Dyamic House Allocatio Sujit Gujar 1 ad James Zou 2 ad David C. Parkes 3 Abstract. We study a dyamic variat o the house allocatio problem. Each aget ows a distict object (a house) ad is able to trade its house while preset i the market. Agets have strict prefereces over houses, ad the market operates without paymets. The goal is to eable a efficiet reallocatio of objects, alog with strategyproofess ad while satisfyig participatio costraits. We first establish coditios uder which a olie mechaism that allows a aget to ifluece the period i which it trades ca be maipulated. This motivates partitio mechaisms i which agets are divided olie ito disjoit feasible tradig groups, with the top tradig cycle algorithm (TTCA) ru separately for each group. I particular, we demostrate good rak-efficiecy for a mechaism that adopts stochastic-optimizatio i determiig how to partitio agets. 1 Itroductio I the house allocatio problem, each of a set of self-iterested agets ows a distict object (a house) ad has strict prefereces o houses [10]. The problem is to fid a reallocatio of objects amogst agets that is robust agaist misreports of prefereces by agets while idetifyig beeficial trades ad without usig moey. The top-tradig cycle algorithm (TTCA) is strategyproof (meaig that truthful reports of prefereces is a domiat strategy for agets) ad fids a allocatio i the core. A allocatio i the core is stable, i the sese that o coalitio of agets ca block the outcome by reallocatig their iitial objects amogst themselves i a way that is weakly better for every aget ad strictly better for at least oe aget i the coalitio. The TTCA is kow to be essetially uique amogst mechaisms for the static house allocatio problem with useful ecoomic properties [8, 6]. I the dyamic model of the house allocatio problem advaced here, each aget has a arrival period ad a departure period ad is oly able to trade with other agets preset simultaeously preset i the market at the same time. For a motivatig example, cosider college housig, with studets o differet leases ad willig to trade durig the moth before their lease expires. Aother example is provided by the problem of reallocatig offices i the workplace, or fidig mutually-beeficial trades (e.g., swaps, three-cycles, etc.) of vacatio property such as time-shares whe ivetory is i the market at differet times. Cotributios. We establish geeral coditios uder which o mechaism i which a aget ca ifluece the set of agets with which it participates i a TTCA (e.g., the period i which it trades ad thus the other agets that it trades with) ca be strategyproof. Give this, we study partitio mechaisms, i which each aget is 1 Idia Istitute of Sciece. email: sujit@csa.iisc.eret.i 2 School of Egieerig ad Applied Scieces, Harvard Uiversity. email: jzou@fas.harvard.edu 3 School of Egieerig ad Applied Scieces, Harvard Uiversity. email: parkes@eecs.harvard.edu assiged olie to a group of agets with which it will egage i a sigle TTCA. We cosider three variatios of partitio mechaisms: (i) trade o departure (DO-TTCA) (ii) trade o departure or whe the populatio size is above some threshold (T-TTCA) (iii) a stochastic optimizatio approach to determie a good partitioig of agets (SO-TTCA). We adopt rak efficiecy as a measure of performace, which is defied as the preferece rak of agets for allocated objects, averaged across all agets ad across istaces sampled from a distributio. This is a meaigful measure of performace for risk eutral agets, each of which has a equal differece i utility for successive houses i its preferece list. The experimetal results show that SO-TTCA ad T-TTCA outperform DO-TTCA by 15-20% i terms of rak efficiecy whe there are large umber of agets ad for uiform arrival time ad patiece distributios. Furthermore, whe the agets arrive accordig to a Poisso process, the SO-TTCA outperforms DO-TTCA by 4-10% i terms of rak efficiecy. For eviromets i which agets ted to be impatiet, T-TTCA has performace competitive with that of SO-TTCA. But T-TTCA is less successful tha SO-TTCA i idetifyig