finally, w xplain how to construct that profil and its corrsponding stratgy. 4.1 Prudnt Profils Our main simplification stp is to show that it is sufficint to considr only profils that buy slops conscutivly on by on. This mans that, at any tim t, at most two slops hav positiv probability. Formally, prudnt profils ar dfind as follows. Dfinition 1 (activ slops, prudnt profils). A slop i is activ at tim t if p i (t) > 0. A continuous profil is calld prudnt if at all tims thr is ithr on or two conscutiv activ slops, i.., if thr xists i {0,...,k} such that p i (t) = 0 for all i i,i + 1. At any givn tim t, at last on slop is activ bcaus i p i(t) = 1 by th dfinition. Considring Eq. (1) as wll, w s that a continuous prudnt profil progrsss from on slop to nxt without skipping any slop in btwn: onc slop i is fully paid for (i.., p i (t) = 1), th algorithm will start buying slop i + 1. W now prov that th st of continuous prudnt profils contains an optimal profil. Intuitivly, th ida is that a non-prudnt profil must hav two non-conscutiv slops with positiv probability at som tim. In this cas w can shift som probability toward a middl slop and only improv th ovrall cost. Thorm 2. If thr xists a continuous c-comptitiv profil p for som c 1, thn thr xists a prudnt c-comptitiv profil p. 8

10 Proof. Lt p = (p 0,...,p k ) b a profil and suppos that all th p i s ar continuous. It follows that B p is also continuous. Dfin bst(t) nxt(t) df = max {i : b i B p (t)}, df = min {i : b i B p (t)}. In words, bst(t) is th highst indx of a slop that is fully within th buying xpnss of p at tim t, and nxt(t) is th highst indx of a slop that is at last partially within th buying xpnss of p at tim t. Obviously, bst(t) nxt(t) bst(t) + 1 for all t. Now, w dfin p as follows: b nxt(t) B p(t) b nxt(t) b bst(t) bst(t) < nxt(t) and i = bst(t), B p(t) b bst(t) p i (t) = b nxt(t) b bst(t) bst(t) < nxt(t) and i = nxt(t), 1 bst(t) = nxt(t) = i, 0 othrwis. Clarly i p i(t) = 1 for vry tim t 0. Furthrmor, p is continuous and thrfor prudnt, bcaus B p is continuous. W now show that th comptitiv factor of p is at most c by proving that B p (t) = B p (t) and R p (t) R p (t) for all t. First, dirctly from dfinitions w hav if nxt(t) > bst(t) thn B p (t) = p bst(t) (t) b bst(t) + p nxt(t) (t) b nxt(t) = b nxt(t) B p (t) b nxt(t) b bst(t) b bst(t) + B p(t) b bst(t) b nxt(t) b bst(t) b nxt(t) = B p (t). If nxt(t) = bst(t) thn obviously B p (t) = B p (t). Considr now rntal paymnts. To prov that R p (t) R p (t) for vry tim t w construct inductivly a squnc of probability distributions p 0,...,p l, whr p 0 = p and p l = p. Suppos w hav constructd p j for j 0. If p j is prudnt w ar don. Othrwis, dfin distribution p j+1 as follows. For any t such that thr ar two non-conscutiv slops with positiv probability, lt i 1 (t),i 2 (t),i 3 (t) b thr slops such that i 1 (t) = min{i : p j i (t) > 0}, i 3(t) = max{i : p j i (t) > 0}, and i 2(t) = i 1 (t) Not that i 2 (t) < i 3 (t) (bcaus p j is not prudnt). Dfin p j i (t) j (t) b i2 (t) b i1 (t) i = i 1 (t), p j+1 i (t) = p j i (t) + p j i (t) p j i (t) j (t) b i2 (t) b i1 (t) + j (t) b i3 (t) b i2 (t) j (t) b i3 (t) b i2 (t) i = i 2 (t), i = i 3 (t), i {i 1(t),i 2 (t),i 3 (t)} whr j (t) > 0 is maximizd so that p j+1 i (t) 0 for all i. Intuitivly, w shift a maximal amount of probability mass from slops i 1 (t) and i 3 (t) to th middl slop i 2 (t). Th fact that j (t) is maximizd 2 In fact, i 2(t) may tak any valu btwn i 1(t) and i 3(t). 9

