Quantifying Inefficiency of Fair Cost-Sharing Mechanisms For

Quantifying Inefﬁciency of Fair Cost-Sharing Mechanisms for Sharing Economy Chi-Kin Chau and Khaled Elbassioni

Abstract—Sharing economy is a distributed peer-to-peer eco- 3) Pass Sharing: Certain anonymous regular passes are nomic paradigm, which gives rise to a variety of social in- validated within a certain fixed period of time (e.g., teractions for economic purposes. One fundamental distributed accesses to public transportation or public facilities), decision-making process is coalition formation for sharing certain replaceable resources collaboratively, for example, sharing hotel which may be shareable among multiple holders, if rooms among travelers, sharing taxi-rides among passengers, they do not overlap in their usage times. Note that this and sharing regular passes among users. Motivated by the example also applies to co-owning or co-leasing physical applications of sharing economy, this paper studies a coalition properties (e.g., houses, cars and parking lots). formation game subject to the capacity of K participants per coalition. The participants in each coalition are supposed to split In this paper, we study a class of distributed decision- the associated cost according to a given cost-sharing mechanism. making processes for sharing economy. In particular, we A stable coalition structure is established when no group of par- consider the problem with a set of participants forming ticipants can opt out to form another coalition that leads to lower coalitions to share certain replaceable resources from a large individual payments. We quantify the inefficiency of distributed decision-making processes under a cost-sharing mechanism by pool of available resources (e.g., hotel rooms, taxicabs, regular the strong price of anarchy (SPoA), comparing a worst-case stable passes), such that any subset of participants can always form coalition structure and a social optimum. In particular, we derive a coalition using separate resources, independent from other SPoA for common fair cost-sharing mechanisms (e.g., equal-split, coalitions. We formulate a coalition formation game, wherein proportional-split, egalitarian and Nash bargaining solutions of participants form arbitrary coalitions of their own accord bargaining games, and usage based cost-sharing). We show that the SPoA for equal-split, proportional-split, and usage based cost- to share the associated cost, subject to a constraint on the maximum number of participants per coalition. sharing (under certain conditions) is Θ(log K), whereas√ the one for egalitarian and Nash bargaining solutions is O( K log K). There are two main aspects investigated in this paper: Therefore, distributed decision-making processes under common fair cost-sharing mechanisms induce only moderate inefficiency. 1) Inefficiency of Distributed Decision-Making: Since there is a capacity per coalition such that not all par- Index Terms—Sharing economy, coalition formation, social and ticipants can form a single coalition, there will exist economic networks, fair cost-sharing mechanisms potentially multiple coalitions and the self-interested participants will opt for the lower payments. Distributed I.INTRODUCTION self-interested behavior often gives rise to outcomes that deviate from a social optimum. A critical question is The rise of “sharing economy” [1] has created a new related to the inefficiency of distributed decision-making paradigm of social and economic networks, which promotes processes. We quantify the inefficiency of distributed distributed peer-to-peer interactions and bypasses traditional decision-making processes by the Strong Price of An- centralized hierarchal service providers and intermediaries. archy (SPoA), a common metric in Algorithmic Game Sharing economy is often facilitated by the advent of perva- Theory that compares the worst-case ratio between the sive information technology platforms, especially by mobile self-interested outcomes (that allow any group of users arXiv:1511.05270v4 [cs.GT] 16 Oct 2017 computing and digital social platforms. One fundamental to deviate jointly) and a social optimum [2]–[5]. distributed decision-making process in sharing economy is 2) Fair Cost-Sharing Mechanisms: Sharing economy can coalition formation for sharing resources and facilities with be regarded as an alternative to the for-profit sector, excess capacity among users collaboratively and efficiently. which resembles cooperative organizations and favors We highlight some examples of sharing economy as follows: distributive justice. When participants share the costs, 1) Hotel Room Sharing: Travelers may share hotel rooms there is a notion of fairness. We aim to characterize the with other fellow travelers, because multiple-occupancy inefficiency of distributed decision-making under com- rooms are more economical. The sharing processes are mon fair cost-sharing mechanisms. First, we consider achieved through private arrangements among travelers. typical fair cost-sharing concepts such as equal-split, 2) Taxi-ride Sharing: Commuters may share taxi-rides be- proportional-split, and usage based cost-sharing. Second, cause of lower taxi fares, despite that the taxicabs may we formulate the cost-sharing problem as a bargaining take a detour to pick-up or drop-off other passengers. game. Thus, the well-known bargaining game solutions (e.g., egalitarian and Nash bargaining solutions) [6] can C.-K. Chau and K. Elbassioni are with Masdar Institute (e-mail: {ckchau, be applied in the context of sharing economy. kelbassioni}@masdar.ac.ae). Abridged version of this paper appears in IEEE Transactions on Control of This paper presents a comprehensive study for the SPoA Network Systems (DOI:10.1109/TCNS.2017.2763747). of a general model of coalition formation, considering various 2

TABLE I: A summary of our results.

Equal-split Proportional-split Egalitarian/Nash Usage Based Existence of Stable Coalition X X X Only in some problems √ Strong Price of Anarchy Θ(log K) Θ(log K) O( K log K) Ω(K) (in general) Θ(log K) (under certain conditions) common fair cost-sharing mechanisms. monotonicity, we show that it is Θ(log K). We also study other cost-sharing mechanisms, such as egalitarian, Nash bargaining A. Our Contributions solutions, and usage based cost-sharing mechanisms that are not considered in [17], [18]. Moreover, [18] is based on the We consider K-capacitated coalitions, where K is the comparison of the utility of a stable matching, while our results maximum number of sharing participants per coalition. A are based on the comparison of the cost of a stable coalition stable coalition structure, wherein no group of participants can structure. Although it is possible to derive some of our results opt out to form another coalition that leads to lower individual (for K = 2) from the previous results, the bounds obtained this payments, is a likely self-interested outcome. The results in way are typically weaker than ours and the gap can increase this paper are summarized as follows (and in Table I). as a linear function of K. 1) The SPoA for any budget balanced cost-sharing mech- Network cost-sharing games with capacitated links and non- anism is O(K). anonymous cost functions [15], [19], [20] are closely related to 2) However, the SPoA for equal-split and proportional-split our problem. Non-anonymous cost functions may depend on cost-sharing is only Θ(log K). the identity of the players in the coalition, so as to capture the 3) The SPoA for egalitarian and Nash bargaining solutions √ asymmetries between the players because of different service is O( K log K). requirements. In [15], a logarithmic upper bound on the price 4) The SPoA for usage based cost-sharing is generally of anarchy considering the Shapley value in network cost- Ω(K). However, we provide natural sufficient conditions sharing games with non-anonymous submodular cost functions to improve the SPoA to be Θ(log K), which apply to was given. Our problem can be modeled by a network cost- the examples of sharing economy in this paper. sharing game with non-anonymous cost functions. In partic- 5) Therefore, distributed decision-making processes under ular, our model is a special case of a K-capacitated network common fair cost-sharing mechanisms induce only mod- cost-sharing game with a simple structure of n parallel links erate inefficiency. and non-anonymous cost functions such that a strategy profile 6) We also study the existence of stable coalition structures. of the users in the network game corresponds to a coalition structure2. However, the key difference of our model from II.RELATED WORK those in [7], [11], [15] is that we consider replaceable re- Our problem belongs to the topic of network cost-sharing sources from a large pool of available resources, such that a and hedonic coalition formation problems [7]–[16]. A study subset of deviating users can always form a new coalition, particularly related to our results is the price of anarchy for independent from other users. This is not true in general stable matching and the various extensions to K-sized coali- network cost-sharing games, when there are limited resources tions [17], [18]. Our coalition formation game is a subclass (e.g., links) that a deviating coalition of players can utilize, and of hedonic coalition formation games [8], [13] that allows it may not be possible to form arbitrary coalitions independent arbitrary coalitions subject to a constraint on the maximum from others. Our model allows us to derive SPoA bounds for number of participants per coalition. This model appears to diverse cost-sharing mechanisms, whereas only specific cost- 1 be realistic in many practical settings of sharing economy . sharing mechanisms (e.g., the Shapley value) were considered Our work differs from typical cooperative games. In our in general network cost-sharing games. coalition formation game, each player joins a coalition that It is also worth mentioning how our results of usage incurs a lower individual payment, under a given cost-sharing based cost-sharing relate to cost-sharing with anonymous cost mechanism. On the other hand, typical cooperative games functions in network design games [7] or connection games generally do not consider a specific cost-sharing mechanism, [11]. On one hand, our model is simpler as we do not assume but find a cost-sharing allocation according to certain axioms. connectivity requirements in a network, but only an abstract One may regard the results about stable matching in [17], setting that allows arbitrary coalitions up to a certain capacity [18] as a special case of K = 2 in our model. How- (but we allow non-anonymous cost functions). On the other ever, unlike the stable matching problem, our model has hand, one of the cost-sharing mechanisms we consider (i.e., additional structure that can be harnessed for tighter results (e.g., monotonicity). For example, according to [17], the price 2 of anarchy for Matthew’s effect (equivalently, proportional- For example, consider taxi-ride sharing. An additional passenger can join a taxi-ride with certain existing passengers who have already formed a coalition, split cost-sharing) can be unbounded. Here with the help of only if all of them will not be worse-off after the change. Otherwise, the existing passengers can always reject the additional passenger by keeping their 1For example, consider taxi-ride sharing. Any passengers can form a current coalition. This is always possible in a network cost-sharing game with coalition to share a taxi-ride, subject to the maximum capacity of a taxicab. n parallel links. 3 usage based cost-sharing) resembles in some sense that used in A K-capacitated social optimum is a coalition structure ∗ [7], [11] if we interpret the resources used by one participant PK ∈ PK that minimizes the total cost: as his chosen path or tree in the network design game. Similar ∗ to the case in [7] (with respect to strong Nash equilibrium), a (K-MINCOALITION) PK ∈ arg min c(P) (1) P∈PK usage based cost-sharing mechanism may not admit a stable ∗ coalition structure. Noteworthily, the SPoA in usage based When K = 2, a social optimum PK can be found in cost-sharing in our model can increase as a linear function polynomial time by reducing the coalition formation problem of K. Nonetheless, we provide general sufficient conditions to a (general graph) matching problem. When K > 2, K- for usage based cost-sharing to induce logarithmic SPoA. MINCOALITION is an NP-hard problem (see Appendix). 2) Canonical Resources: There is often a resource being shared by each coalition (e.g., a hotel room, a taxicab, a III.MODEL pass). The resources are usually replaceable from a large This section presents a general model of coalition formation pool of available resources in the sharing economy. Hence, for sharing certain resources, motivated by the applications of any subset of participants can always form a coalition using sharing economy. Consider an n-participant cooperative game separate resources, independent from other coalitions. For each in coalition form. The coalitions formed by the subsets of coalition G, we consider a canonical resource, which is a class participants are associated with a real-valued cost function. of replaceable resources that can satisfy G, rather than any The participants in a coalition are supposed to split the cost specific resource. The canonical resource shared by a coalition according to a certain payment function. However, as a de- will not be affected by the canonical resources shared by parture from traditional cooperative games, there is a capacity other coalitions. Because of the consideration of canonical per each coalition, such that not all participants can form a resources, our model exhibits different properties than the single coalition. Hence, when given a payment function, the network sharing games with limited resources [15], [19], [20]. participants will opt for coalitions that lead to lower individual Let R(G) be the feasible set of canonical resources that payments subject to a capacity per coalition. can satisfy coalition G. It is naturally assumed that R(H) ⊇ R(G), when H ⊆ G, because the canonical resources that can satisfy a larger coalition G should also satisfy a smaller coali- A. Problem Formulation tion H (by ignoring the participants in G\H). Each canonical The set of n participants is denoted by N .A coalition resource r ∈ R(G) is characterized by a cost cr, and a set structure is a partition of N denoted by P ⊂ 2N , such that of involved facilities F (r). Each facility f ∈ F (r) carries f P f S a cost c , such that c = cr. We do not require G = N and G1 ∩ G2 = ∅ for any pair G1,G2 ∈ P. f∈F(r) G∈P G Let the set of all partitions of N be P. Each element that every participant of utilizes the same facilities. Let G ∈ P is called a coalition. The set of singleton coalitions, Fi(r) ⊆ F (r) be the set of facilities utilized by participant i ∈ G, when r is shared by G. Let r(G) ∈ arg minr∈ (G){cr} Pself , {{i} : i ∈ N }, is called the default coalition structure, R wherein no one forms a coalition with others. be the lowest cost canonical resource for coalition G. Hence, This paper considers arbitrary coalition structures with at we set c(G) = cr(G) and monotonicity is satisfied. If there are most K participants per coalition, which is motivated by multiple lowest cost canonical resources, one is selected by a scenarios of sharing replaceable resources; see Section III-B certain deterministic tie-breaking rule. for examples. The notion of resources will be introduced later in Sec. III-A2, and our model does not always rely on the B. Motivating Examples notion of resources. In practice, K is often much less than n. We present a few motivating examples in sharing economy Let PK {P ∈ P : |G| ≤ K for each G ∈ P} be the set of , to illustrate the aforementioned model. feasible coalition structures, such that each coalition consists of at most K participants. 1) Hotel Room Sharing: Consider N as a set travelers to share hotel rooms. Each traveler i ∈ N is associated with a 1) Cost Function: A cost c(G) (also known as a charac- in out in out tuple (t , t , Ai), where t is the arrival time, t is the teristic function) is assigned to each coalition of participants i i i i departure time, and Ai is the area of preferred locations of G ∈ P ∈ PK , subject to the following properties: hotels. Let K be the maximum number of travelers that can (C1) c(∅) = 0 and c(G) > 0 if G 6= ∅. share a room. A canonical resource is a room booking r, in out in (C2) Monotonicity: c(G) ≥ c(H), if H ⊆ G. associated with a tuple (tr , tr , ar), where tr is the check-in out Monotonicity captures natural coalition formation with in- time, tr is the check-out time, and ar is the hotel location. We creasing cost as the number of participants. The total cost assume that there is a large pool of available rooms for each P location, and we do not consider a specific room. The feasible of coalition structure P is denoted by c(P) , G∈P c(G). (G) For brevity, we also denote c({i}) by ci, where ci is called set R is a set of room bookings shared by a coalition of the default cost of participant i, that is, when i forms no travelers G, if the following conditions are satisfied: coalition with others. When a subset of participants are in- 1) |G| ≤ K; 0 T dexed by N = {i1, i2, ..., ij} ⊆ N , we simply denote the 2) ar ∈ i∈G Ai; in in out out corresponding default costs by {c1, c2, ..., cj}. 3) tr ≤ ti and tr ≥ ti for all i ∈ G. 4