partitios i which each aget teds to have a large umber of tradig parters, which is useful whe agets are patiet. Related Work. For a overview of mechaisms o matchig ad house allocatio, the iterested reader is referred to Sömez ad Üver [11]. I previous models of multi-period house allocatio problems, each aget expresses a preferece o a sequece of allocatios [1, 5]. Our model is differet because a aget s prefereces are expressed oly over its evetual allocatio upo departure ad ot o the sequece of allocatios made while preset i the market. Aother differece is that our agets brig a house to the market upo arrival ad depart with a house, while previous models maitai the same pool of objects throughout ad agets arrive ad leave emptyhaded. Stochastic models of kidey exchages, i which trades are idetified amogst door-patiet pairs, are explored but without icetive cosideratios [2, 12]. For other dyamic models, there is existig work o matchig (with strict prefereces expressed i a bipartite graph) [4], assigmet (which is differet from house allocatio because agets do ot ow a object upo arrival) [13], as well as a cosiderable literature o dyamic mechaisms with moey (see Parkes [7] for a survey). For static problems, there is also computatioal work o the reallocatio of objects through differet types of local trades (e.g., swaps, 3-cycles, etc.), with a view to fidig desirable outcomes such as efficiet or fair allocatios [9, 3]. 2 The Model Let N = {A 1, A 2,..., A } deote the set of agets ad H = {h 1, h 2,..., h } deote the set of distict houses. Aget A i eters
the market with house h i i period a i T, where T = {1, 2,...}, ad departs i d i T. Let Sched N deote the set of all possible arrival ad departure times of N agets. Aget A i has strict prefereces i o the house allocated upo departure. Let (h i h ) (h i h). The preferece profile of all the agets is deoted by = ( 1, 2,..., ) U, where U is the set of all possible strict preferece profiles. Let i deote the preferece profile of agets except A i. A aget kows its prefereces i upo arrival ito the market. We allow for arbitrary misreports of prefereces, but assume that arrival ad departure times are truthfully reported. Let x : N H deote a house allocatio, with aget A i allocated to house x(i). Let X(ρ) deote the set of feasible allocatios give arrival-departure schedule ρ Sched N. A allocatio is feasible for a give schedule ρ if there is a sequece of trades, oe i each period, where the trade i ay give period is oly betwee those agets preset ad is feasible give the allocatio determied through trade i the previous period, ad where each aget allocated to exactly oe house i every period. A olie house allocatio mechaism φ(, ρ) X(ρ), geerates a feasible allocatio give a reported type profile ad arrivaldeparture schedule. The problem is olie, i that a aget s type (a i, d i, i) is ot available util period t = a i ad therefore we require, for all A j N, that φ j (, ρ) = φ j (, ρ ) whe, ad ρ, ρ differ oly i periods after aget A j s departure. A assigmet of houses, x X, Pareto domiates y X, if x i y for all A i ad x j y for some A j. A allocatio y X is Pareto efficiet if there is o allocatio x X that Pareto domiates y. Defiitio 2.1 (Pareto efficiet). A mechaism φ is Pareto efficiet if allocatio x = φ(, ρ) is Pareto efficiet for all preferece profiles ad all schedules ρ. Defiitio 2.2 (Strategyproof (SP)). Let x = φ( i, i, ρ) ad x = φ( i, i, ρ). Mechaism φ is strategyproof if x(i) i x (i), for all A i, all ρ Sched N ad all i U i. Defiitio 2.3 (Idividually Ratioal (IR)). A olie mechaism φ is idividually ratioal if x i i h i, where x = φ(, ρ), for all U, all ρ Sched N. A allocatio x is blocked by a coalitio of agets S N, if there is a feasible allocatio of the houses iitially owed by agets i S amogst themselves that Pareto domiates, for agets i S, the allocatio x. Defiitio 2.4 (Core). A olie mechaism φ is core-selectig if allocatio x = φ(, ρ) is ot blocked by ay coalitio of agets, for ay preferece profile, ad ay schedule ρ. The core implies Pareto