11 mans that w hav ithr that p j+1 i 1 (t) = 0, or p j+1 i 3 (t) = 0, or both hold. Sinc i 3 (t) i 1 (t) dcrass in ach inductiv stp, w may alrady conclud that th numbr of stps rquird, l, is lss than k. Also not that by construction, for all t w hav B p j+1(t) = i pj+1 i (t) b i = i pj i (t) b i = B p j(t). W now prov that th rntal cost improvs ovr th squnc. Fix a tim t, and an indx j in th squnc. Thn w hav j ( (t) j (t) j ) (t) j (t) R p j(t) R p j+1(t) = r i1 (t) r b i2 (t) b i2 (t) + + r i1 (t) b i2 (t) b i1 (t) b i3 (t) b i3 (t) i2 (t) b i3 (t) b i2 (t) ( = j ri1 (t) r i2 (t) (t) b i2 (t) b i1 (t) > 0, r i 2 (t) r i3 (t) b i3 (t) b i2 (t) whr th last inquality follows from th fact that if i < j, thn b j b i r i r j is th x coordinat of th intrsction point btwn th slops i and j. Hnc th rntal cost strictly improvs ovr th squnc for any t, and w ar don. ) 4.2 Tight Profils Our nxt stp is to considr profils that invst in buying as much as possibl undr som spnding rat cap. Our approach is motivatd by th following intuitiv obsrvation. Obsrvation 1. Lt p 1 and p 2 b two randomizd prudnt profils. If B p 1(t) B p 2(t) for vry t, thn R p 1(t) R p 2(t) for vry t. Intuitivly, Obsrvation 1 implis that invsting availabl funds in buying as soon as possibl rsults in lowr rnt, which in turn rsults in mor availabl funds. Nxt, obsrv that any c-comptitiv profil may stop invsting funds in buying if its currnt rnt is c r k, sinc th lowst possibl rnt is r k. This ida is formalizd in th following dfinition. Dfinition 2. Lt p b a c-comptitiv prudnt profil. Th closing tim of th stratgy p, dnotd τp, is dfind by τp = min({t : R p (t) = c r k } { }). Th closing tim τ p is wll dfind for a prudnt profil p, sinc p is continuous. 3 Motivatd by Obsrvation 1, w dfin a class of profils that spnd as much as possibl and as soon as possibl on buying until raching th closing tim. Dfinition 3. Lt c 1. A prudnt c-comptitiv profil p is calld tight if X p (t) = c opt(t) for vry t < τ p. Not that a tight profil nds not invst additional funds in buying slops aftr its closing tim (but it may still do so, subjct to th comptitiv factor constraint). Also not that in th cas of r k = 0, a tight profil must b in stat k with probability 1 at tim s k. On th othr hand, if r k > 0, 3 Not that it may b th cas that τ p =, vn though lim t R p(t) = c r k. 10

12 thr may xist a tight profil p that nvr buys th last slop (but still, its xpctd spnding rat as t tnds to infinity is at most c r k ). It is asy to show that a tight profil can achiv any achivabl comptitiv factor, as provd in th following lmma. Lmma 3. If thr xists a c-comptitiv prudnt profil p for som c 1, thn thr xists a c- comptitiv tight profil p. Proof. Lt p b th profil that satisfis X p (t) = c opt(t) for all t with p k (t) < 1. Clarly, p is tight. W nd to show that p is fasibl, i.., that th rnt paid by p at tim t nvr xcds c d dt