Algorithmica 2001 Springer-Verlag New York Inc.

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1 Algorithica 2001) 30: DOI: /s Algorithica 2001 Springer-Verlag New York Inc Optial Search and One-Way Trading Online Algoriths R El-Yaniv, 1 A Fiat, 2 R M Karp, 3 and G Turpin 4 Abstract This paper is concerned with the tie series search and one-way trading probles In the tie series) search proble a player is searching for the axiu or iniu) price in a sequence that unfolds sequentially, one price at a tie Once during this gae the player can decide to accept the current price p in which case the gae ends and the player s payoff is p Intheone-way trading proble a trader is given the task of trading dollars to yen Each day, a new exchange rate is announced and the trader ust decide how any dollars to convert to yen according to the current rate The gae ends when the trader trades his entire dollar wealth to yen and his payoff is the nuber of yen acquired The search and one-way trading are intiately related Any deterinistic or randoized) one-way trading algorith can be viewed as a randoized search algorith Using the copetitive ratio as a perforance easure we deterine the optial copetitive perforance for several variants of these probles In particular, we show that a siple threat-based strategy is optial and we deterine its copetitive ratio which yields, for realistic values of the proble paraeters, surprisingly low copetitive ratios We also consider and analyze a one-way trading gae played against an adversary called Nature where the online player knows the probability distribution of the axiu exchange rate and that distribution has been chosen by Nature Finally, we consider soe applications for a special case of portfolio selection called two-way trading in which the trader ay trade back and forth between cash and one asset Key Words Tie series search, One-way trading, Two-way trading, Portfolio selection, Online algoriths, Copetitive analysis 1 Introduction 11 Tie Series Search Consider the following eleentary search proble A player is searching for the axiu resp iniu) price of soe asset At each tie period i 1, 2,,n the player obtains a price quotation p i and ust decide whether or not to accept this price Once the player decides to accept soe price p j the gae ends and the player s payoff resp cost) is p j The horizon n ay or ay not be known to the player and if the player has not decided to accept a price during the first n 1 periods he ust accept soe iniu resp axiu) price In essence, in this eleentary search 1 Departent of Coputer Science, Technion Israel Institute of Technology, Haifa 32000, Israel techionacil R El-Yaniv is a Marcella S Geltan Acadeic Lecturer 2 Departent of Coputer Science, Tel Aviv University, Raat Aviv 69978, Israel 3 International Coputer Science Institute, 1947 Center St, Berkeley, CA 94704, USA, and Departent of Electrical Engineering and Coputer Sciences, University of California, Berkeley, CA 94720, USA 4 Departent of Coputer Science, University of Toronto, Toronto, Ontario, Canada M5S 1A4 torontoedu Received October 19, 1998; revised August 12, 1999 Counicated by Ming-Yang Kao Online publication February 1, 2001

2 102 R El-Yaniv, A Fiat, R M Karp, and G Turpin proble the player is confronting the decision proble of when he has acquired sufficient price sapling so that the accepted price is satisfying given what he already knows The eleentary search proble has any variations and extensions If the horizon n is known resp unknown) to the player, then the proble is tered with known duration resp with unknown duration) In the case of unknown duration the player is infored just before the last period so that he can accept the last offered price Throughout this paper we assue a finite known or unknown) duration A search proble is with recall if soe nuber of the ost recent price offers are retained and at any period the player can choose to accept one of the retained offers In another natural extension of the eleentary search proble the player has to pay a sapling cost c i to obtain the ith price quotation In this case the total payoff or return) of a player that accepts the price p j is p j c i 1 i j Applications of the tie series search decision task are required and perfored by virtually all econoic agents and institutions Therefore, search is a ost fundaental feature of econoic arkets Consider the following straightforward applications Job and eployee search In the job search the player is seeking eployent In each period the job seeker obtains one job offer which ay be viewed as the lifetie earning fro the job A sapling cost ay be associated with the generation of each offer and ay include advertising costs, transportation costs, and perhaps the loss incurred fro being currently uneployed Now consider an eployer searching for an eployee We assue that the eployer can test and quantify the suitability of candidates to the job offered Such quantities correspond to price quotations The sapling cost in this case ay be attributed to advertiseent, qualification tests, etc 5 Search for the lowest price of goods Here the player needs to buy soe goods that are sold at different stores at different prices The player can obtain price quotations after identifying the relevant sellers Here sapling fees ay account for traveling or phone costs and for the tie wasted Like job and eployee search this application is of fundaental iportance because optial search strategies deterine the deand function which in turn deterines the nature of the arket itself 12 One-Way Trading Consider a trader who needs to exchange soe initial wealth w 0, given in soe currency say, dollars), to soe other asset or currency say, yen) At the start of each trading period the trader obtains the current price quotation and ust decide whether to accept it or wait for a better price Typically, the cost of sapling a price quotation is negligible as prices are widely available in quotation services Nevertheless, the trader ay be required to pay soe transaction fees for the exchange eg, to a financial institution) 6 Thus stated, the one-way trading proble is a direct application 5 This eployee search is closely related to the well known secretary proble [16], [1] where typically the objective is to accept one or ore secretaries of best ordinal value aong an ordered set of all secretaries 6 These transaction fees can be a function of the aount traded ie, dollars spent) or a function of the price which is equivalent to the aount of yen received) Alternatively or in addition), the fees can include a fixed cost