efficiecy (by cosiderig coalitios of size ) ad IR (by cosiderig coalitios of size oe). To obtai a quatitative measure of efficiecy, we assume i our experimetal aalysis that agets are risk eutral, ad with a uiform differece i utility betwee successive houses i their preferece orders. Based o this, we ca compare the expected utility of two mechaisms i terms of rak-efficiecy. Let rak φ,i (, ρ) deote the rak that A i assigs to the house allocated by mechaism φ give ad ρ, ad defie RANK φ (, ρ) = i rak φ,i (, ρ) Defiitio 2.5. The rak-efficiecy of mechaism φ, rak φ = E,ρ [RANK φ (, ρ)], where the expectatio is take with respect to a distributio o aget prefereces ad arrival-departure schedules. Claim 2.1. No olie house allocatio mechaism with three or more agets ca be Pareto efficiet ad idividually ratioal (IR). Proof. Suppose there exists a olie mechaism φ, which is IR as well as Pareto efficiet. Cosider the followig two period dyamic house allocatio with 3 agets. (Note, A i ows house h i.) A 1 : h 3 1 h 1 1 h 2, a 1 = 1 d 1 = 2 [ ] A 2 : h 1 2 h 2 2 h 3, a 2 = 1 d 2 = 1 [ ] A 3 : h 2 3 h 3 3 h 1, a 3 = 2 d 3 = 2 [ ] I this istace, A 1 h 3, A 2 h 1, A 3 h 2 is the oly feasible, Pareto efficiet ad IR allocatio. As, A 2 departs i period 1, φ should assig h 1 i period 1 oly. If it happes that A 3 : h 3 3 h 1 3 h 2, the the oly allocatio that is feasible, Pareto efficiet ad IR is, A 1 h 1, A 2 h 2, A 3 h 3. For this allocatio, φ should retai h 2 with A 2 at t = 1. But o olie mechaism ca correctly decide at t = 1 whether to assig h 1 or h 2 to A 2 because the prefereces of A 3 are ukow. I cotrast, a serial-dictatorship mechaism, i which agets release owership of their house upo arrival ad receive their most preferred house of those available upo departure (with ties broke at radom), is Pareto efficiet but ot IR. To see that this is Pareto efficiet, ote that the secod aget to be allocated caot receive a better house without the first aget receivig a worse house. This argumet cotiues iductively. Failure of IR is easy to uderstad. Because core is a stroger property tha Pareto efficiecy ad IR we also kow from Claim 2.1 that o olie mechaism ca be coreselectig. 3 Dyamic Top Tradig Cycle Mechaisms I this sectio, we defie the static Top Tradig Cycle Algorithm (TTCA) ad itroduce dyamic geeralizatios, leadig to a result that costrais the use of TTCA for dyamic house allocatio problems. 3.1 The Static TTCA The Top Tradig Cycle Algorithm (TTCA) [10] is strategyproof ad selects a core allocatio for the static house allocatio problem. Defiitio 3.1 (Top Tradig Cycle Algorithm). Every aget poits to its most preferred house. There will be at least oe cycle, ad the agets o ay such cycle (icludig self-loops) receive the house to which they poit. These agets are removed from the system. Now each remaiig aget poits to its most preferred remaiig house. The procedure cotiues till there are o houses left to allocate. Example 3.1 (TTCA). Cosider a problem with 5 agets, with aget A i owig house h i. Let the prefereces of these agets over houses be: A 1 : h 2 1 h 4 1 h 3 1 h 1 1 h 5 A 2 : h 3 2 h 4 2 h 5 2 h 1 2 h 2 A 3 : h 2 3 h 3 3 h 1 3 h 4 3 h 5 A 4 : h 5 4 h 2 4 h 3 4 h 4 4 h 1 A 5 : h 1 5 h 4 5 h 2 5 h 3 5 h 5
The agets poit to their most preferred house as: A 1 A 2 A 3 A 2, A 4 A 5 A 1. Now, A 2 ad A 3 form a cycle. trade ad are removed. Now, the agets poit to the houses as: A 1 A 4 A 5 A 1. This beig a cycle, A 1 gets A 4 s house, A 4 gets A 5 s ad A 5 gets A 1 s. Thus the fial allocatio by TTCA is (A 1 h 4, A 2 h 3, A 3 h 2, A 4 h 5, A 5 h 1 ). We are iterested to explore whether or ot we ca use TTCA as a buildig block for a family of strategyproof olie mechaisms. 