opt(t). Indd, Obsrvation 1 implis that R p(t) R p (t) for vry t, and th lmma follows from th assumption that R p (t) c d dt opt(t). Nxt, w obsrv that any c-comptitiv tight profil must lowr its xpctd rnt to c r k not latr than s k. This proprty is usd in th squl whn constructing tight profils. Obsrvation 2. For any c-comptitiv tight profil, τ p s k. Proof. Assum that τp > s k. Sinc p is tight it follows that X p (s k ) = c opt(s k ). Thrfor, if R p (s k ) > c r k, thn p is not c-comptitiv. 4.3 Constructing Optimal Onlin Stratgis W now us th rsults abov to construct an algorithm that producs th bst possibl onlin stratgy for th multislop problm. Th ida is to guss a comptitiv factor c, and thn try to construct a c-comptitiv tight profil. Givn a way to dcid whthr comptitiv factor c is fasibl, w can apply binary sarch to find th optimal comptitiv ratio c to within any dsird prcision. Th main qustions ar how to tst whthr a givn c is fasibl, and how to construct a profil for a givn c. W answr ths qustions togthr: givn c, w construct a tight c-comptitiv profil in a picwis fashion, until ithr w fail (bcaus c was too small) or until w can guarant succss. In th rmaindr of this sction w dscrib how to construct a tight profil p for a givn comptitiv factor c. W bgin with analyzing th way a tight profil may spnd mony. Considr th situation at som tim t such that t < τp. Lt j b th maximum indx such that s j t. Obsrv that j < k, sinc w hav that τp s k by Obsrvation 2. It follows that d dt opt(t) = r j. Thrfor, th spnding rat of a tight profil at tim t is c r j, and it may spnd at rat c r j until tim s j+1 (or until τp). Hnc, for t (s j,s j+1 ), w hav d dt B p(t) + R p (t) = c d dt opt(t) = c r j. (3) Sinc p is tight and thrfor prudnt, w also hav, assuming bst(t) = i and nxt(t) = i + 1 that B p (t) = p i (t)b i + p i+1 (t)b i+1, and R p (t) = p i (t)r i + p i+1 (t)r i+1. Plugging th abov quations into Eq. (3), w gt d dt p i(t)b i + d dt p i+1(t)b i+1 + p i (t)r i + p i+1 (t)r i+1 = c r j. 11

13 Algorithm 1 Fasibl(c, M): tru if th k-ski instanc M = (b,r) admits comptitiv factor c 1: Lt s i = bi bi 1 r i 1 r i for ach 1 i k 2: Boundary Condition p 1 (0) = 0 3: j 0; i 1 4: loop 5: Dfin p i (t) = c rj ri 1 r i r i 1 + Γ xp( ri 1 ri b i b i 1 t) 6: Try to solv for Γ using Boundary Condition 7: if no solution thn rturn fals possibl scap if not fasibl 8: y p i (s j ) 9: if y < 1 thn 10: Boundary Condition p i (s j ) = y 11: j j + 1 continu at th nxt intrval [s j, s j+1 ] 12: ls 13: Lt x b such that p i (x) = 1 14: Boundary Condition p i+1 (x) = 0 15: i i + 1 mov to nxt slop 16: nd if 17: if i > k or j k thn rturn tru w r don 18: nd loop Sinc p is prudnt, p i (t) = 1 p i+1 (t) and hnc d dt p i(t) = d dt p i+1(t). It follows that d dt p i+1(t) + p i+1 (t) ri+1 r i b i+1 b i = c r j r i b i+1 b i (4) A solution to a diffrntial quation of th form y (x) + αy(x) = β whr α and β ar constants is y = β α + Γ αx, whr Γ dpnds on som boundary condition. Hnc in our cas w conclud that p i+1 (t) = c r r j r i r i+1 i t b + Γ i+1 b i, (5) r i+1 r i and p i (t) = 1 p i+1 (t), whr th constant Γ is dtrmind by th boundary condition. Eq. (5) is our tool to construct p in a