(Note that the monotonicity assumption is satisﬁed: R(H) ⊇ G be pi(G). The utility of participant i is given by: R(G), when H ⊆ G.) In this example, F (r) is be the set of in out ui pi(G) ci − pi(G) (2) days during [tr , tr ] for room booking r, and Fi(r) be the set , of days that i stays in room booking r. For each f ∈ F (r), The following natural properties can be satisﬁed by payment cf is the hotel rate of day f. function pi(·):

2) Taxi-ride Sharing: Consider N as a set of passengers to • Budget Balance: pi(·) is said to be budget balanced, if i ∈ N P share taxi-rides. Each passenger is associated with a i∈G pi(G) = c(G) for every G ⊆ N . s d s d s d tuple (vi, vi , ti, ti ), where vi is the source location, vi is the • Non-negative Payment: pi(·) is said to be non-negative, s d destination location, ti is the earliest departure time, and ti if pi(G) ≥ 0 for every G ∈ P ∈ PK . If non-positive is the latest arrival time. Let K be the maximum number of payment is allowed, then it possible that pi(G) < 0 for passengers that can share a ride. A canonical resource is a ride some i ∈ G. 1 m r, associated with a sequence of locations (v , ..., v ) and a This paper considers the following fair cost-sharing mech- 1 m sequence of arrival timeslots (t , ..., t ) in an increasing order. anisms. Note that only usage based cost-sharing mechanism We assume that there is a large pool of available taxicabs, and takes into account the notion of resources, while the other cost- we do not consider a specific taxicab. The feasible set R(G) sharing mechanisms do not rely on the notion of resources. is a set of rides shared by a coalition of travelers G, if the 1) Equal-split Cost-Sharing: The cost is split equally following conditions are satisfied: among all participants: for i ∈ G, 1) |G| ≤ K; s d (v , v ) eq c(G) 2) All the locations i i i∈G are present in the sequence p (G) (3) (v1, ..., vm); i , |G| t (vs) < t (vd) t (vs) ≥ ts t (vd) ≤ td 3) r i r i , r i i and r i i for all 2) Proportional-split Cost-Sharing: The cost is split pro- i ∈ G t (v) r , where r is the arrival timeslot of ride at portionally according to the participants’ default costs: v location . for i ∈ G, Note that the hotel room sharing problem may be regarded pp ci · c(G) as a one-dimensional version of the taxi-ride sharing problem, pi (G) , P (4) j∈G cj if the preferred location constraint is not considered, and we in out (P c )−c(G) let each tuple (t , t ) in hotel room sharing problem be the pp j∈G j i i Namely, ui pi (G) = ci · P . This ap- j∈G cj source and destination locations. Let F (r) is the set of road proach is also called Matthew’s effect in [17]. segments traversed by ride r, and Fi(r) be the set of road Bargaining Based Cost-Sharing f 3) : One can formulate segments that i travels in ride r. For each f ∈ F (r), let c the cost-sharing problem as a bargaining game with a be the taxi fare for road segment f. feasible set and a disagreement point. In our model, 3) Pass Sharing: Consider N as a set of regular-pass the feasible set is the set of utilities (ui)i∈G, such that P P P holders who want to form coalitions to share some anonymous i∈G ui ≤ i∈G ci − c(G)(⇔ i∈G pi ≥ c(G)). passes. Each user i ∈ N is associated with a set of required The disagreement point is (ui = 0)i∈G, such that each usage timeslots Ti. Let K be the maximum number of sharing participant pays only the respective default cost. There users, so as to limit the hassle of circulating the pass. A are two bargaining solutions in the literature [6]: canonical resource is a pass r, associated with a set of • Egalitarian Bargaining Solution is given by: allowable timeslots T . We assume that there is a large pool r (P c ) − c(G) of available passes, and we do not consider a specific pass. ega j∈G j pi (G) , ci − (5) The feasible set R(G) is a set of passes shared by a coalition |G| of travelers G, if the following conditions are satisfied: Namely, every participant in each coalition has the (P c )−c(G) 1) |G| ≤ K; ega j∈G j same utility: ui pi (G) = |G| for all 2) Ti ∩Tj = ∅ for all i, j ∈ G, i 6= j (i.e., no one overlaps i ∈ G. Note that non-positive payment is possible in their required timeslots); S because it may need to compensate those with 3) i∈G Ti ⊆ Tr. low default costs to reach equal utility at every This setting also applies to sharing physical properties (e.g., participant3. houses, cars and parking lots). Let F (r) are the set of • Nash Bargaining Solution is given by: timeslots required by pass r, and cf be the cost of each nash Y timeslot in F (r). A user needs to cover the cost when he pi (G) ∈ arg max ui pi(G) i∈G (p (G)) uses the pass or shares the cost with other participants when i i∈G i∈G S (6) no one uses it. Hence, let Fi(r) = Ti ∪ Tr\( Tj) , j∈G subject to when i shares pass r in coalition G. X pi(G) = c(G) C. Cost-Sharing Mechanisms i∈G

A coalition of participants G are supposed to share the cost 3 For example, consider G = {i, j, k}, such that cj = ck = c(G) = 1 and ega c(G). Let the cost (or payment) contributed by participant i ∈ ci = 0.1. Then pi (G) = −0.26. 5