3.2 O-lie TTCA We first cosider the simplest possible idea, which is to ru TTCA i every period i which at least oe aget departs ad commit to the allocatio determied for departig agets. Call this mechaism O-TTCA. Claim 3.1. O-TTCA is ot strategy proof whe there are three or more agets. Proof. Cosider a example with 3 agets N = {A 1, A 2, A 3 }, with aget A i owig house h i, ad prefereces: A 1 : h 3 1 h 2 1 h 1 (a 1 = 1, d 1 = 3) [ ] A 2 : h 1 2 h 2 2 h 3 (a 2 = 1, d 2 = 1) [ ] A 3 : h 1 3 h 3 3 h 2 (a 3 = 2, d 3 = 3) [ ] If A 1 reports truthfully, at t = 1, the A 1 A 2 trade will occur ad there will be o trade i periods t {2, 3}. But A 1 ca report h 3 1 h 1 1 h 2. Now o trade occurs i t = 1, ad at t = 3, agets A 1 A 3 trade ad A 1 will receive h 3 which is preferred to h 2. 3.3 Precludig Multiple Trades A sample path ω = (, ρ) is a istace of the dyamic house allocatio problem. At each time t, let ω(t) deote the restrictio of ad ρ to oly those agets with a i < t ad ω(t 1, t 2 ) deote the restrictio to agets with a i {t 1,..., t 2}. Sample path ω(t, t ) is a valid cotiuatio of ω(t) if ω(t, t ) is a istace of agets ad reported prefereces arrivig i [t, t ]. To avoid corer cases, i this sectio we cosider a geeralizatio of the model where there are N classes of equivalet houses; houses i each class are idetical. Agets have strict preferece over classes of houses, ad multiple agets may ow houses i the same class. Agets owig the same house are said to be similar, though they might have differet preferece reports. For TTCA to remai SP, we eed a arbitrary way to break ties amog idetical houses whe lookig at cycles. A atural way would be to break ties with the arrival order of agets, with house belogig to later arrivig agets havig higher priority (ad otherwise at radom). For example, A ows h 1 ad arrives before B, who ows h 2; h 1 ad h 2 are i the same class. If C most prefers that class, i TTCA he would poit to h 2 first. If that s ot available to him, the C would poit to h 1 ad so o. Give a geeralized TTCA with such a tie breakig scheme, we may simulate it with a classical TTCA, cttca, where there are oly distict houses. Cosider aget A participatig i TTCA with preferece, A = c 1 A c 2..., where c i is a class of houses. We ca costruct a aget i cttca with preferece amog houses preset i the system: A= c 11 A c 12... A c 21 A c 22..., where c i1 is the house i class i with the highest tie-break priority, i.e. belogig to the latest arrivig aget. Ruig classical cttca o { } is idetical to ruig TTCA o { } with the tie breakig scheme ad remais SP. We cosider olie mechaisms where agets trade through participatio i TTCA cycles. I order to be feasible, if a TTCA cycle occurs at time t, the all participatig agets must be preset at t. A aget may participate i multiple TTCA cycles. However the example below shows how the ability to participate i multiple TTCA cycles ca easily create icetives for agets to misreport their prefereces. This is familiar from Kurio [5]. Motivatig Example. Cosider a sceario with three agets (A, h A ), (B, h B ), ad (C, h C ). A arrives ad departs i time 1 ad reports preferece h C A h A A h B. B arrives ad departs i time 2 ad reports preferece h A B h B B h C. C arrives ad i 1, departs i 2 ad the mechaism allows C to participate i TTCA at both times. Suppose C has true preferece h B C h C C h A. If C reports his true preferece, the there would be o trade for ay aget. But if C misreports h B C h A C h C, the he ca obtai h A i the first roud, which he ca use to obtai h B i the secod roud. I the rest of this sectio, we restrict our cosideratio to mechaisms that allow each aget to participate i at most oe TTCA. Give ay sample path ω, we may uambiguously state the participatio time of a aget A, deoted t( A ) for report A, ad the time (if ay) whe it participates i TTCA. A aget preset at time t ad who has ot participated i TTCA yet is said to be available. This is still a rich domai of mechaisms; which TTCA a aget participate i ca deped o his arrival/departure ad preferece report. A mechaism is simple if give a sceario ω(0, t) ad aget (A, h 0, A, a, d) preset ad available at t, there exists a set of agets Λ = {A i, h i, i, a i > t, d i}, called the perfect match set for A i sceario ω(0, t), such that (a) if a cotiuatio ω(t+) cotais Λ, the A receives his most preferred house accordig to A uder the