picwis itrativ fashion. W start constructing p from t = 0 using p 1 (t) = c r 0 r 0 r 1 r 0 + Γ r0 r1 b 1 b 0 t, and th boundary condition p 1 (0) = 0. W gt that Γ = r 0(c 1) r 0 r 1, i.., p 1 (t) = r 0(c 1) r 0 r 1 ( r0 r1 b 1 b 0 t 1), and this holds for all t min(s 1,t 1 ), whr t 1 is th solution to p 1 (t 1 ) = 1. In gnral, Eq. (4) rmains tru so long as thr is no chang in th spnding rat and in th slop th profil p is buying. Th spnding rat changs whn t crosss s j, and th profil starts buying slop i + 2 whn p i+1 (t) = 1. W can now dscrib our algorithm. Givn c, Algorithm Fasibl ithr constructs a c-comptitiv tight profil p, or dtrmins that such a profil dos not xist. It starts with th boundary condition 12

14 p 1 (0) = 0 and rvals th first part of th profil as shown abov. Thn, whn th spnding rat changs, or whn thr is a chang in bst(i), it movs to th nxt diffrntial quation with a nw boundary condition. Aftr at most 2k such itrations it ithr computs a c-comptitiv tight profil p or discovrs that such a profil is infasibl. (Not that th profil w construct may continu buying aftr its closing tim.) Sinc w can tst for succss using Algorithm Fasibl, w can apply binary sarch to find th optimal comptitiv ratio to any dsird prcision. W not that it is asy to construct a stratgy that corrsponds to any givn prudnt profil p, as dscribd in Sction 3. W conclud with th following thorm. Thorm 3. Givn any ε > 0 and an instanc of k-slop ski rntal, a (c + ε)-comptitiv randomizd stratgy can b computd in O(k log 1 ε ) tim, whr c is th bst possibl comptitiv ratio for any randomizd onlin stratgy for that instanc. W rmark that it can b shown that for tight profils, th closing tim is a monoton dcrasing function of th comptitiv factor, and thrfor, th smallst comptitiv factor of a tight stratgy is attaind whn th closing tim is xactly s k. (In this cas, th total cost of th stratgy is xactly c opt(t) for all tims t.) 5 Cas Study: 3-Slop Ski Rntal In this sction w prsnt a fw mpirical rsults for som instancs of 3-slop ski rntal. W show how th bst stratgis look lik in a fw typical cass, and plot th comptitiv ratio as a function of th slops. Th rsults appar to indicat that finding a closd-form solution vn for thr slops is a vry challnging task. 5.1 Exprimntal Rsults for 3-Slop Ski Rntal Considr instancs with thr slops. Thr ar ssntially thr intrsting scnarios as follows. Lt t 1 dnot th tim at which th optimal tight stratgy is in slop 1 with probability 1, namly p 1 (t 1 ) = 1. Th scnarios ar: (i) t 1 s 1, (ii) t 1 > s 1, and (iii) p 1 (t) < 1 for vry t. W considr on spcific instanc for ach of ths scnarios. W fix th first and scond slops in all instancs: b 0 = 0, r 0 = 2, r 1 = 0.5, and b 1 = 0.5. W also fix th intrsction points in all instancs: s 1 = 1/3 and s 2 = 1. Th only variabl is th third slop. r 2 = 0.1 and b 2 = 0.9. Th bst stratgy is dpictd in Figur 3a. In this xampl stat 1 is fully paid for at tim t < s 1 = 1/3, and hnc th randomizd stratgy starts to buy slop 2 vn bfor slop 1 turns to b th optimal offlin choic. Th comptitiv ratio in this xampl is approximatly r 2 = 0.3 and b 2 = 0.7. Th bst stratgy is dpictd in Figur 3b. In this cas stat 1 is fully paid for at tim t 0.663, i.., th first slop bcoms optimal (at s 1 ) bfor th bst randomizd stratgy starts buying slop 2. Th comptitiv ratio is in this cas is approximatly