One can impose an additional constraint of non- in our proofs. In the following we denote the K-th harmonic PK 1 negative payments: pi(G) ≥ 0 for all i ∈ G. number by HK , s=1 s . 4) Usage Based Cost-Sharing: Also known as Shapley Theorem 1. Recall the default coalition structure Pself , cost-sharing [7]. We consider a cost-sharing mechanism {i} : i ∈ N . We have that takes into account the usage structure of resources ∗ of participants. Recall that r(G) denotes a lowest-cost K · c(PK ) ≥ c(Pself ) (9) canonical resource for coalition G. Let N f (r(G)) be Consider a budget balanced payment function p (·). Let Pˆ the set of participants that share the same facility f in i K be a respective K-capacitated stable coalition structure. Then, r(G). The cost is split equally among the participants who utilize the same facilities: c(Pself ) ≥ c(PˆK ) (10) X cf pub(G) (7) Hence, the SPoA for pi(·) is upper bounded by SPoAK ≤ K. i , |N f (r(G))| f∈Fi(r(G)) ∗ Proof. First, by monotonicity, we obtain for any G ∈ PK For example, in taxi-ride sharing, passengers will split P P i∈G ci i∈G ci the cost equally for each road segment with those pas- c(G) ≥ max{ci} ≥ ≥ (11) i∈G |G| K sengers traveled together in the respective road segment. Hence,

X 1 X c(Pself ) D. Stable Coalition and Strong Price of Anarchy c(P∗ ) = c(G) ≥ c = (12) K K i K G∈P∗ i∈N Given payment function pi(·), a coalition of participants G K is called a blocking coalition with respect to coalition structure ∗ Since PK is a stable coalition structure, then pi(G) ≤ ci P if all participants in G can strictly reduce their payments ∗ for every G ∈ PK . Otherwise, every i ∈ G can strictly reduce when they form a coalition G to share the cost instead. A his payment by forming a singleton coalition individually. coalition structure is called stable coalition structure, denoted Lastly, since p (·) is a budget balanced payment function, ˆ i by PK ∈ PK , if there exists no blocking coalition with it follows that respect to PˆK . The existence of a stable coalition structure X X X ˆ depends on the cost-sharing mechanism (see Appendix). c(Pself ) = ci ≥ pi(G) = c(PK ) (13) i∈N ˆ i∈G Note that a stable coalition structure is also a strong Nash G∈PK equilibrium4 in our model. However, there is a difference between the case of sharing canonical resources and that of limited resources. For sharing canonical resources, an addi- However, we will show that the SPoA√ for various cost- tional member can join an existing coalition to create a larger sharing mechanisms is O(log K) or O( K log K). coalition, only if all of the participants in the new coalition To derive an upper bound for the SPoA, the following will not be worse-off after the change. Otherwise, the existing lemma provides a general tool. First, define the following coalition can always reject the additional member by keeping notation for a non-negative payment function pi(·): the current canonical resource. However, for sharing limited PK resources, a coalition may be forced to accept an additional s=1 pis (Hs) α {pi(·)}i∈N , max , (14) member, even they will be worse-off, because they cannot c(·),H1⊃···⊃HK c(H1) find a new resource to share with. In this case, a strong Nash where H1, ..., HK are a collection of subsets, such that each equilibrium may not be a stable coalition structure. Hs , {is, ..., iK } for some i1, ..., iK ∈ N . Note that α(·) is Define the Strong Price of Anarchy (SPoA) as the worst- non-decreasing in K. See Appendix for a proof. case ratio between the cost of a stable coalition structure and that of a social optimum over any feasible costs subject to Lemma 1. Suppose pi(·) is a budget balanced non-negative (C1)-(C2): payment function. Given a K-capacitated stable coalition ˆ c(PˆK ) structure PK , and a feasible coalition structure P ∈ PK , SPoAK , max (8) ˆ c(P∗ ) then c(·), PK K c(Pˆ ) K ≤ α{p (·)} Specifically, the strong price of anarchy when using specific c(P) i i∈N cost-sharing mechanisms are denoted by SPoAeq, SPoApp, K K ˆ SPoAega, SPoAnash, SPoAub, respectively. Thus, if PK is a worst-case stable coalition structure and K K K ∗ P = PK is a social optimal coalition structure, then we obtain an upper bound for the SPoA with respect to {pi(·)}i∈N : IV. PRELIMINARY RESULTS Before we derive the SPoA for various cost-sharing mecha- SPoAK ≤ α {pi(·)}i∈N nisms, we present some preliminary results that will be useful 1 Proof. Let P = {G1, ..., Gh}. Define H1 , G1. Then there i1 ∈ H1 Gˆ1 ∈ Pˆ 4 exists a participant 1 1 and a coalition 1 K , such A strong Nash equilibrium is a Nash equilibrium, in which no group of 1 1 1 1 that i ∈ Gˆ and p 1 (H ) ≥ p 1 (Gˆ ); otherwise, all the players can cooperatively deviate in an allowable way that benefits all of its 1 1 i1 1 i1 1 1 1 members. participants in H1 would form a coalition H1 to strictly reduce 6

ˆ eq their payments, which contradicts the fact that PK is a stable Proof. Applying Lemma 1 with pi = pi , we obtain coalition structure. eq eq 1 1 1 1 SPoAK ≤ α {pi (·)}i∈N (18) Next, define H2 , H1 \{i1}. Note that H2 is a feasible K coalition, because arbitrary coalition structures with at most 1 X c(Hs) K participants per coalition are allowed in our model. By the = max (19) c(·),H1⊃···⊃HK c(H1) K − s + 1 1 1 ˆ1 ˆ s=1 same argument, there exists i2 ∈ H2 and a coalition G2 ∈ PK , 1 ˆ1 1 ˆ1 K such that i2 ∈ G2 and pi1 (H2 ) ≥ pi1 (G2). X 1 2 2 ≤ = HK (20) Let G = {it , ..., it }, for any t ∈ {1, ..., h}. Continuing s t 1 Kt s=1 t this argument, we obtain a collection of sets {Hs}, where each Ht {it , ..., it } satisfies the following condition: which follows from the monotonicity of cost function, s , s Kt c(Hs) ≤ c(H1). for any t ∈ {1, ..., h} and s ∈ {1, ..., Kt}, there t t t t Gˆ ∈ Pˆ i ∈ Gˆ p t (H ) ≥ exists s K , such that s s and is s ˆt A. Tight Example pit (G ) s s eq We also present a tight example to show that SPoAK = Hence, the SPoA, SPoAK , with respect to {pi(·)}i∈N is Θ(log K). There are K · K! participants, indexed by upper bounded by t N = {is | t = 1, ..., K!, s = 1, ..., K} Ph PKt ˆt c(Pˆ ) pit (G ) K = t=1 s=1 s s (15) For any non-empty subset G ⊆ N , we define the cost c(G) c(P) Ph t=1 c(Gt) as follows: Ph PKt t t t pit (H ) • Case 1: If G ⊆ {i , ..., i } for some t ∈ {1, ..., K}, then ≤ t=1 s=1 s s (16) 1 K Ph t we set c(G) = 1. t=1 c(H1) (k−1)·(K−s+1)+1 k·(K−s+1) • G ⊆ {i , ..., i } PKt t Case 2: If s s for pit (H ) K! s=1 s s some k ∈ {1, ..., } and s ∈ {1, ..., K}, then we ≤ max t ≤ α {pi(·)}i∈N (17) K−s+1 t∈{1,...,h} c(H1) set c(G) = 1. • Case 3: Otherwise, c(G) = |G|. because α(·) is non-decreasing in K. It is evident that the preceding setting of cost function c(·) satisfies monotonicity, because c(G) = 1 for cases 1 and 2, otherwise c(G) = |G| ≥ |H| ≥ c(H) for all H ⊆ G. When Note that [14] uses an approach called summability similar K = 3, the tight example is illustrated in Fig. 1. to that of Lemma 1. Informally, a payment function (or cost- sharing mechanism) pi(·) is said to be α-summable if for every subset H of participants and every possible ordering σ on H, the sum of the payments of the participants as they are added one-by-one according to σ is at most α · c(H). However, [14] relies on the notion of cross-monotonicity for proving summability. A payment function pi(·) is said to satisfy cross- 0 0 monotonicity, if for any G ⊆ G , pi(G ) ≤ pi(G). [15] showed that if a payment function satisfies cross-monotonicity in a network cost-sharing game, then summability can bound the price of anarchy. Also, cross-monotonicity implies that a Nash equilibrium is a strong Nash equilibrium. Nonetheless, our Fig. 1: An illustration of tight example when K = 3, model is simpler than network cost-sharing games; Lemma 1 c({it }) = c({it }) = c({it }) = 1 c({it , it , it }) = where 1 2 3 , 1 2 3 shows that α {pi(·)} can be used to bound SPoAK without t t t t t t c({i1, i2}) = c({i2, i3}) = c({i1, i3}) = 1 for t = 1, ..., 6. the assumption of cross-monotonicity. In particular, many 1 2 3 4 5 6 Also, c({i2, i2}) = c({i2, i2}) = c({i2, i2}) = 1 and payment functions may violate cross-monotonicity (e.g., egal- 1 2 3 4 5 6 c({i1, i1, i1}) = c({i1, i1, i1}) = 1. The coalitions in orange itarian, Nash bargaining solution, equal-split and proportional- ˆk dotted lines {Gs } are a stable coalition structure, whereas the split), and hence, the approach in [14] will not apply to these ∗ coalitions in blue dashed lines {Gt } are a social optimum. payment functions. ˆk (k−1)·(K−s+1)+1 k·(K−s+1) Let Gs , {is , ..., is }, where s ∈ K! ∗ t t {1, ..., K}, k ∈ {1, ..., K−s+1 }. And let Gt , {i1, ..., iK }, V. EQUAL-SPLIT COST-SHARING MECHANISM where t = {1, ..., K!}. See an illustration in Fig. 1. t By equal-split cost-sharing, if is ∈ G, then Theorem 2. For equal-split cost-sharing, the SPoA is upper  1 , if G ⊆ Gˆk  |G| s bounded by eq  1 ∗ p t (G) = , if G ⊆ Gt (21) is |G| SPoAeq ≤ H = Θ(log K)  1 K K > |G| , otherwise 7