3 Optial Search and One-Way Trading Online Algoriths 103 of the tie series search proble Nevertheless, the trader can partition the initial wealth w 0 and exchange it sequentially in parts, each part in a different exchange rate Clearly, this ore general trading schee ay result in higher returns As we shall later see, trading in parts is equivalent to a randoized search One-way trading algoriths can be applied in various econoic situations For instance, consider a fund anager who decides to change the position of soe portfolio and enter or exit) soe arket In this case w 0 is the part of his wealth allocated to the new position Another natural application arises when an individual, for the purpose of eigrating to a foreign country, sells local property in order to exchange the local currency received to the foreign currency 13 Previous Work Related to Tie Series Search The search proble has received considerable attention in atheatical econoics and operations research Traditionally, a Bayesian approach has been eployed: An optial search strategy is sought under the assuption that the prior distribution of prices is given and stationary In order to allow for analytical results it is usually further assued that price quotations are independent observations of this distribution Although the stationarity and independence assuptions are soewhat siplistic, this basic paradig has given rise to a rich theory that relies on tools fro the theory of optial stopping [11] It is beyond the scope of this paper to survey the Bayesian work related to search 7 Not surprisingly, the Bayesian approach derives search algoriths that are heavily dependent on the price prior distribution Nevertheless, one striking feature of Bayesian optial search algoriths applicable to any proble variants) is that they have the following structure Based upon the proble paraeters, and, in particular, the assued prior distribution, there is a single fixed critical nuber called the reservation price such that the optial policy is to reject all prices below the reservation price and to accept any offer above it 8 The least acceptable assuption of classical Bayesian search odels is that the probability distribution of prices is fully known to the player Several odels attept to relax this assuption For exaple, Rosenfield and Shapiro [28] studied cases where the price distribution is itself a rando variable distributed according to probability law with soe known oents eg, the price distribution is known to be noral with ean distributed according to soe other probability law) In this paper we attept to circuvent alost all distributional assuptions by resorting to copetitive analysis defined in Section 14) Our only assuption is that the price generating process has finite known or unknown) support 14 Copetitive Analysis of Online Probles Let P I, F, U) be a profit axiization proble where I is a set of possible inputs; for each I I, FI ) is the set of feasible outputs; U is a utility function such that for all I and O FI ), UI, O) R Consider any algorith ALG for the proble P Given any input I, ALG coputes a feasible output O FI ) We denote the profit or return) of ALG on the input instance 7 The reader is referred to the excellent surveys by Lippan and McCall that discuss Bayesian solutions for a variety of search proble variants [23], [24] 8 In the case of search with known duration the reservation price can change dynaically

4 104 R El-Yaniv, A Fiat, R M Karp, and G Turpin I by ALGI ) UI, O) Typically, each input can be represented as a finite sequence I i 1, i 2,,i n and a feasible output O can also be represented as finite sequence O o 1, o 2,,o n An algorith ALG coputes online if for each j 1,,n 1, ALG ust copute o j before i j+1 is given An algorith is offline if it can produce a feasible output given the entire input sequence We denote an optial offline algorith by OPT By definition, for each input sequence I the return of OPT is OPTI ) sup O FI ) UI, O) An online algorith is c-copetitive if for any I I, 1) ALGI ) 1 OPTI ) c In this case we also say that ALG attains a copetitive ratio c The least copetitive ratio that ALG attains is called the copetitive ratio of ALG The copetitive ratio is thus a worst-case perforance easure Any c-copetitive online algorith is guaranteed to return at least a fraction 1/c of the optial offline profit no atter how unfortunate or erratic the future will be The use of the copetitive ratio to easure perforance of online algoriths is called copetitive analysis 9 Copetitive analysis has been used in the coputer science literature for over 20 years At first it was used iplicitly for the online approxiation of NP-hard probles see, eg, [17], [19], [20], and [30]) Soewhat later, the seinal work of Sleator and Tarjan [29] on virtual eory and dictionary anageent put forth the use of the copetitive ratio as a general perforance easure for online decision aking Soeties it is convenient to view the copetitive analysis of an online) proble as a two-person gae between the online player and an adversary The online player chooses an online algorith ALG and infors the adversary of his choice The adversary then chooses an input sequence I The payoff to the adversary is the resulting perforance ratio OPTI )/ALGI ) and the payoff to the online player is inus this quantity ie, the gae is zero-su) As is generally the case with two-person zero-su gaes, a randoized strategy is required to obtain optial expected) copetitive perforance Extending the definition of the copetitive ratio to randoized algoriths is straightforward We siply substitute in 1) E[ALGI )] for ALGI ) where the expectation is taken with respect to the distributions used by ALG Since in the gae corresponding to this definition the adversary is ignorant of the outcoes of the rando choices ade by the online algorith this adversary is called oblivious 10 Indeed, it is often the case that randoization against an oblivious adversary) draatically iproves the copetitive perforance see the classical results of [8] and [15] regarding etrical task systes and virtual eory anageent) As we shall later see randoization epowers the online player also in the search proble 15 Copetitive Analysis: A Discussion The ain attraction in using the copetitive ratio for analyzing online algoriths is that there is no need to rely on statistical odeling of input sequences Indeed, it is often extreely difficult to devise realistic statistical 9 The ter copetitive ratio was tered by Karlin et al [21] 10 Other kinds of adversaries that generate input sequences adaptively based on the outcoes of the online rando choices) have been considered see [5])