sceario ω(0, t) + ω(t+), ad (b) if B is preset i ω(0, t) ad ot similar to A the B does ot trade with ay aget i Λ. Two observatios about simple mechaisms: [O1] Give a sceario ω(0, t) ad ay aget A preset ad available i period t ad with d A > t, there exists a cotiuatio ω 1 where the perfect match set for A arrives ad A receives h 1, his most preferred house. [O2] Give a sceario ω(0, t), ad a aget A preset ad available i t with d A > t, there exists a cotiuatio where for each of the other preset agets, B, ot similar to A, a perfect match set for B arrives, ad these are the oly arrivig agets. I this sceario, A would have o cadidate for trade sice all other origially preset agets ot similar to it would be matched up with perfect match sets ad A ca ot trade with aother aget s perfect match sets by defiitio. A ca ot trade with a similar aget sice they have idetical houses. Later we will demostrate that our mechaisms are simple. Now we state the requiremets for simple mechaisms to be strategyproof. Claim 3.2. If a olie house allocatio mechaism is SP ad simple ad aget A participates i TTCA i period t( A) for some report A, the fixig sceario ω(0, t) i regard to all agets except A, aget A cotiues to participate i period t( A) for all reports A. Proof. Cosider ay ω (fixig the istace for all agets except A.) Let t 1 be the earliest of {t( A)} for all reports A for aget A, ad A be the correspodig preferece report. Let ω(0, t 1 ) be the
restrictio of ω up to t 1. We wat to prove that if there exists A with participatio time > t 1, the we ca costruct a sceario where A has icetive to misreport. We cosider various classes of A : (1) If A has h 1 A h 0, the A reportig A must participate i TTCA at t 1. Suppose A does ot participate at t 1. By [O2], there exists a cotiuatio ω 1 (t 1 +) from t 1 oward where A eds up keepig his origial house h 0. I this sceario, A would have beefited from misreportig 1 A i order to obtai h 1. (2) Now cosider if A has the true preferece A = h 2 A... A h 1 A...h 0 A..., for ay feasible h 2 / {h 1, h 0}. By above, A participates at t 1. There are two possibilities. i. (A, A) is allocated h 2 at t 1. By (1) this implies i particular that all A that rak h 2 highest must trade at t 1, ad ot just prefereces of type A. ii. (A, A) is ot allocated h 2 at t 1. Suppose there exists A= h 2 A... that does ot trade at t 1. The there is a cotiuatio ω 2(t 1+) from t 1 oward where A reportig A receives h 2 by [O1]. The A would have beefited from reportig A istead of A uder this sceario. Sice h 2 is arbitrary, this shows that all preferece reports must participate at t 1. This shows that a simple SP mechaism caot use aget s reported preferece to decide at which time to let it participate i TTCA. 4 4 Partitio Mechaisms Give the aalysis i the previous sectio, we cosider ow a special class of mechaisms that esure that each aget oly participates i a sigle TTCA, ad moreover determies the group of agets with which a aget ca trade without cosiderig the reported type profile of a aget. Defiitio 4.1 (Partitio mechaism). Partitio the agets ito groups, {(P 1, t 1), (P 2, t 2),..., (P k, t k )}, such that all the agets i P j N are preset at t = t j ad each aget is i exactly oe group. The partitio is costructed i a way that is idepedet of the agets reported prefereces. Ru TTCA o tradig set P j i period t j, for every j {1,..., k}. We have the followig easy claim: Claim 4.1. A partitio mechaism is strategyproof. This follows immediately from the strategyproofess of TTCA give that each aget is placed i a sigle TTCA evet ad this placemet is idepedet of its report (ad recall that arrival ad departure times are ot maipulable i our model.) 