eq ˆk eq ∗ t t 1 Note that pit (Gs ) = pit (Gt \{i1, ..., is−1}) = K−s+1 . subject to c(H1) ≥ ... ≥ c(HK ) and c(Hs) ≥ s s K t ˆk ∗ max{c , ..., c } (by monotonicity), and c(H ) ≤ P c (by Hence, i1 will not switch from coalition G1 to Gt . Neither s K s t=s t t ˆk ∗ t t Lemma 2). Without loss of generality, we assume c(H ) = 1. will is switch from coalition Gs to Gt \{i1, ..., is−1}. 1 One can check that {Gˆk} are a stable coalition structure, Hence, cs ≤ c(Hs) ≤ c(H1) = 1. s PK whereas {G∗} are a social optimum. Hence, the SPoA is lower Let sˆ be the smallest integer, such that t=ˆs+1 ct ≤ 1 t PK bounded by (and hence t=s ct > 1 for any s ≤ sˆ). If s ≥ sˆ, we csc(Hs) K! K eq obtain PK ≤ cs by our assumption. If s < sˆ, we obtain P P ˆk K ct t=1 s=1 pit (Gs ) 1 t=s eq s X csc(Hs) cs SPoAK ≥ K! = = HK PK ≤ PK . Therefore, P ∗ K − s + 1 t=s ct t=s ct t=1 c(Gt ) s=1 (22) K sˆ−1 pp X X cs SPoA ≤ max cs + (28) K K PK {cs}s=1 c B. Proportional-split Cost-Sharing Mechanism s=ˆs s=1 t=s t sˆ−1 X cs Given cost function c(·), we deﬁne a truncated cost function ≤ 2 + ≤ 2 + log K (29) PK c˜(·) as follows: s=1 t=s ct

( P which follows from Lemma 3 and csˆ ≤ 1. c(G), if c(G) ≤ j∈G cj c˜(G) , P P (23) pp j∈G cj, if c(G) > j∈G cj Note that one can strengthen the SPoA by SPoAK ≤ HK P for K ≤ 6. We can apply the same tight example in Sec. V-A Note that c˜(G) ≤ j∈G cj for any G. pp pp eq pp to show that SPoAK = Θ(log K) because pi (G) = pi (G) Let SPoAK (c(·)) be the SPoA with respect to cost function in this example. c(·) specifically. Lemma 2. For proportional-split cost-sharing, Lemma 3. If 0 ≤ cs ≤ 1 for all s ∈ {1, ..., K} and PK s=ˆs cs ≥ 1 for some sˆ ≤ K, then SPoApp(c(·)) = SPoApp(˜c(·)). K K sˆ−1 X cs ˆ ≤ log K Proof. First, we show that if P is a stable coalition structure, PK ˆ P s=1 t=s ct then for any G ∈ P, we have c(G) ≤ j∈G cj. If we assume P ˆ Proof. First, since 1 ≤ 1 for any positive numbers c(G) > j∈G cj for some G ∈ P, then for all i ∈ G, x+y x+δ x, y, δ, such that δ ≤ y, we obtain pp ci · c(G) Z y pi (G) = P > ci (24) y 1 j∈G cj ≤ dδ = log(x + y) − log x (30) x + y 0 x + δ Namely, every i ∈ G can strictly reduce his payment by form- It follows that ing a singleton coalition individually. This is a contradiction K K ˆ cs X X to the fact that P is a stable coalition structure. ≤ log( ct) − log( ct) (31) ∗ PK Second, we note that if P is a social optimum, then for t=s ct t=s t=s+1 ∗ P ∗ any G ∈ P , we have c(G) ≤ j∈G cj. Otherwise, P does Hence, not attain the least total cost by including G. ˆ ˆ sˆ−1 K K Therefore, we obtain c(P) = c˜(P) and SPoApp(c(·)) = X cs X X c(P∗) c˜(P∗) K ≤ log( cs) − log( cs) (32) pp PK c SPoAK (˜c(·)). s=1 t=s t s=1 s=ˆs Theorem 3. For proportional-split cost-sharing, the SPoA is ≤ log K − log(1) = log K (33) upper bounded by PK PK because s=1 cs ≤ K and s=ˆs cs ≥ 1. SPoApp ≤ log K + 2. K VI.BARGAINING BASED COST-SHARING MECHANISMS Proof. By Lemma 2 we may assume without loss of generality First, we show that the SPoA for egalitarian bargaining P that c(G) ≤ j∈G cj for any G. Applying Lemma 1 with solution and Nash bargaining solution (irrespective of the pp pi = pi , we obtain constraint of non-negative payments) are equivalent. However, there is a difficulty, when we apply Lemma 1 to obtain an up- SPoApp ≤ α{ppp(·)} (25) K i i∈N per bound for the SPoA – we can only obtain an upper bound K 1 X pp as O(K) by the payment function of egalitarian bargaining = max pi (Hs) (26) c(·),H ⊃···⊃H c(H ) s solution, whereas the payment function of Nash bargaining 1 K 1 s=1 solution under the constraint of non-negative payments is where Hs = {is, ..., iK } for some i1, ..., iK ∈ N , with default not convenient to analyze. But we show a property in Nash costs denoted by {cs, ..., cK }. bargaining solution, namely that there always exists a coalition Since ppp(H ) = csc(Hs) , we obtain is s PK structure that satisfies positive payments and its cost is at most t=s ct √ ( K + 1) from that of a given coalition structure. We then K √ pp 1 X csc(Hs) obtain an upper bound as O( K log K) for the SPoA using SPoAK ≤ max K (27) {c ,c(H )}K c(H ) P this property. s s s=1 1 s=1 t=s ct 8

A. Equivalence between Egalitarian and Nash Bargaining Lemma 6. For egalitarian and Nash bargaining solutions Lemma 4. If the constraint of non-negative payments is (irrespective of the constraint of non-negative payments), we not considered in Nash bargaining solution, then egalitarian obtain ega ega nash and Nash bargaining solutions are equivalent: pi (G) = SPoA (c(·)) = SPoA (c(·)) nash K K pi (G). ega nash = SPoAK (˜c(·)) = SPoAK (˜c(·)). Proof. This follows from the fact that the feasible set of Proof. The proof is similar to that of Lemma 2. First, we show Nash bargaining solutions is bounded by the hyperplane ˆ ˆ P u = P c − c(G). The maximal of Q u (i.e., that if P is a stable coalition structure, then for any G ∈ P, i∈G i i∈G i i∈G i c(G) ≤ P c Nash bargaining solution) is attained at the point u = u for we obtain t∈G t based on a contradiction. If we i j P ˆ any i, j ∈ G. assume c(G) > t∈G ct for some G ∈ P, then for all i ∈ G, (P c ) − c(G) Lemma 5. For Nash bargaining solution (irrespective of the ega t∈G t pi (G) = ci − > ci (41) constraint of non-negative payments), given a stable coalition |G| ˆ ˆ structure P ∈ PK and G ∈ P, then every participant has This is a contradiction, since Pˆ cannot be a stable coalition nash non-negative payment: pi (G) ≥ 0 for all i ∈ G. structure. Second, we note that if P∗ is a social optimum, ∗ P Namely, each stable coalition structure with the constraint then for any G ∈ P , we obtain c(G) ≤ t∈G ct. Therefore, ega nash of non-negative payments coincides with a stable coalition by Corollary 1, we obtain SPoAK (c(·)) = SPoAK (c(·)) = ega nash structure without the constraint of non-negative payments. SPoAK (˜c(·)) = SPoAK (˜c(·)). Proof. Consider a coalition G = {i , ..., i } ∈ Pˆ, with 1 K B. Bounding Strong Price of Anarchy default costs denoted by {c1, ..., cK }. We prove the statement pnash(G) < 0 i ∈ G Theorem 4. For egalitarian bargaining solution and Nash by contradiction. Assume that is for some s . Then, bargaining solution (irrespective of the constraint of non- negative payments), the SPoA is upper bounded by P c − c(G) P c − c(G) j∈G j j∈G:j6=is j √ cs < ⇒ cs < (34) ega nash |G| |G| − 1 SPoAK = SPoAK = O( K log K) (42) Thus, we obtain Proof. First, by Lemma 6, it sufﬁces to consider egalitarian P P bargaining solution with cost function satisfying c(G) ≤ cj − c(G) cs + cj − c(G) j∈G = j∈G:j6=is (35) P c for any G. |G| |G| j∈G j Next, by Lemma 8, there exists a coalition structure P˜, 1 P ( + 1) j∈G:j6=i cj − c(G) nash ˜ ˜ |G|−1 s such√ that pi (G) > 0 for all i ∈ G ∈ P, and c(P) ≤ < (36) ∗ ∗ |G| ( K + 1) · c(PK ), where PK is a social optimum. For a P ˆ cj − c(G) stable coalition structure PK , = j∈G:j6=is (37) |G| − 1 c(PˆK ) √ c(PˆK ) P ≤ ( K + 1) · cj − c(G\{is}) c(P∗ ) ˜ ≤ j∈G:j6=is (38) K c(P) |G| − 1 Let P˜ = {G1, ..., Gh}, We apply the similar argument of where c(G\{is}) ≤ c(G) by monotonicity of the cost func- Lemma 1 to obtain that for any t ∈ {1, ..., h} and s ∈ ˆt ˆ t ˆt tion. For any ik 6= is, {1, ..., Kt}, there exists Gs ∈ PK , such that is ∈ Gs and t t P p t (H ) ≥ p t (Gˆ ). Therefore, c − c(G\{i }) is s is s nash j∈G:j6=is j s pi (G\{is}) = ck − (39) k |G| − 1 Ph PKt ega ˆt c(PˆK ) t=1 s=1 pit (Gs) P c − c(G) = s (43) j∈G j nash ˜ Ph < ck − = p (G) (40) c(P) t=1 c(Gt) |G| ik Ph PKt ega t p t (Hs) t=1 s=1 is ega Namely, all users in G\{is} would reduce strictly their pay- ≤ ≤ α {p (·)} (44) Ph t i ments by switching to the coalition G\{i }. This is a contra- t=1 c(H1) s √ diction to the fact that Pˆ is a stable coalition structure. ega ega Hence, SPoAK ≤ ( K + 1) · α {pi (·)} . Corollary 1. For egalitarian bargaining solution and Nash Let Hs = {is, ..., iK }, with the default costs denoted bargaining solution (irrespective of the constraint of non- by {cs, ..., cK }. Recall that egalitarian bargaining solution is negative payments), their SPoA are equivalent: given by K ega nash P SPoA = SPoA ega ( t=s ct) − c(Hs) K K p (Hs) = cs − is K − s + 1 Proof. This follows from Lemma 4 and Lemma 5. subject to c(H1) ≥ ... ≥ c(HK ) and c(Hs) ≥ PK Given cost function c(·), we deﬁne a truncated cost function max{cs, ..., cK } (by monotonicity), and c(Hs) ≤ √t=s ct (by P eqa c˜(·) by Eqn. (23) in Sec. V-B, such that c˜(G) ≤ j∈G cj for Lemma 6). Finally, it follows that SPoAK ≤ O( K log K) any G. by Lemma 10. 9

We can apply the same tight example in Sec. V-A to show ega nash ega that SPoAK = SPoAK ≥ Θ(log K) because pi (G) = eq pi (G) in this example.

