Manipulating Gale-Shapley Algorithm: Preserving Stability and Remaining Inconspicuous

Transcription

1 Maniulating Gale-Shaley Algorithm: Preserving Stability and Remaining Inconsicuous Rohit Vaish Indian Institute of Science, India Dinesh Garg IIT Gandhinagar, India Abstract We study the roblem of maniulation of the men-roosing Gale-Shaley algorithm by a single woman via ermutation of her true reference list Our contribution is threefold: First, we show that the matching induced by an otimal maniulation is stable with resect to the true references Second, we identify a class of otimal maniulations called inconsicuous maniulations which, in addition to reserving stability, are also nearly identical to the true reference list of the maniulator (making the maniulation hard to be detected) Third, for otimal inconsicuous maniulations, we strengthen the stability result by showing that the entire stable lattice of the maniulated instance is contained inside the original lattice 1 Introduction The theory of two-sided matching has been a cornerstone of market design, insiring a wide array of alications such as school choice [Abdulkadiroğlu et al, 2005a; Abdulkadiroğlu et al, 2005b], entry-level labor markets [Roth and Peranson, 1999], and kidney exchange [Roth et al, 2004] All of these markets are built around a centralized matching rocedure, which solicits the references of the articiating agents (tyically in the form of rank ordered lists) and selects a matching outcome based on these references (ie, decides who gets matched to whom) Arguably the most famous of such rocedures is the Gale- Shaley algorithm [Gale and Shaley, 1962], which takes as inut the references of two sets of agents over each other (commonly referred to as men and women), and after a seuence of roosal and rejection stes, oututs a matching that is stable in the sense that no air of agents refer each other over their assigned artners Over the years, the notion of stability has emerged as the most significant redictor of a market s long term success while markets with stable matching rocedures have successfully ersisted, the unstable ones have failed and are out of use [Roth, 2002] A freuent concern in such markets, however, is that of strategic behavior by individual agents Indeed, Roth [1982] has shown that no stable matching mechanism makes it a dominant strategy for every agent to announce their true references This, in articular, means that the Gale-Shaley algorithm is also vulnerable to maniulation Our interest in this aer is to study the conditions under which the Gale-Shaley matching for the maniulated instance remains stable with resect to the true references It is known from the results of Dubins and Freedman [1981] and Roth [1982] that the roosing side (say, the men) in the algorithm has no incentive to maniulate Therefore, any strategic behavior must occur on the roosed-to side (ie, the women) Our starting oint, therefore, is the following natural uestion: Suose a woman can maniulate by ermuting her true reference list while everyone else is truthful Then, is the resulting Gale-Shaley matching stable with resect to the true references? Our first result (Theorem 3) shows that the answer to the above uestion is YES if the maniulation is otimal That is, if, by misreorting, the maniulator secures the best ossible artner (according to her true references), then the resulting Gale-Shaley matching is stable with resect to the true references of the agents This is an encouraging result for both the market designer (who cares about stability with resect to the true references) and the maniulator (who cares about finding the otimal artner) We comlement this result by showing that sub-otimal maniulations can sometimes lead to instability (Examle 1) Besides otimality, the maniulator might also want to avoid being susected of strategic behavior Indeed, it is reasonable to exect that the market designer has a coarse idea of the true references of the agents, say from ast runs of the algorithm or from survey data In such a setting, strategic misreorting can be easily detected if the announced list of an agent looks significantly different from the market designer s estimate Our second result (Theorem 4) shows that this is an avoidable concern for the maniulator: For any otimal maniulation, there exists an euivalent inconsicuous maniulation which results in the same matched artner, and can be derived from the true list by romoting only one man (and making no other changes) In other words, an otimal maniulation is nearly indistinguishable from the true list Our third result (Theorem 5) strengthens the stability imlication of Theorem 3 for inconsicuous otimal maniula- 437