4.1 Simple Partitio Mechaisms Oe simple example of a partitio mechaism is DO-TTCA, which rus TTCA oly amogst the agets that depart i each period t. 4 It is possible for a simple SP mechaism to use reported prefereces to decide i which partitio of TTCA to let a aget participate i, so log as the partitios are cocurret ad always occur i the same roud. Cosider a family of such mechaism as follows: at time t, take all the preset agets ad arbitrarily divide ito two groups P1 ad P2 (ca be more tha two). For a ew aget A with reported pref, simulate the TTCA of i P1 ad P2 ad see uder which partitio, A would have received better allocatio uder ad the let A joi that partitio. Such mechaisms are strategyproof. Through simulatio, we observe empirically that the performace of the mechaism improves whe agets are allowed to participate i TTCA with a large umber of other agets. So i practice, it s more effective to combie the partitios P1 ad P2 ito oe large partitio P. We therefore examie i detail mechaisms where the TTCA a aget participates i does ot deped o its reported preferece. This cotiues util all agets have arrived, where-up TTCA is ru with the remaiig agets. Oe obvious flaw with DO-TTCA is that it forfeits the chace to execute trades amogst a large umber of agets that are preset but may depart at distict times. The Threshold TTCA (T-TTCA) is desiged to address this problem. Let D(t) = {i N : d i = t}. A active aget i period t is a aget that is preset i the market ad has ot yet participated i a TTCA. Let A(t) deote the active agets i period t. Algorithm 1 T-TTCA INPUT: (N,, ρ) OUTPUT: House allocatio 1: while some agets still to arrive do 2: At each time slot t 3: if D(t) the 4: if A(t) > THRSHD the 5: Execute TTCA with the agets A(t) 6: Mark all the agets i A(t) as iactive. 7: else 8: Execute TTCA with the agets D(t) 9: ed if 10: ed if 11: if all the agets have arrived the 12: ru TTCA with all the preset agets 13: ed if 14: ed while The THRSHD parameter ca be selected for a particular probabilistic model of aget prefereces ad arrival-departures to maximize system performace. 4.2 Stochastic Optimizatio Mechaism SO-TTCA adopts a sample-based stochastic optimizatio method for partitioig the agets. Every aget i A(t) D(t), is icluded i the tradig group i period t. For ay other aget A i A(t) \ D(t), the decisio about whether to iclude the aget is made based o the solutio to K (sampled) offlie schedulig problems. The offlie problem determies a partitio of agets give kowledge of the arrival-departure schedule but without cosiderig aget prefereces. A reasoable heuristic first idetifies the period i which there are maximum umber of agets preset, say t 1. Cosider these agets as oe group i the partitio, say P 1, ad the recurse o the remaiig agets. The ituitio is that it teds to be beeficial to move a aget from a smaller tradig group ito a larger tradig group because this leads to more tradig optios for each participat. A empirical aalysis of the rak-efficiecy for differet methods to costruct partitios supports this ituitio. Let N i deote the umber of times aget A i is scheduled ito period t over these K offlie problems, where for each of these problems a possible arrival-departure schedule for the t agets still to arrive is sampled from a probabilistic model of the domai. Aget A i is allocated to the tradig group i the curret period if this cout N i is greater tha K(1 t ), where t is the umber of agets that have arrived up to ad icludig period t. The effect of comparig to K(1 t ) is that the mechaism is more likely to place a aget ito the tradig group as the umber of agets still to arrive decreases (ad thus the opportuity to trade decreases). 4.3 Partitio mechaisms are simple DO-TTCA: Give aget (A, h 0, A, a, d), the perfect match set is a set of idetical agets (A, h, A, d, d) such that h is A s most preferred house accordig to A ad A raks h 0 the