5 Optial Search and One-Way Trading Online Algoriths 105 odels for possible inputs which are always highly dependent on the particular application) This difficulty is often ore extree in coplex dynaical environents such as econoical systes Strategic financial decision aking is therefore a very attractive doain for copetitive analyses Moreover, in financial decision aking it is often desirable to secure soe inial sure profit rather than expecting higher average profits while being exposed to severe risks Essentially, this is what copetitive analysis offers Nevertheless, this risk aversion property of the copetitive ratio is quite often a drawback since this perforance easure can lead to overly defensive algoriths Indeed, whenever decision akers do have soe side inforation or partial statistical) knowledge on the evolution of input sequences it would be a terrible waste to ignore it, which is precisely what the copetitive ratio does Recently, al-binali [2] generalized the pure copetitive analysis so that it can utilize predictions in the for of partial knowledge on future input sequences) while retaining the natural risk aversion of the copetitive ratio In fact, this generalized copetitive analysis fraework allows for trading-off the associated risk with the potential reward 11 This generalized) copetitive analysis offers a robust yet flexible copleentary fraework to the analysis of financial) decision aking under uncertainty An additional advantage of the copetitive ratio is that it offers a unified easure of perforance under which any two strategies are coparable in soe fundaental sense 16 Paper Organization This paper is organized as follows In Section 2 we study the relationship between randoized search and one-way trading In Section 3 we present a siple deterinistic search algorith of a reservation price policy type, which is optial for deterinistic) search We then show how randoization over the reservation price policies can draatically iprove the copetitive ratio In Section 4 we present a general threat-based policy for one-way trading or search) This general policy yields optial algoriths for a nuber of variants of the one-way trading proble We provide in this section detailed analyses of four proble variants In Section 5 we give a few nuerical exaples of the copetitive ratios obtained in the previous sections, showing that for realistic values of the proble paraeters we can obtain surprisingly low ratios In Section 6 we study a proble variant called a gae against Nature We define Nature as an adversary that chooses the probability distribution of the axiu exchange rate the axiu rate is the optial offline return) Although at the outset it sees that Nature is weaker than our ordinary adversary we prove that with respect to one-way trading) Nature and the ordinary adversary are equivalent in a sense to be ade precise later) In Section 7 we apply our results for the one-way trading gae to a two-way trading gae in which the player can trade currencies back and forth Lastly, in Section 8 we suarize our conclusions and indicate soe directions for future work 2 Randoized Search and One-way Trading Notice that in the search proble the online player ust accept one price and in the one-way trading proble the trader 11 To deonstrate the utility of this new technique al-binali has used soe of the results in this paper based on a preliinary version of our paper) showing how the copetitive perforance of a one-way trader can be boosted when he has the knowledge of useful predictions

6 106 R El-Yaniv, A Fiat, R M Karp, and G Turpin can partition his initial wealth and trade the parts sequentially, each part at a different exchange rate Nevertheless, the search and one-way trading probles are closely related Any deterinistic or randoized) one-way trading algorith that trades the initial wealth in parts can be interpreted as a randoized search algorith and vice versa This follows fro the fact that any randoized) one-way trading algorith is equivalent to a randoized trading algorith that trades the entire wealth at once at soe randoly chosen period) Further, any randoized trading algorith that trades the entire wealth at once is equivalent to a deterinistic algorith that trades the initial wealth in parts Forally we have THEOREM 1 i) Let ALG 1 be any randoized one-way trading algorith Then there exists a deterinistic one-way trading algorith ALG 2 such that for any exchange rate sequence σ, E[ALG 1 σ )] ALG 2 σ ) The reverse stateent is also true: ii) Let ALG 2 be any deterinistic one-way trading algorith Then there exists a randoized algorith ALG 1 that akes only a single trade such that E[ALG 1 σ )] ALG 2 σ ) holds for all σ PROOF Let ALG 1 be any randoized one-way trading algorith In particular and using gae-theoretic terinology), ALG 1 ay be a ixed strategy a distribution over deterinistic algoriths) or a behavioral strategy whose daily transactions are chosen randoly) Since in this online gae between the online player and the adversary) the online player has perfect recall she has no eory restrictions), we know that the set of behavioral strategies is a subset of the set of ixed strategies see [3]) Thus, in any case we can assue without loss of generality that ALG 1 is a ixed algorith, which is a probability distribution {wa)} over A, the set of all deterinistic algoriths For any sequence of prices σ p 1, p 2,,p n, the expected return of ALG 1 is E w [ALG 1 σ )] A aσ ) dwa) With respect to σ consider a deterinistic algorith ALG 2 that on period i spends a fraction A si, a)dwa) of its initial wealth where si, a) is the aount spent by the deterinistic algorith a on period i Thus the return of ALG 2 is ALG 2 σ ) n p i i1 A i1 A A si, a) dwa) n si, a)p i dwa) aσ ) dws) E w [ALG 1 σ )] To prove part ii) consider a deterinistic algorith that trades a fraction s i of its initial wealth at the ith period, i s i 1 Now consider a randoized algorith ALG 1 that with probability s i trades its entire wealth at the ith period Clearly, the expected return of ALG 1 equals the return of the deterinistic algorith ALG 2 COROLLARY 2 It follows that a copetitive) optial deterinistic one-way trading algorith has the sae return as an optial randoized search algorith This iplies that randoization cannot iprove the copetitive perforance in one-way trading

7 Optial Search and One-Way Trading Online Algoriths 107 Although randoization cannot help in one-way trading we will see that randoization is advantageous for search As noted earlier, both in the search and one-way trading proble the online player ay be required to pay a sapling cost and/or transaction cost) for each price quotation and/or dollar traded) In this paper we consider sipler proble variants where there are no such costs Also we ake the assuptions that arbitrary fractions of oney units can be traded REMARK 1 In general, the nature of the sapling/transaction fees will affect the copetitive ratio We state without proofs the following observations i) The copetitive ratio of any one-way trading algorith is independent of transaction costs deterined by a fixed percentage applied to the aount spent In this case the equivalence of Theore 1 obviously holds ii) When we introduce transaction fees which are a fixed percentage applied to the prices, the copetitive ratio will iprove but Theore 1 still holds iii) When fixed transaction costs are introduced the deterinistic copetitive ratio increases and there is no longer an equivalence between deterinistic one-way trading algoriths and randoized search algoriths 3 Copetitive Search Algoriths Throughout this paper we assue that prices exchange rates) are chosen by an adversary) fro the real interval [, M] where 0 < M We define the axiu fluctuation ratio of possible prices to be ϕ M/ Copetitive ratios of algoriths will be deterined in ters of ϕ The paraeters, M,orϕ ay or ay not be known to the online player For a start, suppose that both and M are known to the online player In this case the optial deterinistic search strategy is the following reservation price policy RPP): Accept the first price greater than or equal to p M We call p the reservation price Clearly, the optial reservation price should balance the return ratios offline/online) resulting by the following two events: i) the axiu price encountered, p ax,is p in which case the worst-case return ratio is M/p ; ii) p ax < p in which case the worst-case return ratio is p ax / Therefore, the optial reservation price p is the solution of M/p p/ REMARK 2 It is possible to show that if only the fluctuation ratio ϕ is known but not or M), then no better ratio than the trivial one of ϕ is achievable One can draatically iprove the copetitive ratio by using randoized algoriths We now introduce siple randoized algoriths of exponential threshold type that achieve exponentially better copetitive ratios The basic idea of these algoriths is due to Levin [22] Assue, for siplicity, that ϕ 2 k for soe integer k Fori 0, 1,,k 1, let RPPi) be the deterinistic) reservation price policy with reservation price 2 i Define EXPO to be a unifor probability ixture over the set {RPPi)} k 1 i0 That is, before the start of the gae EXPO chooses one of the RPPi) strategies, each with probability 1/k