2 tions Secifically, we show that every matching (including the Gale-Shaley matching) that is stable with resect to the maniulated instance is also stable with resect to the true references We comlement this result by giving an examle of a non-inconsicuous otimal maniulation, where a matching that is stable with resect to the maniulated instance is not stable with resect to the true references (Examle 2) Why study ermutation maniulations? By far, the most commonly studied model of maniulation in the stable matching literature has been that of truncation [Gale and Sotomayor, 1985; Roth and Rothblum, 1999; Coles and Shorrer, 2014; Jaramillo et al, 2014] This model allows the maniulator to remove a tail of her true reference list and reort the rest of the list unshuffled An esecially attractive feature of truncation-based maniulations is exhaustiveness, which means that any ossible misreresentation can be relicated or imroved by a truncation strategy [Roth and Vate, 1991; Jaramillo et al, 2014] However, an otimal truncation maniulation by an agent can look very different from her true reference list In fact, Coles and Shorrer [2014] have shown that when the true references of all the other agents are generated uniformly at random, the otimal truncation strategy for the maniulator involves the removal of a large fraction of her true list This henomenon is articularly severe for large markets, where the ratio of the length of the reorted list to that of the true list goes to zero as the size of the market grows By contrast, as our results show, an otimal ermutation maniulation can be nearly identical to the true list of the maniulator We consider this feature of ermutation maniulations to be an imortant one, both from a theoretical and a ractical standoint 2 Related Work The literature on ermutation maniulations of Gale-Shaley algorithm has focused rimarily on comutational uestions [Teo et al, 2001; Kobayashi and Matsui, 2009; Kobayashi and Matsui, 2010; Deng et al, 2015; Guta et al, 2016] The earliest result in this direction is by Teo et al [2001], who gave an O(n 3 ) algorithm for comuting an otimal ermutation maniulation of Gale-Shaley algorithm by a single maniulator In conjunction with our result (Theorem 3), this means that not only are otimal maniulations stabilityreserving, but are also efficiently comutable The work of Teo et al [2001] was generalized to the coalitional maniulation setting by Deng et al [2015] For ermutation maniulations, they rovide an O(n 6 ) algorithm for comuting a areto-otimal maniulation for any fixed coalition Additionally, they show that any such maniulation can be made inconsicuous There are two imortant ways in which the work of Deng et al [2015] differs from ours: First, their model only allows for maniulations that are stabilityreserving in the first lace On the other hand, we allow the maniulator to ick any ermutation (not just a stabilityreserving one), and identify a class of maniulations that induce stability Second, their result on the inconsicuousness of coalitional maniulation involves a reduction to the stable roommates roblem [Irving, 1985], and uses sohisticated combinatorial structures like rotations and suitor grahs [Kobayashi and Matsui, 2010] Our roof, on the other hand, is direct and much simler, which we consider an imortant contribution of this work Another set of relevant works is by Kobayashi and Matsui [2009; 2010], who study the following extension version of the maniulation roblem: Suose we are given the reference lists of all men, and a comlete/artial matching The goal is to comute a set (if