Lemma 7. ( [21] Theorem 2) Consider a coalition G = {i1, ..., iK }, with corresponding default costs denoted by {c1, ..., cK }, such that c1 ≥ c2 ≥ ... ≥ cK . If the constraint of non-negative payments is considered, then Nash bargaining solution can be expressed by Fig. 2: Consider G = {i1, i2, i3}. The three circles depict the ( (Pm c )−c(G) f t=1 t sets Fi1 (r(G)), Fi2 (r(G)), Fi3 (r(G)). If c is represented nash cs − m , if s ≤ m pi (G) = by a unit area, then X (L) is the intersection area of L. s 0, otherwise G

Pm−1 ( t=1 ct)−c(G) where m is the largest integer such that cm > . m−1 Given a set of participants {i1, ...iK } and Hs , {is, ..., iK }, P X (L) X (L) Lemma 8. Suppose c(G) ≤ j∈G cj for any G. Given any we simply write s , Hs . coalition structure P ∈ PK , there exists a coalition structure In general, the strong price of anarchy of usage based cost- ˜ nash ˜ P ∈ PK√, such that pi (G) > 0 for all i ∈ G ∈ P, and sharing can be Ω(K). c(P˜) ≤ ( K + 1) · c(P). Theorem 5. For usage based cost-sharing in general settings, Proof. See Appendix. there exists an instance, such that

ub Lemma 10. Let bs , c(Hs). Consider the following maxi- SPoAK = Ω(K). mization problem: Proof. See Appendix. K PK ∗ X ( t=s ct) − bs (M1) y (K) , max cs − (45) {c ,b }K K − s + 1 s s s=1 s=1 A. Monotone Utilization subject to ub Generally SPoAK = Ω(K), but we next present a general K ub X sufficient condition for inducing SPoAK = Θ(log K). The set bs ≤ ct, for all s = 1, ..., K − 1, (46) of facilities utilized by participants Fi(·) are said to satisfy t=s monotone utilization, if for all H ⊆ G, 0 ≤ cs ≤ bs ≤ bs+1 ≤ 1, for all s = 1, ..., K, (47) K X f X f X c ≤ c (51) b1 + Kcs − ct ≥ 0, for all s = 1, ..., K (48) f∈∪i∈H Fi(r(H)) f∈∪i∈H Fi(r(G)) t=1 ∗ Namely, the total cost of facilities utilized by a subset of The maximum of (M1) is upper bounded by y (K) ≤ 1 + participants increases in a larger coalition. Note that monotone HK−1 = O(log K). utilization condition implies monotonicity of cost function. In Lemma 10, constraint (48) captures positive payment for For hotel room sharing problem, a set of days of a par- every participant in H1. ticipant stays in a room booking does not depend on the coalition, and hence, Fi(r(H)) = Fi(r(G)). For taxi-ride Proof. See Appendix. sharing problem, one needs to travel a greater distance in order to pick-up and drop-off other passengers when sharing with VII.USAGE BASED COST-SHARING MECHANISM more passengers, and hence, monotone utilization condition is satisfied. Although pass sharing problem does not generally Recall that r(G) is the lowest cost canonical resource for satisfy monotone utilization condition, we will later prove that G, and (r(G)) is the set of facilities utilized by participant Fi the SPoA for pass sharing problem with uniform average cost i ∈ G in canonical resource r(G). First, for each subset L ⊆ is also Θ(log K). G, we define

X f Theorem 6. Consider usage based cost-sharing, such that XG(L) c (49) , Fi(·) satisﬁes the monotone utilization condition. Then T S f∈ Fi(r(G)) Fi(r(G)) i∈L i∈G\L ub SPoAK ≤ HK = Θ(log K). (52) Namely, XG(L) is the total cost of facilities of canonical resource r(G) that are only used by the coalition L exclusively. Proof. See Appendix. See Fig. 2 for an illustration of XG(L). We can apply the same tight example in Sec. V-A to show Usage based payment function can be reformulated as ub that SPoAK = Θ(log K). We set XG(L) = 1 when L = G, X XG(L) pub(G) = (50) otherwise XG(L) = 0. This can satisfy monotone utilization i |L| ub eq L⊆G:i∈L condition. It follows that pi (G) = pi (G) in this example. 10

B. Pass Sharing cost-sharing (under monotone consumption condition or for Pass sharing problem violates monotone utilization condi- pass sharing problem) is only Θ(log K), where K is the max- tion. But we can bound the SPoA for pass sharing problem imum capacity of sharing participants. Therefore, distributed with uniform average cost cf = 1. decision-making processes under common fair cost-sharing mechanisms induce only moderate inefficiency. Theorem 7. Consider usage based cost-sharing for pass The SPoA for egalitarian and Nash solutions (i.e., f √ sharing problem with uniform average cost c = 1. Then O( K log K)) is not known to be tight. It is interesting ub to see if the gap will be closed. Furthermore, the SPoA SPoA ≤ HK + 1 = Θ(log K) (53) K for specific problems (e.g., hotel room, taxi-ride and pass Proof. See Appendix. sharing problems) may be strictly smaller than Θ(log K).A companion study of empirical SPoA for taxi-ride sharing using VIII.EXISTENCEOF STABLE COALITION STRUCTURES real-world taxi data can be found in [22]. This section completes the study by investigating the existence of stable coalition structures considering different cost- REFERENCES sharing mechanisms. First, we define a cyclic preference as [1] The Economist, “The rise of the sharing economy,” Mar. 2013. (i , ..., i ) (G , ..., G ) i ∈ G ∩G [2] N. Andelman, M. Feldman, and Y. Mansour, “Strong price of anarchy,” sequences 1 s and 1 s , where k k k+1 Games and Economic Behavior, vol. 65, no. 2, pp. 289 – 317, 2009. for all k ≤ s − 1, and is ∈ Gs ∩ G1, such that [3] N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani, Algorithmic Game Theory. Cambridge University Press, 2007.

ui1 (pi1 (G1)) > ui1 (pi1 (G2)), [4] C.-K. Chau and K. M. Sim, “The price of anarchy for non-atomic u (p (G )) > u (p (G )), congestion games with symmetric cost maps and elastic demands,” i2 i2 2 i2 i2 3 Operations Research Letters, vol. 31, pp. 327–334, Sept. 2003. . [5] C.-K. Chau and K. M. Sim, “Analyzing the impact of selfish behaviors . of internet users and operators,” IEEE Communications Letters, vol. 7, u (p (G )) > u (p (G )) pp. 463–465, Sept. 2003. is is s is is 1 [6] A. E. Roth, Game-Theoretic Models of Bargaining. Cambridge Univer- sity Press, 2005. Lemma 11. If there exists no cyclic preference, there always [7] S. Albers, “On the value of coordination in network design,” SIAM exists a stable coalition structure. Furthermore, such a stable Journal on Computing, vol. 38, pp. 2273–2302, 2009. coalition structure can be found in time nO(K). [8] H. Aziz and F. Brandl, “Existence of stability in hedonic coalition formation games,” in Proc. of Intl. 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APPENDIX and Pmt−1 c − c(N ) l=mt−2+1 l t−1 A. Monotone Property of α(·) ck − ≤ 0 mt−1 − mt−2 We show that α(·) is non-decreasing in K. Recall that By Lemma 7, the above conditions can guarantee positive PK s=1 pis (Hs) payments in Nash bargaining solution. We then replace each αK {pi(·)}i∈N , max , (54) c(·),H1⊃···⊃HK c(H1) coalition G ∈ P by a collection of coalitions N1, ..., NR. We call such a coalition structure P˜. where H1, ..., HK are a collection of subsets, such that each Note that for each ik ∈ Nt+1, Hs , {is, ..., iK } for some i1, ..., iK ∈ N . We explicitly Pmt c − c(N ) use subscript K to denote the dependence of K. For brevity, l=mt−1+1 l t ntcmt−1+1 − c(Nt) 0 ck ≤ ≤ (58) consider K = K + 1. Suppose mt − mt−1 nt 0 PK where n |N | = m − m . Without loss of generality, s=1 pis (Hs) t , t t t−1 αK0 = (55) we assume c(G) = 1. Let c¯ c and C c(N ). It c(H1) t , mt−1+1 t , t is evident to see that 1 ≥ c¯1 > c¯2 > ... > c¯R > 0. Since for a cost function c(·) and some subsets H1, ..., HK0 , where c(G) ≤ P c for any G, C ≤ n c¯ for all t ∈ {1, ..., K}. 0 j∈G j t t t |H | = K . Then we construct a new set H ∪ {i} for some R 1 1 We next upper bound P C by the following optimiza- element i, and let c(H ∪ {i}) = c(H ) (which still satisﬁes t=1 t 1 1 tion problem (S1): monotonicity). Then R PK0 X pi(H1 ∪ {i}) + s=1 pis (Hs) (S1) max Ct (59) α ≥ ≥ α 0 (C ,n )R K K t t t=1 t=1 c(H1 ∪ {i}) subject to 0 ≤ Ct ≤ 1, for all t ∈ {1, ..., R} (60) because pi(·) is non-negative. R X nt ≤ K (61) B. Bargaining Based Cost-Sharing Mechanisms t=1 Lemma 7. ( [21] Theorem 2) Consider a coalition G = Ct ≤ nt(¯ct − c¯t+1), for all t ∈ {1, ..., R}, (62) {i1, ..., iK }, with corresponding default costs denoted by where we assume c¯R+1 = 0. Since in (S1) the lower bounds {c1, ..., cK }, such that c1 ≥ c2 ≥ ... ≥ cK . If the constraint on nt are only present in Constraints (62), we assume nt = of non-negative payments is considered, then Nash bargaining Ct , where yt , c¯t − c¯t+1 for t = 1, ..., R, and obtain solution can be expressed by yt R ( Pm ( t=1 ct)−c(G) X nash cs − m , if s ≤ m (S2) max Ct (63) p (G) = (C )R is 0, otherwise t t=1 t=1 subject to 0 ≤ Ct ≤ 1, for all t ∈ {1, ..., R} (64) Pm−1 ( t=1 ct)−c(G) where m is the largest integer such that cm > . R m−1 X Ct ≤ K (65) Lemma 8. Suppose c(G) ≤ P c for any G. Given any y j∈G j t=1 t coalition structure P ∈ PK , there exists a coalition structure ˜ nash ˜ Note that (S2) is simply a fractional knapsack problem. P ∈ PK , such that p (G) > 0 for all i ∈ G ∈ P, and √ i Suppose that (y )R are arranged in a non-increasing order, c(P˜) ≤ ( K + 1) · c(P). t t=1 y1 ≥ y2 ≥ · · · ≥ yR. Let ` be the largest index such that Proof. Given coalition structure P, we construct coalition ` R ˜ X 1 X 1 structure P as follows. For each G ∈ P, we sort the ≤ K and > K (66) yt yt participants G = {i1, ..., im} in the decreasing order of their t=1 t=`+1 default costs, such that c1 ≥ c2 ≥ ... ≥ cm. We next split ∗ R Then the optimal solution (Ct ) to (S2) is given by G into R sub-groups {N1, ..., NR}, where i1, ..., im ∈ N1, t=1 1 Lemma 9. Hence, the optimal value of (S2) is PR C∗ ≤ im +1, ..., im ∈ N2, ··· , im +1, ..., im ∈ NR, such that t=1 t 1 2 R−1 R ` + 1. Note that for k = mt−1 + 1, ..., mt R R Pk X X l=m +1 cl − c({imt−1+1, ..., ik}) yt = (¯ct − c¯t+1) =c ¯1 ≤ 1 (67) c − t−1 > 0 (56) k t=1 t=1 k − mt−1