8 108 R El-Yaniv, A Fiat, R M Karp, and G Turpin THEOREM 3 Levin) Let ϕ 2 k for soe integer k Algorith EXPO is cϕ) log ϕ)- copetitive with cϕ) approaching 1 when ϕ PROOF Let p ax be the posterior axiu price obtained Let j be an integer satisfying 2 j p ax < 2 j+1 The particular choice of p ax and, therefore, of j) is controlled by the adversary Note that j k and since all prices are in [, M], j k if and only if p ax M It will later be ade clear that the choice p ax M and, thus, of j k) is not optial for the adversary so we can assue that j k 1 For any choice of j algorith EXPO will return on average k j + ) 2 i k k 2 j+1 + k j 2) 1 i j Denote by R j) the return ratio, offline to online, obtained for a particular j Since for any choice of j it is optial for the adversary to choose p ax arbitrarily close to 2 j+1, we have 2 j+1 R j) k 2 j+1 + k j 2 It is not hard to see that the real-valued function R j) obtains its axiu at j k 2 + 1/ln 2 and it follows that the coefficient of k in R j) is alost 1 resulting in a copetitive ratio that is greater than but approaching k log ϕ as ϕ and k grow) REMARK 3 Theore 3 can be extended in two ways First, siilar result can be obtained when ϕ is not a power of 2 Second, exactly the sae bound holds if the player does not know and M but only knows ϕ Algorith EXPO can be odified to work even without knowledge of ϕ Let µ {qi)} i0 be any probability distribution over the natural nubers Consider algorith EXPO µ that acts as follows After p 1, the first price, is revealed to the online player algorith EXPO µ chooses the reservation price p 12 i with probability qi) LEMMA 1 Algorith EXPO µ is 2/q log ϕ )-copetitive against an oblivious adversary, where ϕ is the posterior global fluctuation ratio PROOF Let p ax be the axiu price obtained and assue that p 1 2 j p ax < p 1 2 j+1 for soe integer j In the worst case OPT s return is less than, but arbitrarily close to, p 1 2 j+1 and algorith EXPO µ returns at least q j)p 12 j on average It follows that the expected copetitive ratio of EXPO µ is not saller than 2/q j) By definition, ϕ p ax /p 1 and therefore j log ϕ<j + 1, so log ϕ j and the proof is coplete In order to apply Lea 1 and derive sall copetitive ratios for algorith EXPO µ we need to construct an appropriate probability distribution µ Sall copetitive ratios are obtained by taking appropriate converging infinite sus For exaple, we can use

9 Optial Search and One-Way Trading Online Algoriths 109 the Rieann zeta function ζx) i1 1/i x ) Specifically, for every positive ε the infinite su i1 1/i1+ε converges to the constant ζ1 + ε) It follows that µ ε {1/ζ1 + ε)i + 1) 1+ε } i0 is a probability distribution over the natural nubers Hence by Lea 1, EXPO µ ε attains a copetitive ratio of 2ζ1 + ε) j + 1) 1+ε Olog 1+ε ϕ)) with ϕ being the posterior global fluctuation ratio However, we can do even better Consider the infinite su i 1/i log1+ε i) Lea 1 yields the copetitive ratio of 2) O logϕ) log 1+ε log ϕ) ) Notice however that as ε decreases the constant in the big-o increases Hence, the particular choice of the distribution µ ε can only be optiized if soe bounds on ϕ are known REMARK 4 It is possible to generalize the upper bound given by 2) and achieve an upper bound of O logϕ) log 1+ε log log log ϕ) } {{ } k for every integer k 4 Optial Threat-Based Policy for One-Way Trading As can be seen in the previous section the copetitive ratio Olog ϕ) is attainable by the siple EXPO algorith under inor assuptions on possible prices ie, the global fluctuation ratio is known) It turns out that the ratio Olog ϕ) is within a constant factor of the best possible Nevertheless, to obtain an optial copetitive ratio, soewhat ore involved algoriths and analyses are required The optial algoriths are best described as deterinistic) one-way trading algoriths and we focus on the one-way trading proble for the reainder of the paper The optial perforance is obtained by algoriths that obey the following threatbased policy Let c be any copetitive ratio that can be attained by soe one-way trading algorith For a start, assue that c is known to the trader For each such c the corresponding threat-based policy consists of the following two rules RULE 1 so far Consider trading dollars to yen only when the current rate is the highest seen RULE 2 Whenever you convert dollars, convert just enough to ensure that a copetitive ratio c would be obtained if an adversary dropped the exchange rate to the iniu possible rate 12 and kept it there throughout the gae Note that these two rules apply to all but the last trading day when, by the rules of the gae, the trader ust trade all reaining dollars to yen 12 The iniu possible rate is defined with respect to the inforation known to the trader That is, it is if is known, and it is p/ϕ if only ϕ is known and p is highest price seen so far