it exists) of the reference lists of all women such that the men-roosing Gale-Shaley matching contains the given matching They show that this roblem is NP-comlete in general (when the given matching is artial), but olynomial time solvable if the given matching is comlete Finally, the recent work of Guta et al [2016] studies the uestion of total stability of a maniulation strategy when all women are strategic individuals and all men are truthful A maniulation strategy is totally stable if (i) it is stabilityreserving, and (ii) it constitutes a stable Nash euilibrium, meaning that any unilateral deviation from the strategy is either non-imroving or induces instability It is shown that checking a given strategy for total stability can be done in olynomial time 3 Preliminaries Problem setu An instance M, W, of the stable marriage roblem consists of a set M of n men, a set W of n women, and a reference rofile = m1,, mn, w1,, wn } consisting of the reference lists of all men and women The reference list of each man m M, denoted by m, is a strict total order over the set of all women (the reference lists of women are defined analogously) We will use the shorthand w 1 m w 2 to denote either w 1 m w 2 or w 1 = w 2 We let w denote the reference lists of all men and women excet woman w Thus, = w, w } Stable matchings A matching refers to a function µ : M W M W, where µ(m) W for all m M, µ(w) M for all w W, and µ(m) = w if and only if µ(w) = m A matching µ admits a blocking air with resect to if there is a air of agents (m, w) such that w m µ(m) and m w µ(w) A matching µ is stable if it admits no blocking air with resect to We let S denote the set of all matchings that are stable with resect to Gale-Shaley algorithm A matching algorithm takes as inut a reference rofile and oututs a matching In this aer, we study the well-known deferred accetance algorithm [Gale and Shaley, 1962], also known as Gale-Shaley algorithm In articular, we will focus on the men-roosing version of this algorithm, abbreviated as GS algorithm We let µ = GS( ) denote the matching outut by the menroosing Gale-Shaley algorithm for the inut rofile Briefly, GS algorithm roceeds in rounds, and each round consists of two hases: (i) a roosal hase, where each single man rooses to his favorite woman from among those 438

3 who haven t rejected him yet, and (ii) a rejection hase, where each woman with multile roosals in hand rejects all roosals excet the one that she likes best The algorithm terminates when no single agents remain Gale and Shaley [1962] showed that GS algorithm always terminates with a stable matching Moreover, this matching is simultaneously the best for all men from among all stable matchings (and, as McVitie and Wilson [1971] later observed, the worst for all women) Theorem 1 recalls these results Theorem 1 (Men-otimality of GS) Let be a reference rofile, and let µ = GS( ) Then, µ S Moreover, for any µ S, µ(m) m µ (m) for all m M and µ (w) w µ(w) for all w W We let Pro(w, ) denote the set of all men who roose to w during the run of GS algorithm on Further, we let Pro(w,, i) denote the i th favorite man of w (according to w ) in the set Pro(w, ) Thus, Pro(w,, 1) = µ(w) for µ = GS( ) Stable lattice Given a reference rofile and two matchings µ and µ, define the join function µ = µ µ as follows: for each m M and w W, µ(m) if µ(m) m µ µ (m) = (m) µ (m) otherwise, and µ µ (w) = (w) if µ(w) w µ (w) µ(w) otherwise Similarly, define the meet function µ = µ µ for all m M and w W as: µ µ (m) = (m) if µ(m) m µ (m) µ(m) otherwise, and µ(w) if µ(w) w µ µ (w) = (w) µ (w) otherwise The following result from [Knuth, 1997], attributed to John Conway, shows that the join and meet of any air of stable matchings are also stable Theorem 2 (Lattice of stable matchings) Let be a reference rofile and let µ, µ S Then, µ, µ S Maniulation A matching algorithm is said to be maniulable by an agent w if there exists a air of reference rofiles and, differing only in the references of w, such that µ (w) w µ(w), where µ and µ are the matchings before and after the maniulation resectively The agent w is referred to as the maniulator In this study, we