Algorithm 2 SO-TTCA INPUT: (N,, ρ) OUTPUT: House allocatio 1: while some agets still to arrive do 2: At each time slot t 3: if D(t) the 4: S = D(t) A(t) 5: Geerate K samples of arrival-departure for the agets still to arrive. 6: For each of these K scearios, call schedule routie. 7: For each aget i A(t) \ D(t) 8: if {k K s.t. scheduled period of i = t} > (1 t the 9: S S {i} 10: ed if 11: Execute TTCA with the agets from S 12: ed if 13: if all the agets have arrived the 14: ru TTCA with all the preset agets 15: ed if 16: ed while {1,..., T }. The departure time for aget A i is uiformly distributed o {a i, a i + T }. We adopt T = 30 ad if it happes 8 that, d i > T, we put d i = T. [E4 ] No-uiform prefereces: some houses are more demaded tha the other. We first associate a popularity idex with each house, with the popularity idex for house A j defied as the probability desity f(x) of a Normal distributio with mea 1, stadard deviatio 0.3, ad evaluated at x = j. Give a popularity idex assiged to each house, we geerate a preferece profile for aget A i by samplig houses accordig to popularity, with houses sampled without replacemet with probability proportioal to the popularity of a house. The sample order defies a aget s prefereces, with the first house the most-preferred, the secod house sampled the secod most-preferred, ad so o [13]. The arrival-departure are set as i [E1]. I T-TTCA, we varied the THRLD parameter aroud the expected umber of the agets arrivig i each period, ad experimetally observed that the rak efficiecy is optimized with a threshold that is set to the expected umber of agets arrivig i each period. For [E3], this is THRLD = ad for [E1], [E2] ad [E4] this is T THRLD = λ. highest ad h secod. Here is the umber of agets similar to A preset i the system. I ay cotiuatio that cotais such A s, A s ad A will be i the same TTCA at time d ad oe A will trade with A. Oly agets similar to A may trade with a A. T-TTCA: Give aget (A, h 0, A, a, d) ad ay sceario ω(0, t) such that A is available at t, with t < d. Fix some threshold, THRSHD. A perfect match set for A is a set of m agets {A i, h i, i, t + 1, t + 1} who arrive ad depart i time t + 1, where m > T HRSHD. The first of agets A i are idetical to A costructed for DO-TTCA. All other agets rak their ow house h i first. They are dummy agets who do ot trade ad oly serve to trigger the threshold. I ay cotiuatio that cotais {A i}, at time t + 1, there are more available agets tha THRSHD, ad hece all preset agets participate i TTCA. Agets A ad oe A would trade ad A gets his top house h. SO-TTCA: Give aget (A, h 0, A, a, d) ad ay sceario ω(0, t) such that A is available at t, with t < d. Suppose there are t agets still yet to arrive. A perfect match for A is the same set of agets {A i } as costructed i the case of T-TTCA, with m = t. I ay cotiuatio cotaiig {A i}, there are o additioal future arrivals. Hece i the offlie schedulig problem, all agets preset at t + 1 would be put ito the same partitio. I this TTCA cycle, A would trade with A ad obtai h. 5 Simulatio Results We perform experimets to evaluate the rak-efficiecy of O-TTCA, DO-TTCA, T-TTCA, ad SO-TTCA uder various simulated eviromets. The prefereces over houses are sampled uiformly at radom for all the agets except i eviromet [E4]. For each eviromet ad each mechaism, all results are averaged over 2000 radom problem istaces. By waitig time (or patiece) we mea the umber of periods a aget is preset i the market. [E1 ] A Poisso process with arrival rate λ is ru util agets arrive. Each aget s waitig time is expoetially distributed with parameter µ. We adopt λ =, ad µ = 0.01λ. 8 [E2 ] Same as [E1] except µ = 0.1λ [E3 ] Arrival time for every aget is uiformly distributed o Figure 1. Rak-efficiecy agaist the umber of agets i eviromet [E1] for patiet agets with λ = /8 ad µ = 0.01λ. We plot the rak-efficiecy, for eviromets [E1] to [E4] i Figures 1 4 respectively. I each figure, we also iclude the rakefficiecy for the O-TTCA mechaism as a referece. Recall that the O-TTCA mechaism is ot strategyproof, ad sigificatly less costraied i that it allows for tradig i every period. We observe that whe agets arrive ito the system by a Poisso process, SO-TTCA improves rak-efficiecy over DO-TTCA by 4-10% depedig upo the waitig times of the agets. Specifically, whe the µ = 0.1λ, SO-TTCA improves by 4% ad for µ = 0.01λ it improves over DO- TTCA by 10%. T-TTCA performs well oly whe the waitig times are lower (i.e., for higher µ, [E2]). Whe the waitig times are high (i experimet [E1]), T-TTCA fails to capture the future periods i which there may be large umber of agets accumulated. Eve if there are may active agets i the curret period, they may be patiet ad willig to wait for some future period. I [E3], the SO-TTCA ad T-TTCA mechaisms have much better performace tha DO-TTCA, improvig rak-efficiecy by 15-
