Secretary Problems via Linear Programming

Transcription

1 Secretary Problems va Lnear Programmng Nv Buchbnder 1, Kamal Jan 2, and Moht Sngh 3 1 Mcrosoft Research, New England, Cambrdge, MA. 2 Mcrosoft Research, Redmond, WA, USA. 3 McGll Unversty, Montreal, Canada. Abstract. In the classcal secretary problem an employer would lke to choose the best canddate among n competng canddates that arrve n a random order. Ths basc concept of n elements arrvng n a random order and rrevocable decsons made by an algorthm have been explored extensvely over the years, and used for modelng the behavor of many processes. Our man contrbuton s a new lnear programmng technque that we ntroduce as a tool for obtanng and analyzng mechansms for the secretary problem and ts varants. The lnear program s formulated usng judcously chosen varables and constrants and we show a one-toone correspondence between mechansms for the secretary problem and feasble solutons to the lnear program. Capturng the set of mechansms as a lnear polytope holds the followng mmedate advantages. Computng the optmal mechansm reduces to solvng a lnear program. Provng an upper bound on the performance of any mechansm reduces to fndng a feasble soluton to the dual program. Explorng varants of the problem s as smple as addng new constrants, or manpulatng the objectve functon of the lnear program. We demonstrate these deas by explorng some natural varants of the secretary problem. In partcular, usng our approach, we desgn optmal secretary mechansms n whch the probablty of selectng a canddate at any poston s equal. We refer to such mechansms as ncentve compatble and these mechansms are motvated by the recent applcatons of secretary problems to onlne auctons. We also show a famly of lnear programs whch characterze all mechansms that are allowed to choose J canddates and gan proft from the K best canddates. We beleve that lnear programmng based approach may be very helpful n the context of other varants of the secretary problem. 1 Introducton In the classcal secretary problem an employer would lke to choose the best canddate among n competng canddates. The canddates are assumed to arrve n a random order. After each ntervew, the poston of the ntervewee n the total order s revealed vs-á-vs already ntervewed canddates. The ntervewer has to decde, rrevocably, whether to accept the canddate for the poston or

2 to reject the canddate. The objectve n the basc problem s to accept the best canddate wth hgh probablty. A mechansm used for choosng the best canddate s to ntervew the frst n/e canddates for the purpose of evaluaton, and then hre the frst canddate that s better than all prevous canddates. Analyss of the mechansm shows that t hres the best canddate wth probablty 1/e and that t s optmal [8, 18]. Ths basc concept of n elements arrvng n a random order and rrevocable decsons made by an algorthm have been explored extensvely over the years. We refer the reader to the survey by Ferguson [9] on the hstorcal and extensve work on dfferent varants of the secretary problem. Recently, there has been a nterest n the secretary problem wth ts applcaton to the onlne aucton problem [13, 3]. Ths has led to the study of varants of the secretary problem whch are motvated by ths applcaton. For example, [15] studed a settng n whch the mechansm s allowed to select multple canddates and the goal s to maxmze the expected proft. Imposng other combnatoral structure on the set of selected canddates, for example, selectng elements whch form an ndependent set of a matrod [4], selectng elements that satsfy a gven knapsack constrant [2], selectng elements that form a matchng n a graph or hypergraph [16], have also been studed. Other varants nclude when the proft of selectng a secretary s dscounted wth tme [5]. Therefore, fndng new ways of abstractng, as well as analyzng and desgnng algorthms, for secretary type problems s of major nterest. 1.1 Our Contrbutons Our man contrbuton s a new lnear programmng technque that we ntroduce as a tool for obtanng and analyzng mechansms for varous secretary problems. We ntroduce a lnear program wth judcously chosen varables and constrants and show a one-to-one correspondence between mechansms for the secretary problem and feasble solutons to the lnear program. Obtanng a mechansm whch maxmzes a certan objectve therefore reduces to fndng an optmal soluton to the lnear program. We use lnear programmng dualty to gve a smple proof that the mechansm obtaned s optmal. We llustrate our technque by applyng t to the classcal secretary problem and obtanng a smple proof of optmalty of the 1 e mechansm [8] n Secton 2. Our lnear program for the classcal secretary problem conssts of a sngle constrant for each poston, boundng the probablty that the mechansm may select the th canddate. Despte ts smplcty, we show that such a set of constrants suffces to correctly capture all possble mechansms. Thus, optmzng over ths polytope results n the optmal mechansm. The smplcty and the tghtness of the lnear programmng formulaton makes t flexble and applcable to many other varants. Capturng the set of mechansms as a lnear polytope holds the followng mmedate advantages. Computng the optmal mechansm reduces to solvng a lnear program.