only consider ermutation maniulations, ie, w is a ermutation of w Besides, we focus only on the maniulation of GS algorithm, and the maniulator is assumed to be a woman, who knows the references of all the other agents We will often use = ( w, w) and µ = GS( ) to denote the maniulated rofile (with resect to ) and the resulting GS matching resectively The matching µ and the set S will be referred to as the induced matching and the induced lattice resectively The set S will be referred to as the original lattice We will say that a maniulation w with resect to is stability-reserving if µ S Otimal maniulation Given a reference rofile, an otimal maniulation of GS algorithm by an agent w with resect to a rofile refers to a reference list w such that (i) µ (w) w µ(w), and (ii) µ (w) w µ (w) for any other reference list w, where µ = GS( w, w) and µ = GS( w, w) Inconsicuous euivalent of a maniulation Given a reference rofile, we call w an inconsicuous euivalent of a maniulation w (of GS algorithm by woman w) if (i) w can be derived from the true reference list w by moving at most one man, and (ii) µ (w) = µ (w), where µ = GS( ) and µ = GS( ) 4 Main Results Our first result (Theorem 3) shows that otimal maniulation of GS algorithm by a single woman reserves stability Theorem 3 (Otimal maniulation is stability-reserving) Let w be an otimal maniulation with resect to for woman w and let µ = GS( ) Then, µ S This is a ositive result, since the stability of the resulting matching is not affected by otimal strategic behaviour of a single agent Examle 1 comlements this result by showing that sub-otimal strategic behavior can lead to instability Examle 1 (Sub-otimal maniulation can be unstable) Consider the stable marriage instance shown below True references of men True references of women m 1: w 1: m 2: w 2: m 3: w 3: m 4: w 4: w 1 maniulates w 1 maniulates sub-otimally ( ) otimally ( ) w 1: w 1: w 2: w 2: w 3: w 3: w 4: w 4: The to row shows the true references of the agents, and the bottom row shows two different maniulations by w 1 with resect to The underlined numbers denote the matched artners of the women under the men-roosing Gale-Shaley algorithm Notice that the maniulation under is sub-otimal for w 1 because she matches to a strictly better artner (according to her true reference list w1 ) under Besides, the matching induced under is not stable with resect to the true references since (m 1, w 1 ) is a blocking air On the other hand, the maniulation under is otimal, and the induced matching is stable with resect to the true references As mentioned earlier in Section 1, it is still ossible that the list reorted by the maniulator (as art of otimal maniulation) might look very different from her true list Indeed, for the stable marriage instance in Examle 1, notice 439

4 that excet for the most referred man (namely m 1 ), the otimal maniulation w 1 by w 1 is a comlete reversal of her true reference list w Theorem 4 addresses this concern by showing that any otimal maniulation has an inconsicuous euivalent which results in the same matched artner for the maniulator (thus maintaining otimality) while differing from the true list in the ositioning of only one man (thus making the maniulation hard to be detected) Theorem 4 (Inconsicuous euivalent of otimal maniulation) Let w be an otimal maniulation with resect to for woman w Let w be another reference list derived from her true list w by romoting the man Pro(w,, 2) to the osition right after Pro(w,, 1) while making no other changes Then, w is the inconsicuous euivalent of w, ie, µ (w) = µ (w), where µ = GS( ) and µ = GS( ) Going back to Examle 1, it can be observed that w 1 can achieve her otimal artner (ie, m 1 ) by reorting the list m 1 m 4 m 2 m 3, derived from her true list w by romoting the man m 4 and making no other changes Our final result strengthens the stability imlication of Theorem 3 for inconsicuous otimal maniulations Recall from Theorem 3 that the men-roosing Gale-Shaley matching for the otimally maniulated instance is stable with resect to the true references This result can be alternatively interreted in the lattice terminology as