10 110 R El-Yaniv, A Fiat, R M Karp, and G Turpin At the outset it is not clear how to follow such a policy, in particular, how to follow Rule 2 that requires trading a quantity that equals the aount of just enough dollars in order to ensure a copetitive ratio of c For now assue that it is possible to copute the quantities prescribed by Rule 2 and assue an algorith that follows this policy Such an algorith converts dollars to yen based on the threat that the exchange rate will drop peranently to the iniu possible rate For each attainable copetitive ratio c the corresponding threat-based algorith can be shown to be c-copetitive This can be intuitively justified as follows Consider the first trade exchange rate is p 1 ) Since the current exchange rate is the highest seen so far the algorith considers a trade Since the copetitive ratio c is attainable by soe deterinistic trading algorith, there exists soe s 0 such that the ratio c will still be attainable if s dollars are traded to yen Further, the chosen aount of dollars s is such that the ratio c is so far guaranteed even if there will be a peranent drop of the exchange rate and no further trades will be conducted except for one last trade converting the reaining dollars with the iniu possible exchange rate) In particular, there is no need to consider any exchange rate which is saller than p 1 Siilar arguents can be used to justify the choice of the aounts for the rest of the trades and thus intuitively this policy induces a c-copetitive algorith A foral analysis follows REMARK 5 Denote the iniu possible exchange rate by Notice that as long as the exchange rates are not larger than c, the threat-based trader should not trade any dollars to yen except of course for the last day, when the trader ust trade all reaining dollars to yen) This follows fro Rule 2 because a copetitive ratio of c is always attainable when the axiu rate is c even if all dollars are exchanged at rate We now develop soe basic properties of threat-based trading These properties will facilitate analyses of the threat-based policy for the following variants of the one-way trading gae Variant 1: Known duration ie, n is known) with and M known Variant 2: Unknown duration with and M known Variant 3: Known duration and known ϕ Variant 4: Unknown duration and known ϕ For each of the above variants, we identify the corresponding optial threat-based online algorith and deterine its copetitive ratio Fix any two positive reals and M with < M, and any integer n > 1 An exchange rate sequence, σ, is an eleent of [, M] n Thus, σ p 1, p 2,,p n, and for each 1 i n, p i [, M] is called the exchange rate of the ith day giving the nuber of yen that can be traded for one dollar on that day Let n) denote the set of all such exchange rate sequences A deterinistic one-way trading algorith is a function, D: n) {1, 2,,n} [0, 1], satisfying the following properties: D is nonincreasing; for each σ n), Dσ, 0) 1, and Dσ, n) 0 Dσ, i) is defined to be the nuber of reaining dollars just after the ith day when the algorith trades its dollars in accordance with σ To easure the perforance of D we

11 Optial Search and One-Way Trading Online Algoriths 111 ake use of soe ore definitions For each σ n), let 3) s i def Dσ, i 1) Dσ, i), i 1,,n s i is called the ith transaction and it thus gives the aount of dollars traded to yen on the ith day by the algorith DGivenD ) and the s i, we define Y D as follows: Y D σ, 0) def 0, i Y D σ, i) def s j p j, j1 i 1,,n Thus, Y D σ, i) gives the nuber of accuulated yen just after the ith transaction has been perfored Clearly, Y D is nondecreasing and the onotonicity of D and Y D ) corresponds to the one-way requireent The return of D on the sequence σ, denoted by P D σ ), is defined to be the quantity Y D σ, n) Anonline one-way trading algorith is a one-way trading algorith D such that the ith transaction is solely based on past and present exchange rates, and the paraeters known to the online player Assue that the online player knows that there are at ost n trading days This corresponds to Variant 1 or 3 13 Consider the threat-based strategy Rule 1 requires that once a transaction has been ade at soe exchange rate, further transactions will be ade only at higher exchange rates; rates that are the sae or lower will be ignored Hence, both OPT and the threat-based algorith conduct transactions only when the exchange rate sequence reaches a new high Therefore, in a worst-case analysis of the perforance of the threat-based algorith, we ay assue that the exchange rate sequence consists of an initial segent of successive axia of length k n In addition, we can assue that the first rate p 1 is larger than the iniu possible rate ties r where r is the target copetitive ratio of the threat-based algorith see Reark 5) That is, iniu possible rate) r p 1 < p 2 < < p k axiu possible rate) However, in order to realize a threat, the adversary ay choose k < n and then take p k+1 p k+2 p n iniu possible rate) Given a proble variant 1 or 3), let ALG be the optial threat-based algorith for that variant The algorith ALG will be associated with the function D ), as defined above, and throughout this section we abbreviate Dσ, i) resp Y σ, i))tod i resp Y i ) Recall that the player starts with D 0 1 dollars and Y 0 0 yen Also, recall that s i as defined in 3)) denotes the ith transaction Specifically, s i D i 1 D i It is easy to see that p i s i Y i Y i 1 Let r be the target copetitive ratio that ALG is trying to achieve LEMMA 2 If ALG is an r-copetitive threat-based algorith then for every i 1, 4) s i p i r Y i 1 + D i 1 iniu possible rate)) r p i iniu possible rate)) 13 The results for Variants 2 and 4 will be then derived fro the results for Variants 1 and 3, respectively, by taking the liit n

12 112 R El-Yaniv, A Fiat, R M Karp, and G Turpin and 5) PROOF 6) Y i + D i iniu possible rate) p i r Since ALG is r-copetitive, by Rule 2 it ust be that p i Y i + D i iniu possible rate) r Here the denoinator represents the return of ALG if an adversary dropped the exchange rate to the iniu possible rate, and the nuerator is the return of OPT for such an exchange rate sequence That is, 7) p i Y i 1 + s i p i ) + D i 1 s i ) iniu possible rate) r Since by Rule 2) ALG ust spend the inial s i that satisfies inequality 7) and since the left-hand side in 7) is decreasing with s i we ust replace the inequality with equality: p i Y i 1 + s i p i ) + D i 1 s i ) iniu possible rate) r Equation 4) is obtained by solving this equality for s i Equation 5) is obtained by replacing the inequality in 6) with equality and rearranging 41 Analysis of Variant 1: Known Duration with, M Known To begin our analysis for Variant 1, we ake use of 4) and 5) specializing the to the case in which the iniu possible rate is as assued for Variant 1 This yields the following expressions for the daily transactions: 8) and for i > 1, s 1 1 r p 1 r p 1, s i 1 r pi r Y i 1 + D i 1 ) 9) p i Then, by 5) at i 1 instead of i we have Y i 1 + D i 1 p i 1 /r, so we obtain, for i > 1, 10) s i 1 r pi p i 1 p i In the above forulas for the daily transactions, r is the target copetitive ratio that the algorith is attepting to achieve Clearly the algorith cannot attain an arbitrarily sall r For exaple, if r 1 we see that s 1 1 forula 8)) and the algorith will spend its entire wealth on the first rate thus failing to achieve a copetitive ratio of 1 with any continuation of the exchange rate sequence that increases above the first rate) Hence, we can obtain fro these forulas r-copetitive threat-based) algoriths only by using sufficiently large values r Our goal now is to identify the sallest achievable copetitive ratio