3 Provng an upper bound on the performance of any mechansm reduces to fndng a feasble soluton to the dual program. Explorng varants of the problem s as smple as addng new constrants, or manpulatng the objectve functon of the lnear program. We next demonstrate these deas by explorng some natural varants of the secretary problem. Incentve Compatblty. As dscussed earler, the optmal mechansm for the classcal secretary problem s to ntervew the frst n/e canddates for the purpose of evaluaton, and then hre the frst canddate that s better than all prevous canddates. Ths mechansm suffers from a crucal drawback. The canddates arrvng early have an ncentve to delay ther ntervew and canddates arrvng after the poston n e + 1 have an ncentve to advance ther ntervew. Such a behavor challenges the man assumpton of the model that ntervewees arrve n a random order. Ths ssue of ncentves s of major mportance especally snce secretary problems have been used recently n the context of onlne auctons [13, 3]. Usng the lnear programmng technque, we study mechansms that are ncentve compatble. We call a mechansm for the secretary problem ncentve compatble f the probablty of selectng a canddate at th poston s equal for each poston 1 n. Snce the probablty of beng selected n each poston s the same, there s no ncentve for any ntervewee to change hs or her poston and therefore the ntervewee arrves at the randomly assgned poston. We show that there exsts an ncentve compatble mechansm whch selects the best canddate wth probablty and that ths mechansm s optmal. Incentve compatblty s captured n the lnear program by ntroducng a set of very smple constrants. Surprsngly, we fnd that the optmal ncentve compatble mechansm sometme selects a canddate who s worse than a prevous canddate. To deal wth ths ssue, we call a mechansm regret-free f the mechansm only selects canddates whch are better than all prevous canddates. We show that the best ncentve compatble mechansm whch s regret free accepts the best canddate wth probablty 1 4. Another ssue wth the optmal ncentve compatble mechansm s that t does not always select a canddate. In the classcal secretary problem, the mechansm can always pck the last canddate but ths soluton s unacceptable when consderng ncentve compatblty. We call a mechansm must-hre f t always hres a canddate. We show that there s a must-hre ncentve compatble mechansm whch hres the best canddate wth probablty 1 4. All the above results are optmal and we use the lnear programmng technque to derve the mechansms as well as prove ther optmalty. In subsequent work [6], we further explore the mportance of ncentve compatblty n the context of onlne auctons. In ths context, bdders are bddng for an tem and may have an ncentve to change ther poston f ths may ncrease ther utlty. We show how to obtan truthful mechansms for such settngs

4 usng underlyng mechansms for secretary type problems. Whle there are nherent dfferences n the aucton model and the secretary problem, a mechansm for the secretary problem s used as a buldng block for obtanng an ncentve compatble mechansm for the onlne aucton problem. The J-choce, K-best Secretary Problem. Our LP formulaton approach s able to capture a much broader class of secretary problems. We defne a most general problem that we call the J-Choce, K-best secretary problem, referred to as the (J, K)-secretary problem. Here, n canddates arrve randomly. The mechansm s allowed to pck up to J dfferent canddates and the objectve s to pck as many from the top K ranked canddates. The (1, 1)-secretary problem s the classcal secretary problem. For any J, K, we provde a lnear program whch characterzes all mechansms for the problem by generalzng the lnear program for the classcal secretary problem. A sub-class that s especally nterestng s the (K, K)-secretary problem, snce t s closely related to the problem of maxmzng the expected proft n a cardnal verson of the problem. In the cardnal verson of the problem, n elements that have arbtrary non-negatve values arrve n a random order. The mechansm s allowed to pck at most k elements and ts goal s to maxmze ts expected proft. We defne a monotone mechansm to be an mechansm that, at any poston, does not select an element that s t best so far wth probablty hgher than an element that s t < t best so far. We note that any reasonable mechansm (and n partcular the optmal mechansm) s monotone. The followng s a smple observaton. We omt the proof due to lack of space. Observaton 1 Let Alg be a monotone mechansm for the (K, K)-secretary problem that s c-compettve. Then the mechansm s also c-compettve for maxmzng the expected proft n the cardnal verson of the problem. Klenberg [15] gave an asymptotcally tght mechansm for the cardnal verson of the problem. However, ths mechansm s randomzed, and also not tght for small values of k. Better mechansms, even restrcted to small values of k, are helpful not only for solvng the orgnal problem, but also for mprovng mechansms that are based upon them. For example, a mechansm for the secretary knapsack [2] uses a mechansm that s 1/e compettve for maxmzng the expected proft for small values of k (k 27). Analyzng the LP asymptotcally for any value n s a challenge even for small value k. However, usng our characterzaton we solve the problem easly for small values k and n whch gves an dea on how compettve rato behaves for small values of k. Our results appear n Table 1. We also gve complete asymptotc analyss for the cases of (1, 2), (2, 1)-secretary problems. 1.2 Related Work The basc secretary problem was ntroduced n a puzzle by Martn Gardner [11]. Dynkn [8] and Lndley [18] gave the optmal soluton and showed that no other