15 1.0 Prob 1.0 Prob t (a) Exampl 1: r = (2,0.5, 0.1), b = (0,0.5, 0.9) t (b) Exampl 2: r = (2,0.5, 0.3), b = (0,0.5, 0.7). 1.0 Prob t (c) Exampl 3: r = (2,0.5, 0.45), b = (0, 0.5, 0.55). Figur 3: Exampls with thr slops: s 1 = 1/3 and s 2 = 1; th p i s do not chang aftr s 2. r 2 = 0.45 and b 2 = Th bst stratgy is dpictd in Figur 3c. Hr th optimal stratgy buys only a part of stat 1, and nvr rachs stat 2. In this cas th optimal off-lin stratgy bhavs similarly to th optimal stratgy in th cas of two slops. Sinc r 1 is vry clos to r 2, th optimal stratgy buys slop 1 until tim s 1, and thn vry slowly improvs its rnt by buying mor of slop 1. Th comptitiv ratio in this cas is approximatly W not that th difficultis posd by multislop ski rntal whn compard to th classical ski rntal problm ar dmonstratd by th abov thr xampls: whil thr is only on bst onlin bhavior in th cas of two slops, thr ar svral possibl bhaviors in th cas of thr slops. 5.2 Comptitiv Ratio In Figur 4 w prsnt a visualization of th comptitiv ratio as a function of th slops. Sinc thr ar many possibl paramtrs, w us thr-dimnsional figurs, whr th variabls ar r 0 (x axis) and r 1 (y axis). Th third dimnsion corrsponds to th optimal comptitiv ratio, and it is rprsntd by th dpth of th shad. As abov, w assum that s 2 = 1 (i.., b 1 = 1 r 1 and b 2 = 1 r 2 ). In Figur 4a th rnt rats ar r 0 (1.1,4),r 1 (0.1,0.9), and r 2 = 0, whil in Figur 4b th rats ar r 0 (1,1.9),r 1 (0.4,0.97), and r 2 = 0.2. As xpctd, th comptitiv ratio in both figurs is always lss than

16 r 1 r r 0 (a) Comptitiv ratio for r 0 (1.1, 4), r 1 (0.1, 0.9), and r 2 = 0. (b) Comptitiv ratio for r 0 (0.4, 0.97), and r 2 = 0.2. r 0 (1,1.9), r 1 Figur 4: Optimal comptitiv ratio for two sts of instancs. 6 An -Comptitiv Stratgy for th Non-Additiv Cas In this sction w considr th non-additiv multislop ski rntal problm. W prsnt a simpl randomizd stratgy which improvs th bst known comptitiv ratio [5] from 2/ln to Our tchniqu is a simpl application of randomizd rpatd doubling that is usd xtnsivly in comptitiv analysis of onlin algorithms, but whos origin prdats th concpt of comptitiv algorithms (s,.g., [7]). Bfor prsnting th stratgy lt us considr th dtails of th non-additiv modl. Augustin t al. [1] dfind a gnral non-additiv modl in which a transition cost b ij is associatd with vry two stats i and j, and showd that on may assum without loss of gnrality that b ij = 0 if i > j and that b ij b j for vry i < j. Obsrv that w may furthr assum that b ij = b j for vry i and j, sinc th optimal (offlin) stratgy rmains unchangd. It follows that th stratgis from [2, 3, 5] that wr dsignd for th cas of buying slops from scratch also work for th gnral non-additiv cas. W propos using th following simpl onlin stratgy. Th ida may b asir to undrstand by comparison to th dtrministic stratgy [3]: If w knw that th optimal cost was x, w would hav known what slop to choos in th first plac. Sinc th onlin stratgy dosn t know x ahad of tim, it gusss first that x = opt(s 1 ); if that