13 Optial Search and One-Way Trading Online Algoriths 113 Let σ be an exchange rate sequence and let r > 1 be any real We say that the threatbased algorith A r, as defined by forulas 8) and 10) applied with r,isr-proper with respect to σ if i) the su of daily transactions coputed by A r, when the exchange rate sequence is σ, is not larger than 1 the initial wealth); and ii) the resulting ratio of optial offline return over online return A r σ ) with respect to σ is not larger than r LEMMA 3 Let σ be any exchange rate sequence If A r is r-proper with respect to σ, then for any r r, A r is r -proper PROOF As noted, and without loss of generality, suppose that σ p 1,,p k,,, with < p 1 < p 2 < < p k For each day i, the daily dollar transactions s i as calculated by A r are not larger than the respective aounts s i as calculated by A r Specifically, for i 1 we have, using 8), s 1 s 1 p 1 1 p 1 r 1 ) 0 r Siilarly, fro 10) we have for i > 1, s i s i p i p i 1 p i 1 r 1 ) 0 r Therefore, i s i i s i, and since A r is r-proper, i s i 1 and, therefore, i s i 1 By the definition of the threat-based algorith and since the copetitive ratio r is attainable, for every day i k, A r chooses a transaction that guarantees a copetitive ratio of r even in a case of a peranent drop to Therefore, A r is r -proper with respect to σ Let σ p 1,,p k,,, be an input sequence For any k, we want the daily transactions to satisfy k i1 s i 1 Suppose for the oent that k is known to the online player In this case, the optial copetitive ratio for a threat-based algorith ust be deterined such that there will be no dollars reaining after the last purchase on day k) In other words, the optial copetitive ratio of the threat-based strategy with k known) has the property that k s i 1 i1 Substituting for the s i fro 8) and 10) we obtain 11) 1 1 r After solving 11) for r we write p 1 r p r k i2 p i p i 1 p i 12) r r k) p 1, p 2,,p k ) def 1 + p 1 k p i p i 1 p 1 p i i2

14 114 R El-Yaniv, A Fiat, R M Karp, and G Turpin As shown in the following lea, we can deterine an attainable copetitive ratio of threat-based algorith in an n-day gae by axiizing r k) p 1,,p k ) over all choices of k n and p 1 < p 2 < < p k M Define 13) r n, M) sup r k) p 1, p 2,,p k ) k n, p 1 <<p k M LEMMA 4 Let σ be any exchange rate sequence Then the threat-based algorith A rn,m) is r n, M)-proper with respect to σ PROOF As usual, suppose that σ p 1,,p k,,, Let r r k) p 1, p 2,, p k ) By construction, the threat-based algorith A r is r-proper for σ Since r n, M) r, by Lea 3 A rn,m) is r n, M)-proper with respect to σ By Lea 4, r n, M) is an achievable copetitive ratio for the proble The rest of this section is devoted to calculating r n, M) Our analysis proceeds as follows: we first fix k, p 1, and p k, and axiize over {p i } k 1 i2 Then we axiize over p k, next over p 1, and, lastly, over k Without loss of generality we assue that k > 1 since for the choice k 1, a copetitive ratio of 1 is trivially achieved The sequence of following leas lead to the evaluation of 13) LEMMA 5 For fixed k > 1, p 1 and p k, k k 1) 1 ax p 1 <p 2 < <p k 1 <p k i2 p i p i 1 p i and the axiu is obtained when for every 2 i k, ) ) p1 1/k 1) p k 14) p i p i 1 p i 1 ) p1 1/k 1) p k PROOF For each i, set x i p i Hence, k i2 p i p i 1 p i k i2 k 1 x i x i 1 x i k However, by the geoetric-arithetic ean inequality, x 1 /x 2 + x 2 /x 3 + +x k 1 /x k ) k 1 ax p i, 2 i k 1 i2 i2 x i 1 x i ) 1/k 1) x1 x2 xk 1 x 2 x 3 x k x1 x k ) 1/k 1), and equality is obtained if and only if all the ters in the left-hand side are equal Hence, k p i p i 1 ) k x i 1 p i x i k 1) in x i, 2 i k 1 i1

15 Optial Search and One-Way Trading Online Algoriths 115 k 1) 1 k 1) 1 x1 x k ) 1/k 1) ) p1 p k ) ) 1/k 1) Fro Lea 5 we iediately obtain sup r k) p 1, p 2,,p k ) sup r k) p 1, p k ), k n, k n, p 1 < <p k M p 1 <p k M where r k) p 1, p k ) def 1 + p 1 k 1) 1 p 1 ) ) p1 1/k 1) p k It is readily seen that r k) p 1, p k ) is increasing with p k Therefore, it is axiized when p k takes its axiu possible value, M, and sup p1,p k rk)p 1, p k ) reduces to sup p1 r k) p 1 ) where 15) r k) p 1 ) def 1 + p 1 k 1) 1 p 1 Abbreviate ax p1 r k) p 1 ) by r k) ) ) p1 1/k 1) M LEMMA 6 ax p1 r k) p 1 ) exists; let p be a nuber in [, M] such that r k) p ) ax p1 r k) p 1 ) Then p is unique and 16) PROOF r k) p ) We use the following substitutions: kp k + p ) u p 1 ) 1/k 1), v M ) 1/k 1) After siplification, we can write the derivative of r k) p 1 ) as follows: 17) dr k) p 1 ) dp 1 uk + ku k 1)v p 2 1 v Consider the nuerator of 17) For every positive v and k > 1, the equation 18) u k + ku k 1)v 0 has a unique positive root, u In other words, there exists p u ) k 1 +, that is, a stationary point of r k) p 1 ) It is straightforward to check that the quantity d 2 r k) p 1 )/dp 2 1 ) p ) is negative and thus r k) p ) is a axiu