5 Number of elements allowed to be pcked by the mechansm Compettve rato 1 1/e = Table 1. Compettve rato for Maxmzng expected proft. Expermental results for n = 100. strategy can do better (see the hstorcal survey by Ferguson [9] on the hstory of the problem). Subsequently, varous varants of the secretary problem have been studed wth dfferent assumptons and requrements [20](see the survey [10]). More recently, there has been sgnfcant work usng generalzatons of secretary problems as a framework for onlne auctons [2 4, 13, 15]. Incentves ssues n onlne mechansms have been studed n several models [1, 13, 17]. These works desgned mechansms where ncentve ssues were consdered for both value and tme strateges. For example, Hajaghay et. al. [13] studed a lmted supply onlne aucton problem, n whch an auctoneer has a lmted supply of dentcal goods and bdders arrve and depart dynamcally. In ther problem bdders also have a tme wndow whch they can le about. Our lnear programmng technque s smlar to the technque of factor revealng lnear programs that have been used successfully n many dfferent settngs [7, 12, 14, 19]. Factor revealng lnear program formulates the performance of an algorthm for a problem as a lnear program (or sometmes, a more general convex program). The objectve functon s the approxmaton factor of the algorthm on the problem. Thus solvng the lnear program gves an upper bound on the worst case nstance whch an adversary could choose to maxmze/mnmze the approxmaton factor. Our technque, n contrast, captures the nformaton structure of the problem tself by a lnear program. We do not apror assume any algorthm but formulate a lnear program whch captures every possble algorthm. Thus optmzng our lnear program not only gves us an optmal algorthm, but t also proves that the algorthm tself s the best possble. 2 Introducng the Technque: Classcal secretary (and varants) In ths secton, we gve a smple lnear program whch we show characterzes all possble mechansms for the secretary problem. We stress that the LP captures not only thresholdng mechansms, but any mechansm ncludng probablstc mechansms. Hence, fndng the best mechansm for the secretary problem s equvalent to fndng the optmal soluton to the lnear program. The lnear program and ts dual appear n Fgure 1. The followng two lemmas show that the lnear program exactly characterzes all feasble mechansms for the secretary problem.

6 (P) max 1 n n =1 p (D) mn n =1 x s.t. s.t. 1 n p 1 1 j=1 pj 1 n n j=+1 xj + x /n 1 n p 0 1 n x 0 Fg. 1. Lnear program and ts Dual for the secretary problem Lemma 1. (Mechansm to LP soluton) Let π be any mechansm for selectng the best canddate. Let p π denote the probablty of selectng the canddate at poston. Then p π s a( feasble soluton to the lnear program (P),.e, t satsfes the constrants p π 1 1 ) j< pπ j for each 1 n. Moreover the objectve value 1 n n =1 pπ s at least the probablty of selectng the best canddate by π. Proof. Let p π be the probablty n whch mechansm π selects canddate. Any mechansm cannot ncrease ts chances of hrng the best canddate by selectng a canddate that s not the best so far, therefore we may consder only such mechansms. We now show that p π satsfes the constrants of lnear program. p π = P r[π selects canddate canddate s best so far] P r[canddate s best so far] P r[π dd not select canddates {1,..., 1}] canddate s best so far] 1 However, the probablty of selectng canddates 1 to 1 depends only on the relatve ranks of these canddates and s ndependent on whether canddate s best so far (whch can be determned after the mechansm have done ts choces regardng canddates 1 to 1). Therefore, we obtan p π 1 (1 j< pπ j ), whch proves our clam. Now we show that the objectve functon of the lnear program s at least the probablty wth whch π accepts the best canddate. Snce the mechansm cannot dstngush whether the th canddate s the best canddate so far or best canddate over all, the probablty that the mechansm hres canddate gven that the best canddate s n the th poston equals the probablty the mechansm hres canddate gven that the best canddate among canddates 1 to s n the th poston. Snce the th canddate s best so far wth probablty 1/, the latter probablty s at least p π. Summng over all n postons we get that π hres the best canddate wth probablty at least 1 n n =1 pπ. Lemma 1 shows that the optmal soluton to (P) s an upper-bound on the performance of the mechansm. The followng lemma shows that every LP soluton actually corresponds to a mechansm whch performs as well as the objectve value of the soluton. Lemma 2. (LP soluton to Mechansm) Let p for 1 n be any feasble LP soluton to (P). Then consder the mechansm π whch selects the canddate wth probablty f canddate s the best canddate so far and p (1 j< p j)