follows: Theorem 3 shows that the men-otimal extreme of the lattice S always lies inside the original lattice S One might wonder whether the same holds for the rest of the maniulation lattice as well Theorem 5 shows that this is indeed the case for inconsicuous otimal maniulations Theorem 5 (Lattice containment result) Let w be an otimal maniulation with resect to for woman w, and let w be its inconsicuous euivalent (as described in Theorem 4) Then, S S Examle 2 comlements Theorem 5 by showing that the imlication does not extend to non-inconsicuous otimal maniulations Examle 2 Consider the following stable marriage instance True references of men True references of women m 1: w 1: m 2: w 2: m 3: w 3: m 4: w 4: m 5: w 5: Ot + Non-Incons Ot + Incons maniulation ( ) maniulation ( ) w 1: w 1: w 2: w 2: w 3: w 3: w 4: w 4: w 5: w 5: The to row shows the true references of the agents, while the bottom row shows two different otimal maniulations and for the maniulator w 1 The underlined numbers denote the matched artners of the women under the menroosing Gale-Shaley algorithm First, we briefly describe why and are both otimal Observe that during the run of GS algorithm on the true rofile, w 1 receives roosals only from m 4 and m 5 Therefore, any maniulation by w 1 must involve swaing the relative ordering of m 4 and m 5 By rejecting m 4 in favor of m 5, w 1 forces m 4 to roose to w 3, thereby dislacing m 3 As a result, m 3 is forced to roose to w 1 This already is an imrovement for w 1 over her original artner m 4 However, w 1 can do even better by retending to refer m 5 over m 3 in the maniulated list This forces m 3 to roose to w 4, thereby dislacing m 2, who in turn is forced to roose to w 1, giving her a more referred match Notice that w 1 can do no better, since rejecting m 2 in favor of m 5 will not dislace m 1 in the reference list of w 2 Thus, m 2 is the otimal artner for w 1 Let us now consider the matching φ defined as φ = (m 1, w 5 ), (m 2, w 1 ), (m 3, w 2 ), (m 4, w 3 ), (m 5, w 4 ) It is easy to verify that φ S but φ / S and φ / S since (m 1, w 1 ) constitutes a blocking air in each case Therefore, the stable lattice induced by an otimal maniulation (in this case ) need not be comletely contained inside the original lattice 5 Proofs of Main Results 51 Proof of Stability of Otimal Maniulation (Theorem 3) We start by stating a lemma (Lemma 6) that will be useful in the roof of Theorem 3 The lemma states that if a fixed man m rooses to a fixed woman w during the run of GS algorithm, then the same holds when w moves that man u/down in her list while making no other changes The roof is omitted due to sace constraints (Notice that the lemma alies only to the fixed man m who was moved in the reference list of w, and not to any other man who roosed to w but was not moved) Lemma 6 Let and be two reference rofiles that are comletely identical excet for the references of a fixed woman w Let w be derived from w by moving a fixed man m and making no other changes Then, m Pro(w, ) m Pro(w, ) We now rovide the roof of Theorem 3 Proof (of Theorem 3) Suose, for contradiction, that µ / S Then, there must exist a air (m, w ) that blocks µ with resect to That is, w m µ (m ) and m w µ (w ) We first claim that w = w Indeed, by the stability of GS algorithm, we have µ S Hence, either µ (m ) m w or µ (w ) w m or both If w w, then by construction w = w Similarly, m = m Therefore, we have that either µ (m ) m w or µ (w ) w m or both, which contradicts the blocking air condition Hence, w = w Next, consider a reference list w for w derived from w by romoting the man m to the to osition and making no other changes Let = ( w, w) and µ = GS( ) Since w = w, we have that m w µ (w) and w m µ (m ) 440