Pmt By the arithmetic mean-harmonic mean inequality (i.e., c − c({i , ..., i }) ` y l=mt−1+1 l mt−1+1 mt+1 P t ` c − ≤ 0 (57) t=1 ` ≥ P` 1 ), we obtain imt+1 t=1 y mt − mt−1 t ` where t ∈ {1, ..., R} and m0 = 0. By monotonicity of c(·) and ` ` X yt 1 ≤ ` ≤ ≤ (68) the ordering on ci, Eqns. (56)-(57) imply that each ik ∈ Nt K P 1 ` ` t=1 yt t=1 satisﬁes √ Pmt+1 c − c(N ) Hence, it follows that ` ≤ K, and the maximum of (S2) is l=mt−1 l t √ ck − > 0, upper bounded by K + 1. mt − mt−1 12

Therefore, this completes the proof by the optimum value z∗ of the following linear program is at most 1 − 1 . R √ √ K X X X ˜ c(Nt) ≤ ( K+1)·c(G) ⇒ c(P) ≤ ( K+1)·c(P). K PK ∗ X (K − s + 1)cs − t=s ct G∈P t=1 G∈P z , max (80) (69) cs K − s + 1 s=1 subject to Lemma 9. The fractional knapsack problem is deﬁned by ( PK ) t=1 ct − 1 max 0, ≤ cs ≤ 1, for all s ∈ {1, ..., K}. R K X (FKP) max Ct (70) (81) (C )R t t=1 t=1 We write subject to 0 ≤ Ct ≤ 1, for all t ∈ {1, ..., R} (71) R K PK X Ct X (K − s + 1)cs − ct ≤ K (72) f(c , ..., c ) t=s (82) y 1 K , K − s + 1 t=1 t s=1 K R Suppose that (yt) are positive and arranged in a non- X t=1 = csαs, (83) increasing order, y1 ≥ y2 ≥ · · · ≥ yR. Let ` be the largest s=1 index such that Ps 1 ∗ ` R where αs , 1− t=1 K−t+1 . Denote by cs, for s = 1, ..., K, X 1 X 1 ≤ K and > K (73) the optimal values maximizing f(c1, ..., cK ) subject to the yt yt t=1 t=`+1 constraints given in Eqn. (81). Note that

∗ R K K s Then the optimal solution (Ct )t=1 to (FKP) is given by X X X 1 αs = K − (84)  K − s + 1  1, if t ∈ {1, ..., `}, s=1 s=1 t=1 ∗  P` 1 K K C = min{K − , 1}, if t = ` + 1, X 1 X t j=1 yj = K − 1 = 0 (85)  K − s + 1 0, if t ∈ {` + 2, ..., R} t=1 s=t Proof. The proof follows from a well-known result in knap- Claim 1. Let (c1, ..., cK ) 6= 0 be a basic feasible solution sack problem (for example, see [23] Theorem 2.2.1). (BFS) of LP Eqn. (80). Then, there exists a partition S1 ∪So of K−h−1 [K] such that and cs = 1 for all s ∈ S1, and cs =c ¯ , K−h Lemma 10. Let bs , c(Hs). Consider the following maxi- mization problem. for all s ∈ So, where h , |So| ≤ K − 1.

K PK Proof. A BFS (c1, ..., cK ) of the LP Eqn. (80) is determined ∗ X ( t=s ct) − bs (M1) y (K) max cs − (74) by exactly K equations among the inequalities in Eqn. (81). , K {cs,bs}s=1 K − s + 1 n PK o s=1 t=1 ct−1 We ﬁrst note that we must have max 0, K = subject to PK t=1 ct−1 PK K K ; otherwise, t=1 ct < 1 implies that cs = 0 for X s bs ≤ ct, for all s ∈ {1, ..., K − 1}, (75) all (since the inequalities in Eqn. (81) in this case reduce to t=s 0 ≤ cs ≤ 1 and the BFS will pick one of these two inequalities 0 ≤ c ≤ b ≤ b ≤ 1, for all s ∈ {1, ..., K}, for each s). s s s+1 (76) PK t=1 ct−1 K Let S1 , {s : cs = 1} and So , {s : cs = K }, X PK c −1 b1 + Kcs − ct ≥ 0, for all s ∈ {1, ..., K}. (77) t=1 t and note that S1 ∪ So = {1, ..., K}. Write x , K = t=1 hx+K−h−1 K . Then x =c ¯. The maximum is upper bounded by y∗(K) ≤ 1 + PK−1 1 = s=1 s Let (c , ..., c ) be a BFS of Eqn. (81) deﬁned by the O(log K). 1 K partition So ∪S1. Then substituting the value of cs in Eqn. (82) Proof. Clearly, it is enough to bound (M1) subject to the and using Eqn. (84) we obtain relaxation: X X f(c1, ..., cK ) =c ¯ αs + αs

0 ≤ cs ≤ bs ≤ bs+1 ≤ 1, for all s ∈ {1, ..., K}, (78) s∈So s∈S1 K P αs X X s∈S1 b1 + Kcs − ct ≥ 0, for all s ∈ {1, ..., K}. (79) = (1 − c¯) αs = (86) |S1| t=1 i∈S1

As the coefﬁcients of bs in (M1) are positive, it is clear that Note that αs > αs+1 for all s = 1, ..., K − 1. It follows that setting bs = 1, for all s, will maximize (M1) without violating the choice S1 = {1} maximizes f(c1, ..., cK ) and the lemma the Constraints (78) and (79). It is then enough to show that follows. 13

C. Usage Based Cost-Sharing Mechanism Theorem 5. For usage based cost-sharing in general settings, Theorem 6. Consider usage based cost-sharing, such that there exists an instance, such that Fi(·) satisﬁes the monotone utilization condition. Then ub SPoAK = Ω(K). ub SPoAK ≤ HK = Θ(log K). (92) Proof. We construct a similar instance to the one in Sec. V-A. ub t Proof. Applying Lemma 1 with pi = pi , we obtain There are K · K! participants, indexed by N = {is | t = K 1, ..., K, s = 1, ..., K!}. For any non-empty subset G ⊆ N 1 X SPoAub ≤ α{pub(·)} = max pub(H ) and L ⊆ G, we deﬁne XG(L) as follows: K i i∈N is s c(·),H1⊃···⊃HK c(H1) t t s=1 • Case 1: If G ⊆ {i1, ..., iK } for some t ∈ {1, ..., K}, then where Hs = {is, ..., iK }, with the corresponding default costs – If |G| = 1, then XG(L) = 1. 0 t denoted by {cs, ..., cK }. Without loss of generality, we assume – If |G| > 1, then let s = min{s | i 0 ∈ G}, and s c(H ) = 1 X (L) X (L) pub(H ) = 1 . Recall that s , Hs and is s  1 t , if L = {i } P Xs(L)  2 s L⊆H :i ∈L |L| . 1 t s s XG(L) = 2 , if L = G\{is} (87) Note that the monotone utilization condition is equivalent  0, otherwise to saying that, for all s ∈ {1, ..., K − 1} and K ≥ t > s, X X See an illustration of an example for setting XG(L) in Xt(L) ≤ Xs(L) (93) Fig. 3. L⊆Ht L⊆Hs:L∩Ht6=∅ ub Hence, we can bound SPoAK by the maximum value of the linear optimization problem (P1):