guss provs to b wrong (namly th gam did not nd by tim s 1 ), th guss of th optimal cost is updatd to x 2x, if that provs wrong th guss is doubld onc again tc. It is not hard to s that this way, th ovrall cost to th onlin stratgy is at most 4 tims th optimal cost, whr half of th worst-cas cost is attributd to th gam nding immdiatly aftr doubling th guss. In th randomizd stratgy, that componnt is mad smallr by choosing randomly th initial guss; th multiplication factor valu is optimizd so as to gt th bst xpctd comptitiv ratio. Spcifically, dfin B 1 = opt(s 1 )/α X, whr α > 1 is th multiplication factor w dtrmin latr, and X is a ral numbr chosn uniformly at random from [0,1) (B 1 will b th initial guss for th 15

17 optimal cost, namly w guss that th optimal stratgy is to b in th first slop). Our (j +1)st guss df is B j+1 = B 1 α j. Dfin, for j 1, τ j = opt 1 (B j ), i.., τ j is th tim at which th optimal off-lin cost is B j. Also dfin τ 0 = 0. In cas th last slop is flat (i.., r k = 0), dfin τ j = s k for all j such that B j > b k. Dfin, for j 1, i(j) to b th stat of th optimal stratgy if th gam nds at tim τ j (if τ j = s i for som i, thn th optimal stat is ithr i 1 or i; w dfin i(j) = i 1 in this cas). As hintd abov, th undrlying rational is that in th tim intrval [τ j 1,τ j ), w guss that th optimal cost is B j. This lads to th following concrt stratgy: At tim τ j 1, if th gam hasn t ndd yt, movs to slop i(j). Not that this mans that at tim 0, th stratgy buys slop 0, bcaus B 1 opt(s 1 ) and hnc i(1) = 0. Th ky to analyz th comptitiv factor of this stratgy is th following lmma. Lmma 4. Th xpctd cost of th stratgy dscribd abov is at most α lnα tims th optimum. Proof. If th gam nds bfor tim τ 1, thn th onlin stratgy is optimal, bcaus th stratgy uss slop 0 until tim τ 1 < s 1. So suppos that th gam nds at tim τ and lt l > 1 b such that τ [τ l 1,τ l ). Thn th xpctd cost of th onlin stratgy is at most l [ ] E opt(τ j ) α E α 1 opt(τ l) j=0 [ ] α 2 X = E α 1 opt(τ) = α α 1 1 x=0 α x dx opt(τ) = α ln α opt(τ). Th first inquality follows from th dfinition of τ j. Th following quality is basd on th obsrvation that th random variabl opt(τ l )/opt(τ) is idntical to th random variabl α 1 X. Thorm 4. Thr xists a randomizd stratgy for non-additiv multislop ski rntal whos cost on ach instanc is nvr mor than tims th bst possibl. Proof. Us th stratgy abov with α =. Lmma 4 yilds th rsult. 7 Conclusion W hav studid randomizd algorithms for th multislop ski rntal problm. W gav a charactrization of an optimal randomizd stratgy and an fficint algorithm for finding such a stratgy for any givn additiv instanc (i.., configuration of slops). By considring a sub-optimal stratgy, w hav shown that th comptitiv factor is nvr mor than th factor for th classical two-slops cas, namly 1. Obtaining a bttr bound on th comptitiv factor as a function of th slops paramtrs rmains an opn problm. For th non-additiv cas, w prsntd a simpl randomizd stratgy that guarants comptitiv ratio of. 16

