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Home , Stable marriage problem, Stable matching theory

This paper was presented as part of the Mini-Conference at IEEE INFOCOM 2011

Seen As Stable Marriages

Hong Xu, Baochun Li Department of Electrical and Computer Engineering University of Toronto

Abstract—In this paper, we advocate the use of stable agents from both sides to find the optimal solution, which is framework in solving networking problems, which are tradi- computationally expensive. tionally solved using utility-based optimization or . Born in , stable matching efficiently resolves conflicts of interest among selfish agents in the , with a simple and elegant procedure of deferred acceptance. We illustrate through (w1,w3,w2) (m1,m2,m3) one technical case study how it can be applied in practical m1 w1 scenarios where the impeding complexity of idiosyncratic factors makes defining a utility function difficult. Due to its use of (w2,w3,w1) (m1,m3,m2) generic preferences, stable matching has the potential to offer efficient and practical solutions to networking problems, while its m2 w2 mathematical structure and rich literature in economics provide many opportunities for theoretical studies. In closing, we discuss (w2,w1,w3) open questions when applying the stable matching framework. (m3,m2,m1) m3 w3 I. INTRODUCTION Matching is perhaps one of the most important functions of Fig. 1. A simple example of the marriage market. The matching shown is a stable matching. markets. The , introduced by Gale and Shapley in their seminal work [1], is arguably one of the most Many large-scale networked systems, such as peer-to-peer interesting and successful abstractions of such markets. There networks, consist of numerous autonomous components. The are men and women, looking for partners, as shown in Fig. 1. interests of these selfish agents are not aligned, and individual Each has a ranking, or ,overagentsoftheopposite rationality needs to be taken into account. While game theory is gender. Given marriages that assign each man to a woman, the widely adopted in such scenarios, its existing use often relies on following is certainly not desirable with respect to individual some rigorous definition of utility. Oftentimes it is difficult to rationality: there exists a pair of man and woman who both quantize and unify the effects of various factors into a single prefer each other to their assigned partners. Such a pair is utility function, not to mention the stringent requirement of unstable in the sense that they have a clear incentive to break complete information that is implicitly assumed. In practice, up from the current marriage and marry each other instead. these techniques are not widely adopted due to these difficulties Therefore, a good marriage does not induce any such unstable though theoretically sound. pairs: it is stable. When the problem is extended beyond a one- In this paper, we advocate the use of stable matching as to-one setting (such as in the college admissions problem [1]), ageneralframeworktotacklenetworkingproblems,where it is more generally referred to as stable matching. preferences are used to model each agent’s interest, and stability Matching problems also exist pervasively in networking, serves as the instead of optimality. As a case ranging from assigning channels to users and flows in wireless study, we present a problem related to content-eyeball ISP scheduling, to mapping video segments to servers in video-on- peering, and illustrate how it can be modelled and solved as demand streaming systems. In many cases, network elements avariantofstablematchingproblems.Theuseofthestable are controlled by some central entity, and a metric of utility matching framework is a substantial and likely controversial may be easily defined. The problem can then be formed into shift from utility-based optimization or game-theoretic solution an optimization problem that can be systematically solved in methods. Merits of the stable matching framework lie in the acentralizedordistributedmanner,withoutpayingrespectto competitiveness of outcomes, generality of the preferences, individual rationality. efficiency and simplicity of its algorithmic implementations, However, as technologies evolve and networks become com- and most importantly, its overall practicality. Stable matchings plex, optimization becomes monolithic, sometimes even in- are in the of the market that cannot be improved upon ept, to be used in practice. Its effectiveness hinges upon the by a coalition of agents [2]. The generic preferences embrace availability of complete and accurate information, which may many heterogeneous and complex considerations that different not be realistic in practical settings. Parameter values serving agents may have. The classical deferred acceptance as inputs may be distorted by delayed, noisy, or inaccurate can be applied in a centralized manner with little complexity. In measurements, which may lead to a significant “drift” in a addition, we propose a new decentralized mechanism that yields computed optimal solution from the actual optimum. Further, exactly the same outcome without centralized coordination at an optimization algorithm needs to enumerate combinations of the helm.