K X X Xs(L) (P1) max (94) K {Xs(·)}s=1 |L| s=1 L⊆Hs:is∈L subject to X X Xt(L) ≤ Xs(L), Fig. 3: An illustration of an example for setting L⊆Ht L⊆Hs:L∩Ht6=∅ 00 0 for all 1 ≤ s < t ≤ K, (95) XG(L) in case 1. Suppose s < s < s. t t t X X{it } {i } = 1,X{it ,it } {i } = X{it ,it } {i 0 } = X (L) ≤ 1, for all s ∈ {1, ..., K}, (96) s s s s0 s s s0 s 1 t t t 1 X{it ,it ,it } {i 00 } = X{it ,it ,it } {i , i 0 } = . L⊆H1 s s0 s00 s s s0 s00 s s 2 Otherwise, XG(L) = 0. Xs(L) ≥ 0, for all s ∈ {1, ..., K},L ⊆ Hs (97)

(k−1)·(K−s+1)+1 k·(K−s+1) For s ∈ {1, ..., K} and L ⊆ Hs, we deﬁne • Case 2: If G ⊆ {is , ..., is } for K! ( some k ∈ {1, ..., K−s+1 } and s ∈ {1, ..., K}, then 1 |L| , if is ∈ L, ρ(L, s) , (98) 1, if L = G 0, otherwise XG(L) = (88) 0, otherwise Then the dual problem to (P1) can be written as follows: • Case 3: Otherwise, G is a partition of m sets G = H ∪ 1 (D1) min z (99) · · ·∪Hm, where each Ht is a maximal subset that satisﬁes λ(·),z either Case 1 or Case 2. Then, subject to m t−1 X X X XG(L) = XHs (L ∩ Hs) (89) λ(s, t) − λ(t, s) ≥ ρ(L, t), s=1 s=1 t+1≤s≤K:L∩Hs6=∅ ˆk (k−1)·(K−s+1)+1 k·(K−s+1) for all t ∈ {2, ..., K},L ⊆ H , (100) Let Gs , {is , ..., is }, where s ∈ t {1, ..., K}, k ∈ {1, ..., K! }. And let G∗ {it , ..., it }, X K−s+1 t , 1 K ρ(L, 1) + λ(1, s) ≤ z, for all L ⊆ H1, ˆk where t ∈ {1, ..., K!}. One can check that {Gs } form a stable 2≤s≤K:L∩Hs6=∅ ∗ coalition structure, whereas {Gt } are a social optimum. Note (101) t t t that if Hs = {is, ..., iK }, then λ(s, t) ≥ 0, for all 1 ≤ s < t ≤ K, z ≥ 0 (102)

ub t 1 ∗ ∗ pit (Hs) ≥ (90) We next provide a primal-dual feasible pair (X , λ ) whose s 2 PK 1 objective value is s=1 s . To better understand this proof, it Therefore, the price of anarchy is lower bounded by may be instructive to look at the example when K = 3 in PK! PK ub t Table II. t=1 s=1 pit (Hs) K SPoAub ≥ s ≥ (91) K PK! t 2 Primal solution: t=1 c(H1) 14

For s ∈ {1, ..., K} and L ⊆ Hs, we set For L ⊆ H1, we sum the equations in Constraint (108) for L ∩ Ht for all t ∈ {2, ..., K} to obtain ∗ 1, if L = Hs, K Xs (L) , (103) X 0, otherwise ρ(L ∩ Ht, t) (113) t=2 K t−1 K For s ∈ {1, ..., K − 1}, s < t ≤ K, we obtain X X X X = λ∗(s, t) − λ∗(t, s) (114)

X t=2 s=1 t=2 t+1≤s≤K:L∩Hs6= 1 =X∗(H ) = X∗(L) (104) ∅ t t t K t−1 K K L⊆H X X X X t ≥ λ∗(s, t) − λ∗(t, s) (115) ∗ X ∗ =Xs (Hs) = Xs (L) (105) t=2 s=1 t=2 s=t+1 L⊆H :L∩H 6= K−1 K K K s t ∅ X X X X = λ∗(s, t) − λ∗(t, s) (116) Also, we obtain s=1 t=s+1 t=2 s=t+1 K X ∗ X ∗ ∗ = λ (1, s) (117) X1 (L) = X1 (H1) = 1 (106) s=2 L⊆H1 It follows that ∗ Hence, X satisfies Constraint (95) and Constraint (96). K K X ∗ X ρ(L, 1) + λ (1, s) ≤ ρ(L ∩ Ht, t) (118) Dual solution: s=2 t=1 We first claim that there is a set of numbers λ∗(s, t) ≥ 0, |L| X 1 X 1 for 1 ≤ s < t ≤ K that satisfy Constraint (100) as equalities. = = ≤ z∗ (119) |L ∩ Ht| t To show this, we study the linear optimization problem (D2): 1≤t≤K:it∈L t=1 Hence, Constraint (101) is satisfied. X (D2) min λ(s, t) (107) Optimality: λ(·) 1≤s

subject to K ∗ K X X Xs (L) X 1 ∗ t−1 = = z (120) X X |L| |Hs| λ(s, t) − λ(t, s) = ρ(L, t), s=1 L⊆Hs:is∈L s=1

s=1 t+1≤s≤K:L∩Hs6=∅ for all t ∈ {2, ..., K},L ⊆ Ht, (108) λ(s, t) ≥ 0, for all 1 ≤ s < t ≤ K (109) Theorem 7. Consider usage based cost-sharing for pass sharing problem with uniform average cost cf = 1. Then and its dual: ub SPoAK ≤ HK + 1 = Θ(log K). (121) K X X Xs(L) (P2) max (110) Proof. Recall that Ti is the a set of required usage timeslots X (·) |L| s s=2 L⊆H :i ∈L of user i, Tr is the set allowable timeslots of pass r, and s s S subject to Fi(r(G)) = Ti ∪ Tr\( j∈G Tj) . For pass sharing problem, X X we note that XG(L) = 0 if 1 < |L| < |G|. Hence, Xt(L) ≤ Xs(L) + 1, X (G) L⊆Ht L⊆Hs:L∩Ht6=∅ ub G pi (G) = + XG({i}) (122) for all 1 < s < t ≤ K, (111) |G| X f Xt(L) ≤ 1, for all 1 < t ≤ K (112) Since the average cost c = 1, XG({i}) = |Ti|, and XG(G) = S L⊆Htu |Tr\( j∈G Tj)|. ˆ Let Hs , {is, ..., iK } and X , maxs∈{1,...,K} Xs(Hs). ub By Constraint (112), the primal problem (P2) is bounded; Applying Lemma 1 with pi = pi , we obtain ∗ it is also feasible since Xs (L) for s ∈ {2, ..., K} given in K ˆ Eqn. (103) satisﬁes Constraint (111) and Constraint (112). It ub 1 X X SPoAK ≤ max + |Tis | follows that the dual (D2) is also feasible and bounded, that c(·),H ⊃···⊃H c(H ) |H | 1 K 1 s=1 s is, there exist numbers λ∗(s, t) satisfying Constraint (108). (123) ∗ PK 1 ∗ ∗ ˆ Let z = s=1 s . We claim that (λ , z ) is a feasible Note that X ≤ c(H1) because of monotonicity of cost PK solution to the dual problem Eqn. (99). Evidently, we need function, and s=1 |Tis | ≤ c(H1), because the coalition of only to check that it satisﬁes Constraint (101). users H1 = {i1, ..., iK } cannot overlap in their required usage 15

X1({1,2}) X1({1,3}) X1({1,2,3}) X2({2,3}) (P1) max X1({1}) + 2 + 2 + 3 + X2({2}) + 2 + X3({3}) subject to λ(1, 2) : X2({2}) + X2({3}) + X2({2, 3}) ≤ X1({2}) + X1({3}) + X1({1, 2}) + X1({1, 3}) + X1({2, 3}) + X1({1, 2, 3}) λ(1, 3) : X3({3}) ≤ X1({3}) + X1({1, 3}) + X1({2, 3}) + X1({1, 2, 3}) λ(2, 3) : X3({3}) ≤ X2({3}) + X2({2, 3}) z : X1({1}) + X1({2}) + X1({3}) + X1({1, 2}) + X1({1, 3}) + X1({2, 3}) + X1({1, 2, 3}) ≤ 1 X1({1}),X1({2}),X1({3}),X1({1, 2}),X1({1, 3}),X1({2, 3}) ≥ 0, X1({1, 2, 3}),X2({2}),X2({3}),X2({2, 3}),X3({3}) ≥ 0

(D1) min z subject to X2({2}): λ(1, 2) ≥ 1 X2({3}): λ(1, 2) − λ(2, 3) ≥ 0 1 X2({2, 3}): λ(1, 2) − λ(2, 3) ≥ 2 X3({3}): λ(1, 3) + λ(2, 3) ≥ 1 X1({1}) : 1 ≤ z X1({2}): λ(1, 2) ≤ z X1({3}): λ(1, 2) + λ(1, 3) ≤ z 1 X1({1, 2}): 2 + λ(1, 2) ≤ z 1 X1({1, 3}): 2 + λ(1, 2) + λ({3}, 1, 3) ≤ z X1({2, 3}): λ(1, 2) + λ(1, 3) ≤ z 1 X1({1, 2, 3}): 3 + λ(1, 2) + λ(1, 3) ≤ z λ(1, 2), λ(1, 3), λ(2, 3) ≥ 0

(D2) min λ(1, 2) + λ(1, 3) + λ(2, 3) subject to X2({2}): λ(1, 2) = 1 X2({3}): λ(1, 2) − λ(2, 3) = 0 1 X2({2, 3}): λ(1, 2) − λ(2, 3) = 2 X3({3}): λ(1, 3) + λ(2, 3) = 1 λ(1, 2), λ(1, 3), λ(2, 3) ≥ 0