18 Acknowldgmnt W thank Sffy Naor and Niv Buchbindr for stimulating discussions. Rfrncs [1] J. Augustin, S. Irani, and C. Swamy. Optimal powr-down stratgis. In 45th IEEE Symp. on Foundations of Computr Scinc, pags , [2] Y. Azar, Y. Bartal, E. Furstin, A. Fiat, S. Lonardi, and A. Rosén. On capital invstmnt. Algorithmica, 25(1):22 36, [3] Y. Bjrano, I. Cidon, and J. S. Naor. Dynamic sssion managmnt for static and mobil usrs: a comptitiv on-lin algorithmic approach. In 4th Intrnational Workshop on Discrt Algorithms and Mthods for Mobil Computing and Communications, pags ACM, [4] A. Borodin and R. El-Yaniv. Onlin Computation and Comptitiv Analysis. Cambridg Univrsity Prss, [5] P. Damaschk. Narly optimal stratgis for spcial cass of on-lin capital invstmnt. Thortical Computr Scinc, 302(1-3):35 44, [6] R. Flischr. On th bahncard problm. Thortical Computr Scinc, 268(1): , [7] S. Gal. Sarch Gams. Acadmic Prss, [8] S. Irani, R. K. Gupta, and S. K. Shukla. Comptitiv analysis of dynamic powr managmnt stratgis for systms with multipl powr savings stats. In Dsign, Automation and Tst in Europ Confrnc and Exhibition, pags , [9] A. R. Karlin, C. Knyon, and D. Randall. Dynamic TCP acknowldgmnt and othr storis about /( 1). Algorithmica, 36(3): , [10] A. R. Karlin, M. S. Manass, L. A. McGoch, and S. S. Owicki. Comptitiv randomizd algorithms for nonuniform problms. Algorithmica, 11(6): , [11] A. R. Karlin, M. S. Manass, L. Rudolph, and D. D. Slator. Comptitiv snoopy caching. Algorithmica, 3(1):77 119, [12] Z. Lotkr, B. Patt-Shamir, and D. Rawitz. Ski rntal with two gnral options. Information Procssing Lttrs, 108(6): , [13] A. W. Marshall and I. Olkin. Inqualitis: Thory of Majorization and Its Applications. Acadmic Prss, [14] A. Myrson. Th parking prmit problm. In 46th IEEE Symp. on Foundations of Computr Scinc, pags ,

19 APPENDIX: Arbitrary to Continuous Stratgis In this sction w show that if thr xists a c-comptitiv profil p with finitly many discontinuitis, thn thr xists a continuous c-comptitiv profil. 4 Th ida in th proof blow is as follows. Lt p b a non-continuous profil, and considr a point τ of discontinuity of B p (t). Sinc B p (t) is monoton non-dcrasing, it must b th cas that B p (t) jumps by som constant > 0 at tim τ. It follows that th instantanous comptitiv ratio, dfind by c(t) = X p (t)/opt(t), also jumps discontinuously at tim t = τ. W can cancl th discontinuity at τ by smaring this jump ovr an intrval of positiv lngth nding at τ, without incrasing th original comptitiv ratio. Spcifically, w prov th following. Thorm 5. If thr xists a c-comptitiv profil p that contains a finit st of discontinuitis for som c 1, thn thr xists a continuous c-comptitiv profil. Proof. As a first stp w dfin a nw profil p which is a right-continuous vrsion of p. For all τ 0, dfin p i (τ) = lim t τ + p(t) (lim t τ + mans lim t τ, t>τ). Clarly, p is a profil, sinc p i(τ) = i=0 lim p i(t) = lim p i (t) = 1. t τ + t τ + i=0 (Th scond quality holds sinc thr xists som δ > 0 such that p is continuous for t [t δ,t + δ] \ {t}.) Also notic that Eq. (1) continus to hold. Furthrmor, sinc p diffrs from p in only a finit numbr of points, p is also c-comptitiv. Dnot th st of points of discontinuity of p by T = {τ 1,τ 2,...,τ n }. For ach j {1,...,n}, dnot B p (τj ) = lim t τ B p (t) and R p (τj ) = lim j t τ R p (t). Both limits xist bcaus B p and j R p ar monoton and boundd, and by th discontinuity of p at τ j, w hav B p (τ j ) > B p (τj ) and R p (τ j ) < R p (τj ). Now, for ach j {1,...,n}, th continuity of p at all points xcpt T allows us to choos ε j small nough such that th following conditions hold: 1. [τ j ε j,τ j ) T =. (Th only discontinuity of p in th intrval is its right ndpoint.) 