5 Moreover, since m = m, we have that w m µ (m ) Therefore, during the run of GS algorithm on, it must be that m rooses to and is rejected by w before he is matched to µ (m ) That is, m Pro(w, ) By alying Lemma 6 with and as the old and the new rofiles resectively, we get that m Pro(w, ) Since m is at the to osition in w, we must have µ (w) = m This, however, contradicts the otimality of w, since m w µ (w) according to the blocking air condition 52 Proof of Inconsicuous Euivalent of Otimal Maniulation (Theorem 4) Our roof of Theorem 4 crucially relies on the swaing lemma (Lemma 7) This lemma rovides a comlete descrition of the matched artners that the maniulator can secure by swaing any air of adjacent men in her reference list We state the lemma below and omit its roof due to sace constraints It will be convenient to denote the set of non-roosing men by Non-Pro(w, ) Thus, Non-Pro(w, ) = M \ Pro(w, ) Lemma 7 (Swaing lemma) Let and be two reference rofiles differing only in the references of a fixed woman w, and let µ = GS( ) and µ = GS( ) Let w be derived from w by swaing the ositions of an adjacent air of men (m i, m j ) and making no other changes Then, (1) if m i Non-Pro(w, ) or m j Non-Pro(w, ), then µ (w) = µ(w) (2) if m i, m j / Pro(w,, 1), Pro(w,, 2)}, then µ (w) = µ(w) (3) if m i = Pro(w,, 2) and m j = Pro(w,, 3), then µ (w) µ(w), m j } (4) if m i = Pro(w,, 1) and m j = Pro(w,, 2), then Pro(w,, 2) m i, m j } In words, case (1) shows that swaing two non-roosers or a rooser and a non-rooser cannot give a different matched artner Case (2) shows the same result for swaing a air of roosers outside the first and the second best Case (3) shows that the matched artner either stays the same or becomes worse (according to the true reference list) if the second and third-best roosers are swaed Finally, case (4) shows that swaing the first and the second-best roosers can lead to at most one new roosal that is better than the old artner (according to the true reference list) Proof (of Theorem 4) We construct the reference list w by starting from w and erforming a seuence of massaging oerations on it in order to make it resemble the original reference list w, excet for the lacement of the man Pro(w,, 2) Each such oeration involves swaing a air of adjacent men in the current list By reeatedly invoking swaing lemma (Lemma 7), we will argue that w continues to receive a roosal from µ (w) at each intermediate ste, giving us the desired result We start by describing the construction of the list w followed by arguing the correctness of this construction For notational convenience, we will use = Pro(w,, 1) and = Pro(w,, 2) (1) Constructing w: Starting from w, we construct a seuence of reference lists (1) w, (2) w, culminating in w as follows (refer Figure 1): (a) Promoting : The list (1) w is obtained from w by romoting to the osition right after Since any men between and in the list w can only be non-roosers, it follows from case (1) of swaing lemma (Lemma 7) that the run of the algorithm on the rofile (1) = ( w, (1) w ) is identical to that on Thus, Pro(w, (1), 1) = and Pro(w, (1), 2) = Notice that any man above in the list (1) w must be a non-rooser (b) Fixing the art of the list above : Our goal in this ste will be to make the art of the list (1) w above and including resemble that in the list w We achieve this by relacing the set of non-roosers above in (1) w with the set of men above in the true list w It is easy to see that no man in the latter set can be a rooser to w at any stage, or else the otimality of the maniulation is violated Thus, our task involves shuffling around a set of non-roosers in the list (1) w, which, by case (1) of swaing lemma (Lemma 7), does not affect the run of GS algorithm We call the resulting reference list (2) w It is easy to verify that after this ste, the new list (2) w resembles the true reference list w for all ositions above and including, and that Pro(w, (2), 1) = and Pro(w, (2), 2) =, where (2) = ( w, (2) w ) (c) Fixing the art of the list below : The final ste in our construction involves a seuence of reference lists (3) w, (4) w, } The list (k+1) w is derived from (k) w by swaing a air of adjacent men (m i, m j ) in (k) w such that (i) Pro(w, (k), 2) w (k) m i and Pro(w, (k), 2) (k) w m j, and (ii) m i w m j and m j (k) w m i In words, each new list in the seuence is derived from the revious list by swaing a air of adjacent men that are (i) both ositioned below the second-favorite rooser according to the revious list, and (ii) are incorrectly ordered with resect to the true reference list w No other changes are made Notice that this seuence of reference lists must be finite since there can only be a finite number of airs of men that are incorrectly ordered with resect to the true list Let (l) w be the final list in this seuence, and let (k) = ( w, (k) w ) and µ (k) = GS( (k) ) denote the reference rofile and the corresonding GS matching at each ste This finishes the construction of the seuence of reference lists (2) Correctness: We will now show that Theorem 4 holds for 441