16 116 R El-Yaniv, A Fiat, R M Karp, and G Turpin We can rewrite 18) in the for u k 1) v u ) k 1 + k, and if we substitute back for u and v we obtain ) 1/k 1) 19) p M k 1) p + k k 1) p + k 1) The proof of 16) is then coplete by substituting 19) in 15) That is, r k) p ) 1 + p p ) ) 1/k 1) k 1) 1 p M ) 1 + p p k 1) 1 + p )k 1) p + k 1) kp p + k 1) 1 k 1) p + k 1) We can now uniquely characterize the worst-case k-day sequence of exchange rates against the threat-based algorith Let ˆσ k ˆp 1, ˆp 2,, ˆp k denote this sequence By Lea 6, ˆp 1 p We also know, fro an earlier discussion, that ˆp k M In addition, by Lea 5, for all 2 i k 1, 20) so p i p i 1 p i 1 ) p1 1/k 1), p k ) ˆp1 1/k 1) ˆp i 1 ˆp i + 1 M ) ) ˆp1 1/k 1) M The behavior of the threat-based algorith against ˆσ k can also be ade clear now For 1 i k, denote by ŝ i the daily aounts that the threat-based algorith spends when the exchange rate sequence is ˆσ k LEMMA 7 For all 1 i k, ŝ i 1/k PROOF It is readily seen, by Lea 5, that ŝ 2 ŝ 3 ŝ k The proof is copleted by showing that ŝ 1 1/k as follows We substitute the expression of r k) fro 16) for r in 8) That is, ŝ 1 1 r p r k) k) p

17 Optial Search and One-Way Trading Online Algoriths 117 p + k 1) p kp /p + k 1)) kp p p + k 1) p p ) kp p )p + k 1)) 1 k Thus, against the worst-case) exchange rate sequence ˆσ k, the threat-based algorith obeys the conventional wisdo of investent advisers by eploying a dollar-cost averaging strategy, in which an equal nuber of dollars is invested each day The next lea yields a ore inforative characterization of r k) LEMMA 8 21) r k) is the unique solution, r, of ) ) r 1) 1/k r k 1 M PROOF 22) First we show that r k) k 1 ) ) ˆp1 1/k 1) M Consider forula 10) for s i, i > 1 We already know that ŝ i 1/k Therefore, using 20) we obtain 1 k 1 r k) 1 r k) ˆp i ˆp i 1 ˆp i 1 ˆp1 M ) ) 1/k 1) This proves 22) Using 8) we derive the following expression for ˆp 1 : ˆp 1 rk) k 1) 23) k r k) which is obtained when we solve the equation 1/k 1/r k) ) ˆp 1 r k) )/ ˆp 1 )) for ˆp 1 We now substitute the expression for ˆp 1, 23), in 22) and learn that r k) is the solution of the following equation: kr r k) k) ) 1/k 1) ) 1) 24) k 1 k r k) )M ) Starting with 24), the following sequence of equivalent equalities copletes the proof of the lea: 25) r k) kr k) ) 1/k 1) 1) k k, k r k) )M )

18 118 R El-Yaniv, A Fiat, R M Karp, and G Turpin k r k) k k r k) kr k) ) 1) k k r k) )M ) r k) ) 1/k 1) M kr k) ) 1/k 1) 1), k r k) )M ) kr k) ) k/k 1) 1), k r k) )M ) kr k) ) 1/k 1) 1) k r k) )M ) Notice that the right-hand side of 25) is identical to the exponentiated ter of 24) The lea is coplete by using 24) while substituting the left-hand side of 25) for this ter Consider the representation of r k), 21) It is clear that r k) < M/ since the copetitive ratio M/ is attained by the trivial strategy that trades all dollars in the iniu possible rate,, and the threat-based algorith certainly perfors strictly better Hence, r 1)/M ) r 1)/M/ 1) <1, and then it is not hard to see that r k) is strictly increasing with k Therefore, we ust take the pessiistic assuption that k n This yields the following corollary that gives the best attainable copetitive ratio, r n, M), that the threat-based algorith can attain for an n-day conversion gae COROLLARY 4 26) r n, M) is the root, r, of the equation ) ) r 1) 1/n r n 1 M To suarize, we have two ethods of calculating r n, M): Solve dr n) p 1 )/dp 1 0 for its root e as in 17)) and then substitute it into 15) Solve 26) for r The next theore states that r n, M) is the best copetitive ratio that a one-way trading algorith can achieve for the known duration case with n trading days THEOREM 5 Let, M, and n be given Then r n, M) is the lowest possible copetitive ratio for a known duration one-way trading gae with known and M) PROOF Let ALG be any deterinistic algorith different fro the threat-based algorith Using an adversary arguent we show that ALG cannot achieve a ratio saller than r n as defined in 26)) Let ˆσ n ˆp 1, ˆp 2,, ˆp n be the exchange rate sequence that axiizes rp 1, p 2,,p n ) for an n-day gae see the discussion after Lea 6) On the first day we present ˆp 1 to ALGIfALG spends less than 1/n dollars on this rate, then we end the gae Therefore, ALG ust convert the reaining dollars with the iniu possible rate, If this is the case, ALG cannot achieve a ratio saller than r n ; siply because ˆp 1 is chosen such that 1/n is the inial aount that should be spent to guarantee the ratio r n Therefore, we assue that ALG spends on the first day an aount s 1 1/n In this case we continue the gae and present ALG with the next rate, ˆp 2 In general, if