2. {s 1,...,s k } [τ j ε j,τ j ) =. (Th only possibl s i in th intrval is its right ndpoint.) 3. ε j B p (τ j) B p (τ ) c r R p (τ j ) > 1 2 (R p (τ j ε j ) + R p (τ j )). Using ths ε j s, dfin a nw profil p as follows (s Figur 5 for illustration): τ j t p i (t) = ε j p i (τ j ε j ) + t (τ j ε j ) ε j p i (τ j) t (τ j ε j,τ j ) for som j, p i (t) othrwis. 4 Not that th dtrministic 2-comptitiv stratgy for th multislop ski rntal problm from [8] can b viwd as a randomizd stratgy with k discontinuity points. i=0 18

20 B(t) τ j ε j τ j t Figur 5: Obtaining B p from B p. Solid lin rprsnt B p, dashd lin rprsnt changs in B p. p is a profil, sinc for vry tim t [τ j ε j,τ j ] w hav: i=0 ( p i (t) = τj t p i ε (τ j ε j ) + t (τ ) j ε j ) p i j ε (τ j) = τ j t + t (τ j ε j ) = 1. j ε j ε j i=0 and for any othr tim t w gt that k i=0 p i (t) = k i=0 p i (t) = 1. Morovr, p is continuous by construction. It rmains to analyz th comptitiv ratio of p. Dfin c (t) = X p (t)/opt(t). W show that c (t) c for vry t 0. W do this in two stps: first, w considr all tim points that wr not changd btwn p and p. Claim 1. c (t) c for all t / j [τ j ε j,τ j ). Proof. Dfin τ 0 = 0. W prov th claim by showing, using induction on j, that if t [τ j 1,τ j ε j ), thn X p (t) X p (t). Ths ar intrvals covring all points whos dfinition is idntical in p and p. For th bas cas j = 1, obsrv that th total buying cost at any point 0 t τ 1 ε 1 is idntical in p and p. Th rntal rats at ths points in p and p ar thrfor also idntical, and hnc th bas cas is provn. For th inductiv stp, assum that th claim holds for j, and considr t [τ j,τ j+1 ε j+1 ). By induction, X p (τ j ε j ) X p (τ j ε j ). Morovr, w claim that X p (τ j ) X p (τ j ): this is tru sinc B p (τ j ) = B p (τ j ) by construction, and sinc by dfinition of ε j, th rnt paid by p at any point in an intrval [τ j ε j,τ j ] is at last R p (τ j ), whil th avrag rnt paid by p in this intrval is 1 2 (R p (τ j ε j ) + R p (τ j )) < R p (τ j ). Thus X p (τ j) X p (τ j ). To complt th inductiv stp, not that p and p coincid in [τ j,τ j+1 ε j+1 ), and hnc X p (t) X p (t) for all t [τ j,τ j+1 ε j+1 ). Considr now t (τ j ε j,τ j ] for som j. Dnot j = B p (τ j ) B p (τj ). By dfinition, for p w hav d dt X p (t) = d dt (B p (t) + R p (t)) j ε j, and for th optimal stratgy w hav d dt opt(t) = r i for 19

21 som 0 i k. Thrfor, using th claim from abov, w obtain d dt c (t) = > = d dt X p (t) opt(t) X p (t) d dt opt(t) opt(t) 2 j ε j opt(t) X p (t) r i opt(t) 2 j ε j [opt(τ j ) (τ j t) r i ] r i [X p (τ j ) j ε j (τ j t)] 0. j ε j opt(τ j ) r i X p (τ j ) opt(t) 2 opt(t) 2 j ε j opt(τ j ) r i c opt(τ j ) opt(t) 2 Sinc c (t) has a positiv drivativ for all t (τ j ε j,τ j ], w hav that for c (t) c (τ j ), and by th claim abov, c (τ j ) c. 20

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