7 max 1 n n =1 p + q(1 n =1 p) (D) mn n =1 x + q s.t. s.t. 1 n p 1 1 j=1 pj 1 n n j=+1 xj + x /n q 1 n p 0 1 n x 0 Fg. 2. Lnear program and ts Dual for the rehrng secretary problem canddate 1,..., 1 have not been selected,.e., the mechansm reaches canddate. Then π s a mechansm whch selects the best canddate wth probablty 1 n n =1 p. Proof. Frst, notce that the mechansm s well defned snce for any, 1. We prove by nducton that the probablty that the mechansm p (1 j< p j) selects canddate at poston s exactly p. The base case s trval. Assume ths s true untl 1. At step, the probablty we choose s the probablty that we ddn t choose canddates 1 to 1 whch s 1 j< p j tmes the probablty p (1 j< that the current canddate s best so far whch s 1/ tmes pj) whch s exactly p. The probablty of hrng the th canddate gven that the th canddate s the best canddate s equal the probablty of hrng the th canddate gven the th canddate s the best canddate among canddates 1 to. Otherwse, t means that the mechansm s able to dstngush between the event of seeng the relatve ranks and the absolute ranks whch s a contradcton to the defnton of the secretary problem. Snce the th canddate s best so far wth probablty 1/, the latter probablty equals p (the mechansm hres only the best canddate so far). Summng over all possble poston n we get that the mechansm π hres the best canddate wth probablty 1 n n =1 p. Usng the above equvalence between LP solutons and the mechansms, t s easy to show that the optmal mechansm can hre the best canddate wth probablty of no more than 1/e. The proof s smply by constructng a feasble soluton to the dual lnear program. Lemma 3 ([8]). No mechansm can hre the best canddate wth probablty better than 1/e + o(1). Proof. To prove an upper bound of 1/e+o(1) we only need to construct a feasble dual soluton to program (D) wth value 1/e+o(1). Set x = 0 for each 1 n e and x = 1 n (1 n 1 1 j= j ) for n e < n. A smple calculaton shows that x s feasble and has objectve value at most 1 e + o(1). 2.1 Allowed to Rehre One natural extenson of the secretary problem s the case when one s allowed to rehre the best secretary at the end wth certan probablty. That s, suppose