19 Optial Search and One-Way Trading Online Algoriths 119 at the end of the ith day the total aount in dollars that ALG spent is less than i/n we iediately end the gae Otherwise, we continue and present ALG with the next rate, ˆp i+1, etc Let j be the iniu i such that at the end of the ith day, the total aount spent by ALG so far is less than i/n Denote by s i the aount spent by ALG on the ith day, 1 i j Since the gae proceeded to the jth day we know that 1 n s 1, 2 n s 1 + s 2, j 1 n j 1 s i i1 However, by the choice of j, j n > j s i i1 Therefore ALG could have gained ore by spending exactly 1/n on each of the first j 1 days and by spending s j s j + j 1 i1 s i j 1)/n) at the higher rate, ˆp jevenif this is the case, since s j < 1/n, ALG could not guarantee a copetitive ratio of r n since ˆp j is chosen such that exactly 1/n dollars should be spent on the jth day to attain a ratio of r n It follows then that ALG ust coincide with the threat-based algorith, achieving a ratio of r n, or otherwise ALG incurs a higher ratio on this exchange rate sequence REMARK 6 With respect to the proof of Theore 5, notice that the adversary ay end the gae after any day, i, provided that the total aount spent by ALG is less than or equal to i/n For exaple, if ALG spends exactly 1/n dollars on the first day, then by dropping the rest of the sequence to, the adversary forces a copetitive ratio of r n on ALG Thus, there are exactly n types of worst-case sequences against the threat-based algorith Naely, the sequences ˆp 1, ˆp 2,, ˆp i,,,,, i 1, 2,,n } {{ } n i Of course, the adversary ay use other rates saller than ˆp i instead of all the s on the last days, except for the very last day) REMARK 7 It is easy to extend the proof of Theore 5 to the case of randoized algoriths against oblivious adversaries All that is needed is to note that an oblivious adversary can calculate the expected aounts that the algorith will spend on each trading day Then the proof is analogous to the proof of Theore 5 Nevertheless, we already know fro Corollary 2 that randoization cannot help in one-way trading

20 120 R El-Yaniv, A Fiat, R M Karp, and G Turpin 42 A Gae Against a Lenient Adversary As can be seen in the proof of Theore 5, the adversary can always force a copetitive ratio of r n, M) on any algorith Nevertheless, for any practical purpose, it is ost likely the case that we will confront a ore lenient adversary one which deviates fro the worst-case sequence of exchange rates In this section we describe an algorith that always perfor as well as the previous algorith However, on soe exchange rate sequences, those which are not worst-case, the new algorith strictly iproves the offline to online ratio At the start of each trading day, the online player knows the nuber of reaining days, n n and is presented with an exchange rate, x In addition, the player already accuulated Y 0 yen and has D 1 dollars reaining At this stage, the online player can calculate the best attainable copetitive ratio for the reaining days given x, and use it to deterine the aount of dollars to be spent The idea is siply to assue that the current day is the first trading day of an n -day trading period and that the rest of the rates will be chosen by an adversary ie, to axiize the copetitive ratio) Since we consider x to be a first day rate we denote it by p 1 and siilarly, we denote the rest of the worst-case exchange rates by p i, i 2, 3,,n Also, the worst-case) daily transactions will be denoted by s i GivenD, Y, n, and p 1 we now derive a forula for r, the best attainable ratio fro this stage onward, as well as forulas for the s i Note that for usage the online player need only know the quantity s 1 An application of 4) with i 1 yields 27) s 1 p 1 r Y + D) r p 1 ) For i > 1 we obtain fro 10) 28) s i 1 r p i p i 1 p i Since the aounts to be spent su up to D we have n D s 1 + s i i2 p 1 r Y + D) r p 1 ) + 1 r n i2 p i p i 1 p i Solving for r, r p 1 Dp 1 + Y [ 1 + p 1 p 1 n i2 p 1 Dp 1 + Y r n ) p 1,,p n ), p i p i 1 p i ] where r n ) ) is defined by 12) with k n ) Thus, the best ratio, r, is siply written as a noralization of the best ratio for a regular n -day gae D 1, Y 0) with future rates p 2, p 3,,p n To deterine the optial ratio at this stage, we ust optiize over

21 Optial Search and One-Way Trading Online Algoriths 121 all possible future exchange rates Notice however that we need not axiize over p 1 since it is already given Hence, we apply Lea 5 to obtain r r p 1, n, D, Y,, M) def p 1 Dp 1 + Y r n ) p 1 ), where r n ) p 1 ) is given by 15) The iproved, adaptive algorith ay be suarized as follows: Given an exchange rate x when there are l trading days reaining l >0), the online player with D dollars and Y yen calculates r r x,l,d, Y,, M) Ifx r, then the online player akes no transaction see Reark 5) Otherwise, the online player trades s 1 equation 27)) dollars to yen We now exeplify how this adaptive algorith takes advantage of opportunities encountered in the trading period ie, deviations fro the worst-case exchange rate sequence) In this exaple, let n be an arbitrary nuber of days 2), and suppose that the first exchange rate we encounter is the axiu possible rate, M Suppose also that at this stage the algorith holds D dollars and Y yen In contrast to the worst-case algorith described in the previous section, we shall see that the adaptive algorith identifies this opportunity and trades all available D dollars on this fortunate) exchange rate First, by 15), r n ) M) 1, so r M/DM + Y ) Hence, )/ ) s 1 MY + D) MM ) M DM + Y DM + Y MDM ) MM ) D REMARK 8 In general, it can be shown that when using this iproved algorith, the sequence of ratios r that is calculated by the online player in this anner is nonincreasing and if the adversary always deviates fro the worst possible sequence, then the sequence of calculated ratios, r, is strictly decreasing 43 Analysis of Variant 2: Unknown Duration with, M Known Since in this variant the nuber of trading days is not given to the online player, he ust consider an arbitrarily large nuber of days Define r, M) def li n r n, M) Notice that r n is onotone increasing with n Therefore, r is larger than r n for any n and therefore, by Lea 3 the threat-based algorith calculated with r A r )is r -proper for any input sequence σ and therefore r is an attainable copetitive ratio for any finite trading period On the other hand, as r n is the lower bound for each n-day trading gae, the lower bound for Variant 2 approaches fro below) r, M) since the adversary ay choose arbitrarily large n

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