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Elicitation and Approximately Stable Matching with Partial Preferences

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Elicitation and Approximately Stable Matching with Partial Preferences Elicitation and Approximately Stable Matching with Partial Preferences Joanna Drummond and Craig Boutilier Department of Computer Science University of Toronto fjdrummond,[email protected] Abstract cally to minimize this informational burden. For large-scale matching problems involving hundreds or thousands of op- Algorithms for stable marriage and related match- tions on each side (e.g., hospital-resident matching, or paper- ing problems typically assume that full preference reviewer matching), not only is the cognitive burden on par- information is available. While the Gale-Shapley ticipants high, much of this ranking information will gener- algorithm can be viewed as a means of eliciting ally be irrelevant to the goal of producing a stable matching. preferences incrementally, it does not prescribe a For example, in resident matching, one expects some degree general means for matching with incomplete infor- of correlation across hospitals in the assessment of the most mation, nor is it designed to minimize elicitation. desirable candidates, and residents to have correlated views We propose the use of maximum regret to mea- on the desirability of hospitals. Thus, more desirable can- sure the (inverse) degree of stability of a match- didates will tend to be matched to more desirable hospitals, ing with partial preferences; minimax regret to find and “first-tier” participants (hospitals and residents) are wast- matchings that are maximally stable in the pres- ing time and mental energy if they offer precise rankings of ence of partial preferences; and heuristic elicitation lower-tier alternatives, while second-tier participants should schemes that use max regret to determine relevant not bother providing precise rankings of first-tier alternatives. preference queries. We show that several of our In this paper, we develop a framework for incremental pref- schemes find stable matchings while eliciting con- erence elicitation in stable matching problems, as well as siderably less preference information than Gale- procedures for robust matching in the presence of incom- Shapley and are much more appropriate in settings plete preference information. Our goal is two-fold: first, where approximate stability is viable. we want to find stable matchings while requiring that par- ticipants specify only relevant information about their prefer- 1 Introduction ences; second, because full stability may require considerable Matching problems are ubiquitous and find application in a preference information, we want to exploit some form of ap- variety of domains. One of the most widely studied economic proximate stability to further reduce the information burden matching problems is the stable marriage problem [Gale and in practice. To address the latter goal, we use maximum regret Shapley, 1962], in which members of two disjoint groups to bound the potential for defection in a matching with par- (colloquially men and women) express preferences for be- tial preferences, and devise algorithms for computing match- ing matched (married) to members of the opposite group. A ings with minimal maximum (minimax) regret. To address the stable matching ensures that specified partnerships offer no former, we develop elicitation schemes that use the match- incentive for an unmatched pair to defect from the matching. ings computed via minimax regret to find suitable queries. The stable marriage problem is emblematic of a rich set of bi- Our empirical results suggest that our regret-based methods partite matching problems in which elements of each set have and elicitation heuristics find approximate and exact stable some affinity for, or preference over, elements of the other. matchings with much less than complete preference informa- It has direct application to a variety of problems (e.g., labor tion across a variety of preference distributions, and signifi- market matching, school admissions) [Niederle et al., 2008]; cantly outperform the (interactive) GS algorithm. and the classic Gale-Shapley (GS) algorithm [Gale and Shap- 2 Background ley, 1962], and its variants, can be used to compute stable matchings very effectively. We first describe the stable matching problem addressed in While algorithms for many forms of stable matching this paper, briefly discuss prior work on matching with par- are computationally efficient, the informational burden they tial preferences and elicitation, then outline the probabilistic place on participants can be severe: they usually require that preference models used in our experiments. participants rank all potential partners/matches. Of course, Stable Matching. In this paper we focus on the classic sta- algorithms like GS can be used in an interactive fashion to ble marriage/matching problem (SMP) [Gale and Shapley, avoid this, as we outline below, but are not designed specifi- 1962], which we formulate using the