6 w (1) ẉ (2) ẉ (k) ẉ (k+1) w (l) w resembles w m j m i m i m j Figure 1: The seuence of reference lists constructed in the roof of Theorem 4 Here, = Pro(w,, 1) = µ (w) and = Pro(w,, 2) w = (l) That is, we will show that: (a) The list (l) w can be obtained from the true list w by romoting to the osition right after while making no other changes, and (b) Pro(w, (l), 1) = µ (l) (w) = Case (a) follows easily from the above construction Indeed, the stes (1a) and (1b) ensure that (l) w resembles w for all ositions above and including, while ste (1c) incrementally corrects for airs that are out of order with resect to w, thus eventually terminating with a list (l) w that is identical to w excet for the ositioning of the man We rove case (b) by induction The base case consists of showing that µ (3) (w) = Indeed, let (m i, m j ) be the air of adjacent men in (2) w that are swaed in (3) w By construction, we know that m i, m j / Pro(w, (2), 1), Pro(w, (2), 2)} Therefore, from cases (1) and (2) of swaing lemma (Lemma 7), we have that µ (3) (w) = We now roceed to the induction hyothesis That is, we now assume that µ (k) (w) = for all 3 < k K, and show that µ (K+1) (w) = As before, let (m i, m j ) be the air of adjacent men that are swaed in (K) w to obtain (K+1) w By construction, we have that m i, m j / Pro(w, (K), 1), Pro(w, (K), 2)} Therefore, from cases (1) and (2) of swaing lemma (Lemma 7), we have that µ (K+1) (w) = µ (K) (w) Finally, using the induction hyothesis, we get that µ (K+1) (w) = Hence, by induction, we have that µ (l) (w) = This finishes the roof of case (b) and, as a result, of Theorem 4 53 Proof of Lattice Containment (Theorem 5) Proof (of Theorem 5) Let us assume, for contradiction, that there exists a matching φ such that φ S \ S Then, there must exist a air (m, w) that blocks φ with resect to (it follows from the argument in the roof of Theorem 3 that one of the blocking agents must be the maniulator w) Thus, m w φ(w) and w m φ(m) Consider the set S Since φ S, it follows from Theorem 2 that φ(w) w µ (w), where µ = GS( ) Moreover, from Theorem 4, we know that µ (w) = µ (w) for µ = GS( ) Therefore, we also have that φ(w) w µ (w) By inconsicuousness of w, we have that φ(w) w µ (w) Combining this with the blocking air condition above, we get that m w φ(w) w µ (w) Once again, by inconsicuousness of w, we have that m w φ(w) We also know that m = m by construction Therefore, from the blocking air condition, we have that w m φ(m) Combining this with the condition m w φ(w), we get that the air (m, w) blocks φ with resect to, which contradicts the assumtion φ S Therefore, S S 6 Concluding Remarks We studied the roblem of maniulation of Gale-Shaley algorithm by a single agent, and identified a class of maniulations called inconsicuous maniulations which are otimal, stability-reserving, and nearly identical to the maniulator s true reference list Our work motivates several directions for future work First, extensions of the current model to a setting with several self-interested maniulators (seeking an euilibrium strategy) will be interesting to study It will also be interesting to find out if our techniues in articular, swaing lemma can lead to faster algorithms for otimal maniulation by a single agent or a coalition of agents; the current best algorithms for these roblems are O(n 3 ) [Teo et al, 2001] and O(n 6 ) [Deng et al, 2015] resectively Finally, it would be of interest to conduct a similar analysis for other stable matching algorithms, such as ones that comute minimum regret [Knuth, 1997; Gusfield, 1987], egalitarian [Irving et al, 1987; Teo and Sethuraman, 1998] or median [Teo and Sethuraman, 1998] stable matchings Acknowledgements We thank the anonymous reviewers for their helful comments Part of this work was done at IBM Research, Bangalore, India RV thanks Microsoft Research for a travel grant to resent this work at IJCAI