8 (P 1) max 1 n n =1 f (P 2) max 1 n n =1 f (P 3) max 1 n n =1 f s.t. s.t. s.t. p 1/n p 1/n p = 1/n f + ( 1) p 1 f + ( 1) p 1 f + ( 1) p 1 f p f = p f p p, f 0 p, f 0 p, f 0 (Incentve compatble) (Regret free) (Must-hre) Fg. 3. (P1): Characterzes any ncentve compatble mechansm. (P2) characterzes mechansms that are regret free. (P3) characterzes mechansms that are must-hre mechansms. that after the ntervewer has seen all n canddates, he s allowed to hre the best canddate wth certan probablty q f no other canddate has been hred. Observe that f q = 0, the problem reduces to the classcal secretary problem whle f q = 1, then the optmal strategy s to wat tll the end and then hre the best canddate. We gve a tght descrpton of strateges as q changes. Ths can be acheved smply by modfyng the lnear program: smply add n the objectve functon q(1 n =1 p ). That s, f the mechansm dd not hre any canddate you may hre the best canddate wth probablty q. Solvng the prmal and the correspondng dual (see Fgure 2) gve the followng tght result. The proof s omtted. Theorem 2. There s a mechansm for the rehre varant that selects the best secretary wth probablty e (1 q) + o(1) and t s optmal. 3 Incentve Compatblty In ths secton we study ncentve compatble mechansms for the secretary problem. We desgn a set of mechansms M p and show that wth certan parameters these mechansms are the optmal mechansms for certan secretary problems. To ths end, we derve lnear formulatons that characterze the set of possble ncentve compatble mechansms and also analyze the dual lnear programs. The basc lnear formulaton that characterzes all ncentve compatble mechansms appears n Fgure 3. We gve a set of three lnear formulatons. The formulaton (P 1) characterzes all mechansms that are ncentve compatble, (P 2) captures mechansms that are also regret free and (P 3) captures mechansms that are must-hre mechansms. Ths s formalzed n the followng two lemmas. Lemma 4. (Mechansm to LP soluton) Let π be any mechansm for selectng the best canddate that s ncentve compatble. Let p π denote the probablty the mechansm selects a canddate at each poston, and let f π be the probablty the mechansm selects the canddate at poston gven that the canddate at poston s the best canddate. Then: p π, f π s a feasble soluton to the lnear program (P 1).

9 If the mechansm s also regret free then p π, f π s a feasble soluton to the lnear program (P 2). If the mechansm s also must-hre then p π, f π s a feasble soluton to the lnear program (P 3). The objectve value 1 n n =1 f π s at least the probablty of selectng the best canddate by π. Proof. The proof follows the same deas as n the proof of Lemma 1. The condton of ncentve compatblty mples that p = p j = p for any two postons and j. Also, n the orgnal secretary problem, every mechansm could be modfed to be a regret free mechansm. Ths s not true for an ncentve compatble mechansm. Indeed, we have the followng constrant, f p snce the probablty of hrng n the th poston s at least the probablty of hrng n the th poston gven that the canddate s best so far tmes 1/. If the mechansm s also supposed to be regret free then equalty must hold for each. In the must-hre part we demand that the sum of p s 1. The resultng formulaton gven n Fgure 3 s after smplfcaton. Lemma 4 shows that the optmal soluton to the lnear formulatons s an upper-bound on the performance of the mechansm. To show the converse we defne a famly of mechansms that are defned by ther probablty of selectng a canddate at each poston 0 p 1/n, we show that the set of feasble solutons to (P 1) corresponds to the set of mechansms M p defned here. Incentve Compatble Mechansm M p : Let 0 p 1/n. For each 1 n, whle no canddate s selected, do If 1 1 2p, select the th canddate wth probablty 1/p +1 f she s the best canddate so far. 1 If 2p < n, let r = 1/p +1. Select the th canddate wth probablty 1 f her rank s n top r and wth probablty r r f her rank s r + 1. The followng lemma shows that every LP soluton to (P 1) corresponds to a mechansm whch performs as well as the objectve value of the soluton. Lemma 5. (LP soluton to Mechansm) Let p, f for 1 n be a feasble LP soluton to (P 1). Then the mechansm M p selects the best canddate wth probablty whch s at least 1 n n =1 f. Proof. For any p, the optmal values of f are gven by the followng. f = p for 1 1 2p and f = 1 ( 1)p for > 1 2p. For ease of calculatons, we gnore the fact the fractons need not be ntegers. These are exactly the values acheved by the mechansm M p for any value p. Lemma 6. The mechansm ( M p s ) ncentve compatble for each 0 p 1/n and has effcency of pn + pn 2