common “marriage sce- nario.” An SMP consists of set of men M and women W , erence information have received relatively little attention. each of size n, and a set of preference orderings associated GS is typically implemented by having participants submit with each participant: each man m 2 M has a strict total their complete preferences, with the “proposal/acceptance” 0 preference order m over W , with w m w indicating that simulated within the algorithm. However, it can be viewed m prefers w to w0 as a potential partner; each w 2 W has as an interactive elicitation algorithm as well, with proposers a similar ordering over M. We use to denote the set of and acceptors providing only the information needed to run preference rankings or preference profile of the SMP. For any GS. We are unaware attempts to use early termination of GS q 2 M [ W , we use Rq to refer q’s (range of) potential part- to find approximately stable matches. Pini et al. [2011] de- ners, i.e., Rq = W if q 2 M, and Rq = M if q 2 W .A velop several notions of approximate stability, but this is not matching is a function µ : M [ W ! M [ W , such that used to address partial information (and relies on real-valued µ(q) 2 Rq, and q = µ(µ(q)). In other words, µ matches utilities rather than ordinal preferences). Closest in spirit to men and women in a 1:1 correspondence. Given a match- our work is that of Rastegari et al. [2013], who investigate ing µ, we say (m; w) is a blocking pair if w m µ(m) and the problem of minimizing the number of interviews needed m w µ(w). A blocking pair destabilizes µ since m and w to find a stable matching in a labor market. They analyze the prefer each other to their partners in µ, and thus have incen- problem from several perspectives. They show that finding tive to defect (or “run away” with each other). A matching is a minimal certificate (i.e., set of partial preferences that sup- stable if it admits no blocking pairs. ports an employer-optimal matching) is NP-hard. They also The Gale-Shapley (GS) algorithm [Gale and Shapley, provide an MDP-formulation assuming a prior over prefer- 1962] for finding stable matchings proceeds in rounds (we ences (which enumerates all, information states or, in our ter- describe the female-proposing variant). Initially, nobody minology below, “partial preference profiles”), but is other- is engaged. At each round, any unengaged woman pro- wise polynomial in this (exponentially-sized) set. Our work poses to the favorite man in her preference order to whom differs in its focus on computing matches that are approxi- she has not yet proposed. Each man receiving (one or mately stable given partial preferences, computationally ef- more) proposals accepts the proposal of the proposer (in- fective heuristics for elicitation, and anytime evaluation (we cluding his current engaged partner, if any) who is highest also evaluate our schemes experimentally). in his preference ranking—they become engaged—and re- Biro´ and Norman [2012] present a method for stable jects all other proposals. Once a round is reached where matching that can be viewed as an elicitation scheme. In- each woman is engaged, the current set of engagements is dividuals interact randomly, and form a new pair if it offers returned as the matching µ. This mechanism has a num- benefit (relative to their current partners). They show that this ber of remarkable properties: it will converge to a stable is likely to result in an egalitarian solution. However, inter- matching in polynomial time; it is strategy-proof for women preted as an elicitation scheme, it seems to both scale poorly (but not men, vice versa if men propose); and among (the computationally and require far more elicitation rounds than lattice of) all stable matchings it gives every woman her our approach. Other work has considered equilibrium match- most preferred “achievable” partner [Gale and Shapley, 1962; ing with unobservable preferences [Liu et al., 2012]. It is Niederle et al., 2008]. known that the communication complexity of stable match- 2 [ ] This approach has been extended in a variety of ways and ing is Ω(n log n) Chou and Lu, 2010 , hence no elicitation has been applied in many practical settings. Many-to-one scheme can reduce elicitation burden the worst case. extensions are common (e.g., hospital-resident or student- Our work is closely related to previous work on school match [Roth, 1984; Abdulkadiroglu et al., 2005]), and regret-based robust optimization and preference elicitation more flexible forms of preferences can be accommodated. [Boutilier et al., 2006; Braziunas and Boutilier, 2007; Lu and For instance, “incomplete” preference lists can be used to Boutilier, 2011b]. Specifically, we adapt the methods of Lu express acceptability thresholds: a total order over a subset and Boutilier [2011b] for regret-based winner determination of partners is specified, with unranked partners deemed un- and elicitation in voting using ordinal preference rankings to acceptable [Roth et al., 1993]. Indifference (ties) in rank- our robust stable matching problem.
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