7 References [Abdulkadiroğlu et al, 2005a] Atila Abdulkadiroğlu, Parag A Pathak, and Alvin E Roth The New York City High School Match American Economic Review, ages , 2005 [Abdulkadiroğlu et al, 2005b] Atila Abdulkadiroğlu, Parag A Pathak, Alvin E Roth, and Tayfun Sönmez The Boston Public School Match American Economic Review, ages , 2005 [Coles and Shorrer, 2014] Peter Coles and Ran Shorrer Otimal Truncation in Matching Markets Games and Economic Behavior, 87: , 2014 [Deng et al, 2015] Yuan Deng, Weiran Shen, and Pingzhong Tang Coalition Maniulations of the Gale-Shaley Algorithm arxiv rerint arxiv: , 2015 [Dubins and Freedman, 1981] Lester E Dubins and David A Freedman Machiavelli and the Gale-Shaley Algorithm The American Mathematical Monthly, 88(7): , 1981 [Gale and Shaley, 1962] David Gale and Lloyd S Shaley College Admissions and the Stability of Marriage The American Mathematical Monthly, 69(1):9 15, 1962 [Gale and Sotomayor, 1985] David Gale and Marilda Sotomayor Ms Machiavelli and the Stable Matching Problem The American Mathematical Monthly, 92(4): , 1985 [Guta et al, 2016] Sushmita Guta, Kazuo Iwama, and Shuichi Miyazaki Total Stability in Stable Matching Games In 15th Scandinavian Symosium and Workshos on Algorithm Theory (SWAT 2016), ages 23:1 23:12, 2016 [Gusfield, 1987] Dan Gusfield Three Fast Algorithms for Four Problems in Stable Marriage SIAM Journal on Comuting, 16(1): , 1987 [Irving et al, 1987] Robert W Irving, Paul Leather, and Dan Gusfield An Efficient Algorithm for the Otimal Stable Marriage Journal of the ACM (JACM), 34(3): , 1987 [Irving, 1985] Robert W Irving An Efficient Algorithm for the Stable Roommates Problem Journal of Algorithms, 6(4): , 1985 [Jaramillo et al, 2014] Paula Jaramillo, Çaǧatay Kayı, and Fli Klijn On the Exhaustiveness of Truncation and Droing Strategies in Many-to-Many Matching Markets Social Choice and Welfare, 42(4): , 2014 [Knuth, 1997] Donald Ervin Knuth Stable Marriage and Its Relation to Other Combinatorial Problems: An Introduction to the Mathematical Analysis of Algorithms, volume 10 American Mathematical Soc, 1997 [Kobayashi and Matsui, 2009] Hirotatsu Kobayashi and Tomomi Matsui Successful Maniulation in Stable Marriage Model with Comlete Preference Lists IEICE TRANSAC- TIONS on Information and Systems, 92(2): , 2009 [Kobayashi and Matsui, 2010] Hirotatsu Kobayashi and Tomomi Matsui Cheating Strategies for the Gale-Shaley Algorithm with Comlete Preference Lists Algorithmica, 58(1): , 2010 [McVitie and Wilson, 1971] David G McVitie and Leslie B Wilson The Stable Marriage Problem Communications of the ACM, 14(7): , 1971 [Roth and Peranson, 1999] Alvin E Roth and Elliott Peranson The Redesign of the Matching Market for American Physicians: Some Engineering Asects of Economic Design American Economic Review, 89(4): , 1999 [Roth and Rothblum, 1999] Alvin E Roth and Uriel G Rothblum Truncation Strategies in Matching Markets In Search of Advice for Particiants Econometrica, 67(1):21 43, 1999 [Roth and Vate, 1991] Alvin E Roth and John H Vande Vate Incentives in Two-Sided Matching with Random Stable Mechanisms Economic theory, 1(1):31 44, 1991 [Roth et al, 2004] Alvin E Roth, Tayfun Sönmez, and M Utku Ünver Kidney Exchange The Quarterly Journal of Economics, 119(2): , 2004 [Roth, 1982] Alvin E Roth The Economics of Matching: Stability and Incentives Mathematics of oerations research, 7(4): , 1982 [Roth, 2002] Alvin E Roth The Economist as Engineer: Game Theory, Exerimentation, and Comutation as Tools for Design Economics Econometrica, 70(4): , 2002 [Teo and Sethuraman, 1998] Chung-Piaw Teo and Jay Sethuraman The Geometry of Fractional Stable Matchings and its Alications Mathematics of Oerations Research, 23(4): , 1998 [Teo et al, 2001] Chung-Piaw Teo, Jay Sethuraman, and Wee-Peng Tan Gale-Shaley Stable Marriage Problem Revisited: Strategic Issues and Alications Management Science, 47(9): ,