978-1-4244-9921-2/11/$26.00 ©2011 IEEE 586 We do not claim that we are the first to use stable matching The matching shown in Fig. 1 is a stable matching for the given in networking. There exist papers — though very few — that preferences of agents. used the concept of stable matching, as we shall discuss soon. Theorem 1: Astablematchingexistsforeverymarriage In contrast, the contribution of this paper lies in an explicit market. and general discussion of the and its This can be readily proved by the classic deferred acceptance application in networking with concrete case studies, and in algorithm,ortheGale-Shapley algorithm proposed in [1] (with provoking a broader understanding of its theoretic merit and men proposing). In the first round, each man proposes to his practical value as another tool from economics. Broadly, stable first choice if he has any acceptable ones. Each woman rejects matching can be applied in scenarios where coincidences and any unacceptable proposals and, if more than one acceptable conflicts of interest among agents need to be resolved, but proposals are received, holds the most preferred and rejects their considerations are difficult to be modelled quantitatively. It all others. In each round that follows, any man rejected at the naturally fits into cases when agents’ individual rationality has previous round makes a new proposal to his most preferred to be respected. Alternatively, it also applies to cases withno acceptable partner who has not yet rejected him, or makes no hints of selfish agents, but optimization is clearly not possible proposal if no acceptable choices remain. Each woman holds or desirable. her most preferred offer up to this round,andrejectsalltherest. When no further proposals are made, the algorithm stops and II. BACKGROUND AND RELATED WORK matches each woman to the man (if any) whose proposal she We start by introducing the basic theory of stable matching is holding. The woman-proposing version works in the same in the one-to-one marriage model, as shown in Fig. 1. There are way by swapping the roles of men and women. two disjoint sets of agents, M = {m1,m2,...,mn} and W = It is then observed that which side proposes in the algorithm {w1,w2,...,wp},menandwomen.Eachagenthasacomplete has significant consequences. Specifically, the algorithm finds and transitive preference over individuals on the other side, the two extremes among the set of stable matchings. The man- and the possibility of being unmatched [3]. Preferences can be proposing version yields a man-optimal outcome that every represented as rank order lists of the form pm1 = w4,w2,...,∅, man likes at least as well as any other stable matching, and meaning that man m1’s first choice of partner is w4,second the woman-proposing version a woman-optimal one. This is w2 and so on, until at some point he prefers to remain single referred to as the polarization of stable matchings. (i.e. matched to the void set). We use >i to denote the ordering Following [1], the simple marriage model has been extended relationship of agent i (on either side of the market). If i prefers to other matching problems. Because of the richness of the to remain single than being matched to j,i.e.∅ >i j,thenj literature, it is bold to even attempt a cursory survey of the is said to be unacceptable to i,andi’s preference only include existing results here. Instead, we choose to introduce some of the acceptable partners. Preferences are strict if each agent is the different models along with our case studies, and present not indifferent between any two acceptable partners. relevant and important theoretical developments that can be Definition 1: An outcome of the market is a matching µ : applied to solve these problems, while using this section as a M×W→M×W such that w = µ(m) if and only if necessary background for a better understanding of the material. µ(w)=m,andµ(m) ∈W∪∅, µ(w) ∈M∪∅, ∀m, w. From a practical perspective, due to the efficiency of stable This implies that the outcome matches agents on one side to matchings and the simplicity of implementation, the deferred those on the other side, or to the empty set. Agents’ preferences acceptance algorithm has profound influence on market design. over outcomes are determined solely by their preferences for It has been adopted in a number of practical matching markets, their own partners in the matching. prominent examples of which include the National Resident It is clear that we need further criteria to distill a good set Matching Program of U.S. for medical school graduates, many of matchings from all the possible outcomes. The first obvious medical labor markets in Canada and Britain, and recently criterion is individual rationality. school choice systems in Boston and New York City [3]. Definition 2: Amatchingisindividual rational to all agents, There only exists a very limited number of papers in the if and only if there does not exist an agent i who prefers being networking literature that used solutions designed to achieve unmatched to being matched with µ(i), i.e., ∅ >i µ(i). stable matching [4]. In contrast, this paper seeks to presenta The second natural criterion is that a blocking set should not systematic study on the feasibility and unique advantages of occur in a good matching: applying the stable matching framework to possibly a wider Definition 3: Amatchingµ is blocked by a pair of agents range of problems in networking, with an objective of drawing (m, w) if they each prefer each other to the partner they receive attention to and provoking discussions on this practical and at µ.Thatis,w>m µ(m) and m>w µ(w).Suchapairis effective tool from the field of economics. called a blocking set in general. If there is a blocking set in the matching, the agents involved III. CONTENT AND EYEBALL ISP PEERING AS have an incentive to break up and form new marriages. There- MANY-TO-MANY STABLE MATCHING fore such an “unstable” matching is not desirable. In this section, we present a general many-to-many matching Definition 4: Amatchingµ is stable if and only if it is problem of interconnecting Internet Service Providers (ISPs), individual rational, and is not blocked by any pair of agents. where stable matching is arguably the only practical solution.