10 Proof. We prove by nducton that the mechansm M p selects each poston wth probablty p. It s easy to verfy that for = 1 ths s true. For > 1. The probablty the mechansm chooses poston s by our nducton hypothess: r 1 (1 ( 1)p) = 1/p ( 1)p (1 ( 1)p) = 1/p + 1 = p The probablty the mechansm selects the best canddate s related to f. f = p for 1 1/2p, and f = 1 ( 1)p for 1/2p < n. Thus, we get: 1 n n f = 1 1/2p p + n =1 =1 n =1/2p+1 (1 ( 1)p) = 1 ( 1 4pn + pn ) 2 Optmzng the lnear programs (P 1), (P 2) and (P 3) exactly, we get the followng theorem. The optmalty of the mechansms can also be shown by exhbtng an optmal dual soluton. Theorem 3. The famly of mechansms M p acheves the followng. 1. Mechansm M 1/ 2n s ncentve compatble wth effcency of Mechansm M 1/2n s ncentve compatble and regret free wth effcency Mechansm M 1/n s ncentve compatble and must-hre wth effcency 1 4. Moreover, all these mechansm are optmal for effcency along wth the addtonal property. 4 The J-choce K-best secretary problem In ths secton we study a general problem of selectng as many of the top 1,..., K ranked secretares gven J rounds to select. The mechansm s gven J possble rounds n whch t may select a canddate, and t gans from selectng any of the frst K ranked canddates. The classcal secretary problem s exactly 1-choce 1-best secretary problem. Other specal cases nclude cases n whch the mechansm s gven J rounds and get proft only for the best canddate, or gettng a sngle round, but receve proft for any of the best K canddates. Our result s a smple lnear formulaton that characterze all strateges for selectng the canddates. The followng two lemmas show that the above lnear program exactly characterzes all feasble mechansms for the (J, K)-secretary problem. Lemma 7. (Mechansm to LP soluton) Let π be any mechansm for selectng the (J, K)-secretary problem. Let p j (π): The probablty of acceptng the canddate at th poston n the jth round for each 1 n and each 1 j J. q j k (π): The probablty of acceptng the canddate th poston n the jth round gven that the canddate s the kth best canddate among the frst canddates for each 1 n, 1 j J and 1 k K.

11 max F (q) = 1 n n =1 J j=1 K k=1 k l=1 ( n k l)( l 1) 1 ( n 1 k 1) q j l s.t. 1 n, 1 j J p j = 1 mn{,k} k=1 q j k 1 n, 1 k K q 1 k 1 l< p1 1 n, 1 k K, 2 j J q j k 1 n, 1 k K, 1 j J p j, qj k 0 l< pj 1 l< pj Fg. 4. Lnear program for the (J, K)-secretary problem Then (p(π), q(π)) s a feasble soluton and expected number of top K canddates selected s at most F (p(π), q(π)). mn{,k} k=1 q j k Proof. Let us prove the frst type of constrants of the form: p j = 1 It s clear that there s no reason for any mechansm to select a canddate whch s not at least the K best so far. Such a canddate cannot be even potentally one of the K best globally and therefore s not proftable for the mechansm. Thus, the probablty any mechansm selects the th canddate n the jth round s the sum of the probablty of selectng the th canddate n the jth round gven that the canddate s the kth best canddate so far tmes 1/, whch s the probablty that the canddate s the kth best so far. We sum untl the mnmum between and K to get the desred equalty whch holds for every mechansm.let us now prove the thrd type of constrants (the second type follows by the same arguments). Consder any mechansm and some poston and some rounds j. q j k = P r[π selects canddate n round j canddate s kth best so far] P r[π selects exactly j 1 canddates out of cand. {1,..., 1}] canddate s kth best so far] = P r[π selects exactly j 1 canddates out of cand. {1,..., 1}] = l< p j 1 (π) p j (π) l< The nequalty follows snce n order to select canddate n round j the mechansm must have selected exactly j 1 canddates out of the prevous 1 canddates. The followng equalty then follows snce the decsons made by the polcy wth respect to the 1 canddates depend only on the relatve ranks of the 1 canddates, and s ndependent of the rank of the th canddate wth respect to these canddates. The fnal equalty follows snce the event of selectng j 1 canddates contans the event of selectng j canddates, whch concludes our proof. Fnally, let us consder the objectve functon and prove that t upper bounds the performance of the mechansm. For analyss purpose let us consder the probabltes f j k that are defned as probablty of selectng the th canddate n the