Figure 2. Rak-efficiecy agaist the umber of agets i eviromet [E2] for less patiet agets with λ = /8 ad µ = 0.1λ Figure 3. Rak-efficiecy agaist the umber of agets i eviromet [E3] with a uiform arrival-departure model. 20% for large umbers of agets. The performace of SO-TTCA ad T-TTCA are almost idetical uder this model. With the waitig time beig relatively small as compared to experimets i [E1] ad [E2], T-TTCA is able to capture the periods i which more agets are preset simultaeously. I geeral, SO-TTCA is more robust to differet arrival-departure models tha T-TTCA. I additio, SO-TTCA requires o tuig, whereas T-TTCA requires that the threshold is appropriately set. 6 Coclusios I this paper we have cosidered a dyamic versio of the house allocatio problem. We idetify a trade-off betwee efficiecy ad idividual ratioality i the olie versio of the house allocatio problem, ad idetify a requiremet that agets caot be able to trade with differet subsets of agets by chagig their type report. We cosider a family of partitio mechaisms i which a aget s tradig group is determied without regard to its type. A clear beefit is established for T-TTCA ad SO-TTCA over a method that performs tradig upo the departure of oe or more agets, ad the stochastic optimizatio approach (SO-TTCA) outperformig the threshold approach (T-TTCA) for Poisso arrival processes. A iterestig aspect of the use of stochastic optimizatio is that it determies which subset of agets should trade but ot what trade should occur, ad without cosideratio of specific aget types. This preserves strategyproofess. The most iterestig immediate ext steps are to cosider a geeralizatio i which agets ca misreport arrival ad departure, while cotiuig to explore the role for stochastic optimizatio i idetifyig useful tradig groups. Ackowledgmets. The first author wat to ackowledge Prof Y Narahari ad Ifosys Techologies Pvt Ltd for fiacial support. REFERENCES [1] A. Abdulkadiroglu ad T. Sömez, House allocatio with existig teats, J. of Ecoomic Theory, 88(2), 233 260, (1999). [2] P. Awasthi ad T. Sadholm, Olie stochastic optimizatio i the large: Applicatio to kidey exchage, i Proc. IJCAI 09, (2009). [3] Y. Chevaleyre, U. Edriss, ad N. Maudet, Simple egotiatio schemes for agets with simple prefereces: Sufficiecy, ecessity ad maximality, J. Auto. Agets ad Multiaget Systems, 20(2), 234 259, (2010). [4] S. Gujar ad D. C. Parkes, Dyamic Matchig with a Fall-back Optio, i Proc. ECAI, (2010). [5] M. Kurio, House allocatio with overlappig agets: A dyamic mechaism desig approach, Techical report, Dept. of Ecoomics, Uiv. of Pittsburgh, (2009). [6] J. Ma, Strategy-proofess ad the strict core i a market with idivisibilities, It. J. of Game Theory, 23(1), 75 83, (1994). [7] D. C. Parkes, Olie mechaisms, i Algorithmic Game Theory, eds., N. Nisa, T. Roughgarde, E. Tardos, ad V. Vazirai, chapter 16, CUP, (2007). [8] A. E. Roth, Icetive compatibility i a market with idivisible goods, Ecoomics Letters, 9(2), 127 132, (1982). [9] T. Sadholm, A implemetatio of the cotract et protocol based o margial cost calculatios, i Proc. AAAI, pp. 256 256, (1993). [10] L. Shapley ad H. Scarf, O cores ad idivisibility, J. of Math. Eco., 1(1), 23 37, (1974). [11] T. Sömez ad U. Üver, Matchig, allocatio, ad exchage of discrete resources, i Hadbook of Social Ecoomics, eds., J. Behabib, A. Bisi, ad M. Jackso, Elsevier, (2008). [12] U. Üver, Dyamic kidey exchage, Rev. of Eco. Stud., (2009). [13] J. Zou, S. Gujar, ad D. C. Parkes, Tolerable Maipulability i Dyamic Assigmet without Moey, i Proc. AAAI, (2010). Figure 4. Rak-efficiecy agaist the umber of agets i eviromet [E4] with prefereces that are correlated across agets ad deped o the popularity idex of a house.