587 The Internet consists of tens of thousands of ISPs, with also be applied here. The problem with game-theoretic ap- profound heterogeneity. Roughly, they fall into three categories, proach is that some concrete utility function needs to be defined content, eyeball,andtransit [5]. Content ISPs (CPs) specialize (similar to optimization), which is not possible considering the in content delivery, such as Microsoft MSN and Google. enormous complexity involved when making peering decisions. Eyeball ISPs (EPs) are in the business of selling retail Internet The approach is also unlikely to succeed, access to end users, such as Verizon and Comcast. Traditionally due to that are extremely common in real-world it has been understood that CPs and EPs buy transit from tier-1 markets. Stable matching remedies both issues by adopting a ISPs to deliver their traffic, and EPs establish settlement-free general preference framework and a stability solution concept peering links among themselves, provided that their network resistant to of coalitions from both sides [2]. sizes or traffic volumes are roughly equal in order to save costs. This hierarchical picture is becoming increasingly invalid. A. The Model Some CPs have built their own backbones for efficient delivery of content that constitutes the majority of Internet’s traffic [6], ISPs generally have different incentives to form peering and thus become an indispensable part of the network. EPs relationships. As it improves latency which is critical to their also possess increasing bargaining power because of their large revenue, CPs tend to exhibit strong incentives for peering, while customer base. To reduce the bulk of transit costs and improve some EPs may only be willing to enter a paid-peering contract, latency, CPs are now directly peering with EPs, creating a much since it possesses bargaining power under the assumption that flatter Internet pyramid with tier-1 ISPs losing their grip onthe eyeball customers are less vulnerable to switching to another ISP peering ecosystem [7]. EP. Their interests can also be in conflict, especially when one This trend has been increasingly realized and independently ISP is favored by multiple ISPs. observed among practitioners and researchers [6], [7]. Existing Thus, content-eyeball peering can be cast as a many-to-many peering mechanism, however, may not work well for the stable matching problem with a set of CPs C and a set of emerging content-eyeball peering market. As of today, peering EPs E.EachISPhasaquota,whichisthemaximumnumber decision is usually made through bilateral negotiation. These of partners it allows for peering, due to technical overhead decisions may look beneficial from a local perspective, but from and personnel constraints of setting up the connections, and aglobalperspectivetheyareveryunappealing[5].Theresulted business considerations such as paid peering. Each ISP is market inefficiency is hard to rectify, since breaking the contract assumed to have a strict preference ordering of its acceptable involves the risk of legal lawsuit and possibly anti-trust scrutiny, partners. This set of acceptable partners and their rankings and incurs disruption to the Internet. could be produced by each ISP identifying and contacting their The difficulty in establishing a market mechanism is mainly potential partners and evaluating the potential benefits. the idiosyncratic factors involved in making peering decisions, Due to the variety and complexity of economic and business including geographic coverage, traffic volume, routing require- policy factors affecting this evaluation process, we can notand ments, marketing considerations, etc., alongside the common shall not attempt to define a generic preference framework or and conflicting interests of ISPs that will induce market failure anything alike that each ISP adheres to, as in sharp contrast to if not resolved. Existing papers have studied novel settlement most previous work [5]. The only assumption on the structure strategies of ISPs through economical analysis when an ISP of the preferences we make is a responsiveness assumption interconnection topology is given [5], but few has touched the in order to reduce the exponential complexity of expressing issue of how this topology is formed endogenously, especially preferences over all subsets of acceptable ISPs. Specifically, among content and eyeball ISPs. Moreover, the market is Definition 5: For any set of EPs E1 with |E1|