12 jth round gven that the kth best canddate s n the th poston. Note that the man dfference between f j k and q j k s that whle the former consder the kth best canddate overall, the latter only looks from the mechansm s perspectve and therefore looks at the event of the kth best canddate among the frst canddates. It s easy to state the objectve functon usng the frst set of varables as smply the sum over all values of, j and k of f j k dvded by 1/n. To fnsh we smply defne each f j k n terms of q j k whch proves the lemma. Clam. For each 1 n, 1 j J and 1 k K, we must have f j k = ( k 1 )( n l 1 k l ( n 1 l=1 k 1 ) ) q j l The proof s omtted due to lack of space. The proof of Lemma 7 follows drectly from the clam. Lemma 7 shows that the optmal soluton to (P) s an upper-bound on the performance of the mechansm. The followng lemma shows that every LP soluton actually corresponds to a mechansm whch performs as well as the objectve value of the soluton. Lemma 8. (LP soluton to Mechansm) Let (p, q) be any feasble LP soluton to (P). Then consder the mechansm π defned nductvely as follows. For each poston 1 n, If the mechansm has not selected any canddate among poston {1,..., 1} and the rank of canddate among {1,..., } s k for some 1 k K, then q 1 k 1. l< p1 select canddate wth probablty If the mechansm has selected j 1 canddates n postons 1,..., 1 for some 2 j J and the rank of canddate among {1,..., } s k for some 1 k K, then select canddate wth probablty Else do not select canddate. q j k l< pj 1 l< pj Then expected number of top k canddates selected by π s exactly F (p, q). Proof (Sketch). The proof s by nducton on the steps of the mechansm. It can be verfed easly that the procedure above keeps by nducton that p j (π) = p j,qj k (π) = q j k. That s, the probablty the mechansm selects the th canddate n the jth round s the same as the LP. As stated n Lemma 7 there s a correspondence between the values of q j k (π) and f j k (π) whch s the probabltes of hrng the th canddate n the jth round gven that the canddate s the kth best. Thus, the objectve functon of π s exactly F (p, q). We now gve optmal mechansm for the (1, 2) and (2, 1)-secretary problem. Observe that (1, 1)-secretary problem s the tradtonal secretary problem. Theorem 4. There exsts mechansms whch acheve a performance of.

13 1 1. e + 1 e for (2, 1)-secretary problem for the (1, 2) secretary problem. Moreover all these mechansms are (nearly) optmal. Proof. (Sketch) To gve a mechansm, we wll gve a prmal soluton to LP (J, K). The optmalty s shown by exhbtng a dual soluton of the same value. Due to lack of space we only prove the (2, 1) case. n e 3/2 (2,1)-secretary. Let t 1 = that selects the th canddate f th canddate s best so far and t 1 < t 2 and no other canddate has been selected or f t 2 n and th canddate s best so far and at most one canddate has been selected. The performance of ths mechansm s 1 e + 1 e 3 2 where p 1 = 0 for 1 < t 1 and p 1 = t1 1 and p 2 = t 2 t 1 ( 1) 1 ( 1) and 1 n. e 3 2 and t 2 = n e. Consder the followng mechansm. The mechansm corresponds to the prmal LP soluton ( 1) for t 1 n, p 2 = 0 for 1 < t 2 1 t 1 1 j=1 1 for t 2 n, q j 1 = p j for each 1 j 2 Dual Soluton. We frst smplfy the prmal lnear program by elmnatng the q j k varables usng the frst set of constrants. Let y denote the dual varables correspondng to the second set of constrants and z the varables correspondng to the thrd set of constrants. Then the followng dual soluton s of value 1 e + 1 o(1). Set z = 0 for 1 < t 2 and z = 1 n (1 n j=+1 1 j ) for t 2 n. Set y = 0 for 1 < t 1, y = 1 n (1 n j=+1 1 j ) + n 1 j=t 2 n (1 n k=j+1 1 k ) for t 1 < t 2 and y = 1 n (1 n j=+1 1 j ) + n j= 1 n (1 n k=j+1 1 k ) for t 2 n. 5 Further Dscusson Characterzng the set of mechansms n secretary type problems as a lnear polytope possesses many advantages. In contrast to methods of factor revealng LPs n whch lnear programs are used to analyze a sngle algorthm, here we characterze all mechansms by a lnear program. One drecton for future research s tryng to capture more complex settngs of a more combnatoral nature. One such example s the clean problem studed n [4] n whch elements of a matrod arrve one-by-one. Ths problem seems extremely appealng snce matrod constrants are exactly captured by a lnear program. Another promsng drecton s obtanng upper bounds. Whle the lnear program whch characterzes the performance may be too complex to obtan a smple mechansm, the dual lnear may stll be used for obtanng upper bounds on the performance of any mechansm. We beleve that lnear programmng and dualty s a powerful approach for studyng secretary problems and wll be applcable n more generalty. References 1. Baruch Awerbuch, Yoss Azar, and Adam Meyerson. Reducng Truth-Tellng Onlne Mechansms to Onlne Optmzaton. In Proceedngs of ACM Symposum on Theory of Computng, pages , 2003.

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