588 B. Solution Concepts and a Centralized Mechanism The analysis of this mechanism, including the polarization The technical difficulty we encounter here is the choice of of the outcomes, is essentially the same as that in Sec. II stability concepts. Many-to-many matching is a more general with proper generalization. Due to space constraints, we omit model than many-to-one matching. Several stability concepts the details in this paper. Note that since CPs have stronger can be defined. The most common ones in the literature are incentives to peer, and EPs have the benefit of getting paid pairwise stability, corewise stability,andsetwise stability [9]. peering deals, it may be convincing to let CPs be the proposing side and the mechanism produces the CP-optimal outcome. Definition 7: Amatchingµ is pairwise stable if there are no ISPs c and e who are not partners in µ,butbybecoming C. A Distributed Implementation partners, possibly dissolving some of their partnerships given by µ to remain within quotas and keeping other ones, can both The content-eyeball peering market is inherently decentral- obtain a strictly preferred set of partners. ized. Thus a distributed procedure with minimum centralized coordination is highly desirable. Moreover, in reality, an ISP This is the usual stability concept we have seen in the one- would like to keep its preference and the final peering results to-one and many-to-one problems. private for obvious reasons, which cannot be achieved in Definition 8: Amatchingµ is in the core (corewise stable)if the centralized mechanism. In this section, we present a new there is no subset of ISPs who by forming all their partnerships distributed mechanism that preserves information privacy while only among themselves, can all obtain a strictly preferred set producing precisely the same stable matching given the same of partners. problem instance. Our mechanism is an extension of [11] in the Definition 9: Amatchingµ is setwise stable if there is no marriage market. It consists of two procedures, CP and EP as subset of ISPs who by forming new partnerships only among presented in Algorithm 1 and 2, which are executed in each CP themselves, possibly dissolving some of their partnershipsgiven and EP, respectively. Note that the execution is asynchronous. by µ to remain within quotas and keeping other ones, can all Procedure CP,afterinitialization,performsthefollowing obtain a strictly preferred set of partners. loop. If a CP c does not have enough partners confirmed, and Both corewise and setwise stability deal with coalitions of its preference p is nonempty, it iterates by proposing to its multiple ISPs. The key difference is that corewise stability c first choice t,deletingt from p ,andaddingt to list,untilit requires all agents involved in a blocking set to be better off, c has q partners in list.Itthenwaitsonamessagereply.Ifthe while setwise stability only concerns the agents who form new c reply is accept, it does nothing. If the reply is reject, it removes partnerships in the blocking set. More detailed comparison and the sender from p .Ifthesenderisinitslist,itremovesthe discussions of the concepts are available in [9]. c sender from list,whichmeansitsproposaltothesenderfailed. Intuitively, setwise stability is the strongest definition and the Rejection is normally from EPs whom c has proposed to. It other two are special cases of this concept. However, its proper- can also be from EPs that c did not propose to, which we ties and algorithmic implementations are less well understood will explain shortly in the EP procedure. This continues until compared to the usual pairwise stability. This dilemma does aspecialstopmessageisreceivedfroma“registrar,”which not exist for one-to-one and many-to-one cases where the three we assume is a server that delivers all the encrypted messages concepts are equivalent [9], which is not the case for many-to- exchanged between ISPs and thus can detect the quiescence. many matching in general, simply because each agent can have multiple partners and form different coalitions [9]. Algorithm 1 Distributed CP Procedure Therefore, for empirical purposes, we restrict our attention list ←∅, end ← false; to the case where the coalitions are formed between a single while !end do CP and a set of EPs, or a single EP and a set of CPs, but not while |list|

589 proposals from these CPs given the status quo. This reduces the of stable outcomes that favors the proposing side of the market. possible number of proposals from those who will definitely be It is desirable in many cases to find a “fair” stable matching that rejected, and speeds up the algorithm. This also ensures that does not favor either side as an alternative operating point of the when quota was met, EP e can accept a new proposal from c system. There are efforts in the economics literature that pursue as long as c is currently in pe.Ifthisisthecase,thenagain this direction, such as the egalitarian stable matching that all CPs after the newly accepted one in pe are also rejected minimizes the total rank sum of the outcome in the marriage and deleted from pe.TheEP procedure also terminates when model [12]. This is a feasible extension to our framework. it receives the stop message from the registrar. -proofness is also crucial to the effectiveness of sta- The distributed and asynchronous stable matching mecha- ble matching mechanisms. Is it the dominant strategy for agents nism eliminates the need of centralized coordination at the helm to truthfully report, or act according to their preferences and for the matching process. More importantly, it perfectly main- quotas? Can they benefit from manipulating this information? tains the privacy of preference information and the final peering In this regard, some negative results have been proven on the decisions. Each ISP only accesses its own preference, and at theoretical impossibility of achieving truthfulness for both sides the end it has knowledge about only its own peering result. As of the market in some models [2]. On the other hand, many the distributed mechanism shares the same deferred acceptance real-life implementation experiences suggest that the impact of procedure, the final outcome is exactly the same CP-optimal misbehavior is extremely limited, especially when each agent stable matching produced by the centralized mechanism. has a small number of acceptable partners in a thick market [3]. Algorithm 2 Distributed EP Procedure Given this mixed picture, it is interesting to explore the issue in a networking context, and understand the extent to which list ←∅, end ← false; this impacts the suitability of the new framework. while !end do Our attempt of advocating stable matching in favor of con- msg ← getMsg(); ventional optimization or game-theoretic approaches here may switch msg.type seem ambitious. Yet, we believe that stable matching, due to its propose: t ← msg.sender; rich theoretical foundation and practical implementation experi- if |list|

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