X2({2,3}) (P2) max X2({2}) + 2 + X3({3}) subject to λ(1, 2) : X2({2}) + X2({3}) + X2({2, 3}) ≤ 1 λ(1, 3) : X3({3}) ≤ 1 λ(2, 3) : X3({3}) ≤ X2({3}) + X2({2, 3}) + 1

TABLE II: (P1), (D1), (D2), (P2) for K = 3. timeslots, which are within the allowable timeslots of a pass Lemma 11. If there exists no cyclic preference, there always utilized in c(H1). Therefore, exists a stable coalition structure. Furthermore, such a stable O(K) K coalition structure can be found in time n . ub X 1 SPoAK ≤ 1 + max = HK + 1 (124) c(·) s Proof. We include the standard argument for completeness s=1 (see, e.g., [12] for dynamic coalition formation by local improvements). Consider a directed graph G = (NK ,E) on N the set NK , {S ∈ 2 : |S| ≤ K} of subsets of size D. Existence of Stable Coalition Structures at most K. For two sets G1,G2 ∈ NK , we define an edge (G1,G2) ∈ E if and only if there is a participant i ∈ G1 ∩G2 This section investigates the existence of stable coali- such that ui(G1) < ui(G2). Then the existence of a cyclic tion structures considering different cost-sharing mechanisms. preference is equivalent to the existence of a directed cycle in First, we define a cyclic preference as sequences (i1, ..., is) G. Thus if there exists no cyclic preference, then G is acyclic and (G1, ..., Gs), where ik ∈ Gk ∩ Gk+1 for all k ≤ s − 1, and hence has at least one sink. and i ∈ G ∩ G , such that s s 1 Let P be a maximal subset of sinks in G with the property 0 ui (pi (G1)) > ui (pi (G2)), that any two distinct nodes G, G ∈ P are pairwise disjoint. 1 1 1 1 0 u (p (G )) > u (p (G )), Let S be the set of participants covered by P, and G be the i2 i2 2 i2 i2 3 subgraph of G obtained by deleting all the nodes containing . . some participant in S. By induction, there is a stable coalition structure P0 among uis (pis (Gs)) > uis (pis (G1)) 0 0 the set of participants S , N\S. It follows that P ∪ P 16 is a stable coalition structure on the set of all participants. Summing the above equations, one obtains a contradiction 0 > Indeed, if there is a blocking coalition G1 then G1 ∩ S 6= ∅ 0. This completes the proof by Lemma 11. (since otherwise P0 is not stable among the participants in S0). Theorem 11. For Nash bargaining solution, there always But then there must exist i ∈ S such that contain ui(G1) > exists a stable coalition structure, irrespective of the constraint ui(G2), where G2 ∈ P is the coalition containing i. This of non-negative payments. would imply that (G2,G1) ∈ E contradicting that G2 is a sink in G. Proof. First, if the constraint of non-negative payments is not considered, then the existence of a stable coalition structure Theorem 8. For equal-split cost-sharing, there always exists follows from Corollary 1 and Theorem 10. a stable coalition structure. Second, if the constraint of non-negative payments is con- nash Proof. If there exists a cyclic preference, defined by (i1, ..., is) sidered, then ui(pi (G)) ≤ ci. Note that there exists at least nash and (G1, ..., Gs), then one participant i ∈ G for any G ⊆ N , such that pi (G) > 0. Otherwise, P pnash(G) = c(G) = 0 (which we may c(G ) c(G ) i∈G i u (peq(G )) = c − 1 > c − 2 = u (peq(G ))exclude, without loss of generality). Suppose that there exists a i1 i1 1 i1 i1 i1 i1 2 |G1| |G2| cyclic preference, defined by (i1, ..., is) and (G1, ..., Gs). Let eq c(G2) c(G3) eq H ⊆ G be the set of participants with positive payment in ui (p (G2)) = ci − > ci − = ui (p (G3)), t t 2 i2 2 |G | 2 |G | 2 i2 nash 2 3 each Gt, that is, pi (Gt) > 0 for all i ∈ Ht. By Lemma 7, . it follows that . P ( ( cj )−c(Gt) c(G ) c(G ) j∈Ht eq s 1 eq nash , if i ∈ Ht u (p (G )) = c − > c − = u (p (G )) |Ht| is is s is is is is 1 ui(pi (Gt)) = (130) |Gs| |G1| ci, if i 6∈ Ht

Summing the above equations, one obtains a contradiction 0 > nash By Lemma 7, if i 6∈ Ht, ui(p (Gt)) = ci ≤ 0. This completes the proof by Lemma 11. P i P ( cj )−c(Gt) ( cj )−c(Gt) j∈Ht nash j∈Ht . Hence, ui(p (Gt)) ≤ |Ht| i |Ht| Theorem 9. For proportional-split cost-sharing, there always for all i ∈ Gt. exists a stable coalition structure. nash nash Note that if ui(pi (Gt)) > ui(pi (Gr0 )), then nash nash pp pp ui(p (Gr0 )) 6= ci because ui(p (G)) ≤ ci. If there Proof. If ui(p (G1)) > ui(p (G2)), then i i i i exists a cyclic preference, then ci · c(G1) ci · c(G2) c(G1) c(G2) P ci− > ci− ⇒ < ( c ) − c(G ) P P P P j∈H1 j 1 nash cj cj cj cj ≥ u (p (G )) j∈G1 j∈G2 j∈G1 j∈G2 i1 i1 1 (125) |H1| P Thus, if there exists a cyclic preference, then ( cj) − c(G2) >u (pnash(G )) = j∈H2 , i1 i1 2 (131) c(G ) c(G ) c(G ) c(G ) |H2| 1 2 s 1 P P < P < ··· < P < P ( cj) − c(G2) cj cj cj cj j∈H2 nash j∈G1 j∈G2 j∈Gs j∈G1 ≥ u (p (G )) i2 i2 2 (126) |H2| P Summing the above equations, one obtains a contradiction 0 > ( cj) − c(G3) >u (pnash(G )) = j∈H3 , (132) 0. This completes the proof by Lemma 11. i2 i2 3 |H3| . Theorem 10. For egalitarian cost-sharing, there always exists . a stable coalition structure. P ( cj) − c(Gs) j∈Hs ≥ u (pnash(G )) Proof. If there exists a cyclic preference, then is is s |Hs| P (P c ) − c(G ) ( cj) − c(G1) ega j∈G1 j 1 nash j∈H1 u (p (G )) = >ui (p (G1)) = (133) i1 i1 1 s is |G1| |H1| (P c ) − c(G ) j∈G2 j 2 ega Summing the above equations, one obtains a contradiction 0 > > = u (p (G )), (127) i1 i1 2 |G2| 0. This completes the proof by Lemma 11. P ( cj) − c(G2) u (pega(G )) = j∈G2 i2 i2 2 |G2| E. Usage Based Cost-Sharing P ( cj) − c(G3) > j∈G3 = u (pega(G )), (128) In general, usage based cost-sharing can induce cyclic pref- i2 i2 3 |G3| erence, and hence, possibly the absence of a stable coalition . structure. However, we can show the existence of a stable . coalition structure in some special cases. P ( cj) − c(Gs) u (pega(G )) = j∈Gs is is s 1) Pass Sharing: For any K ≥ 2, we can show that there |Gs| P always exists a stable coalition structure in the pass sharing ( cj) − c(G1) > j∈G1 = u (pega(G )) (129) problem, by ruling out any cyclic preference. Without loss of is is 1 |G1| generality, we assume the average cost rate is 1 (i.e., cf = 1). 17

If participants i ∈ G share a pass r, then i’s payment is given by 1 pub(G) = |T | + |T \(∪ T )| (134) i i |G| r j∈G j If i prefers to share in coalition G with pass r rather than on 0 0 ub ub 0 G with pass r , then pi (G) < pi (G ), namely, 1 1 |T \(∪ T )| < |T 0 \(∪ 0 T )|. (135) |G| r j∈G j |G0| r j∈G j

If there exists a cyclic preference defined by (i1, ..., is), (G1, ..., Gs) and (r1, ..., rs), then 1 1 |Tr1 \(∪j∈G1 Tj)| < |Tr2 \(∪j∈G2 Tj)| |G1| |G2| Fig. 4: An illustration for taxi-ride sharing problem with no < ··· stable coalition structure considering usage based cost-sharing. 1 1 < |Trs \(∪j∈Gs Tj)| < |Tr\(∪j∈G1 Tj)|. (136) |Gs| |G1| This generates a contradiction. Hence, there always exists a F. NP-Hardness stable coalition structure. This section studies the hardness of solving K- MINCOALITION. 2) Hotel Room Sharing: When K = 2, we can show that there always exists a stable coalition structure, by ruling out Theorem 12. K-MINCOALITION is NP-hard for K ≥ 3. out in any cyclic preference. Let τi = ti − ti be the interval Proof. First, define the set of coalitions with at most size K length required by participant i, and τ be the length of the N i,j by NK , {S ∈ 2 : |S| ≤ K}. overlapped interval, if participants i, j share a room. Without K-MINCOALITION can be reduced from NP-hard problem loss of generality, we assume the room rate is 1. Then, i’s (EXACT-COVER-BY-K-SETS), defined as follows. Given a payment is given by collection G of subsets of N , {1, ..., n}, each of size K, find a pairwise-disjoint sub-collection G0 that covers N , that ub 1 1 p ({i, j}) = (τi − τi,j) + τi,j = τi − τi,j (137) 0 0 0 S i 2 2 is, S ∩ S = ∅ for all distinct S, S ∈ G and S∈G0 = N . Given an instance G of EXACT-COVER-BY-K-SETS, we ub If i prefers to share with j rather than k, then pi ({i, j}) < construct an instance of K-MINCOALITION as follows. For ub pi ({i, k}), namely, τi,j > τi,k. If there exists a cyclic S ∈ NK , we define c(S) as follows: preference (i1, ..., is), then  1 if S ∈ G,  τi1,is > τi1,i2 > ... > τis−1,is > τi1,is (138) c(S) = 1 if S ∈ NK \G, |S| ≤ K − 1, (140)  2 if S ∈ N \G, |S| = K This generates a contradiction. Hence, there always exists a K stable coalition structure. One can check that (C1) and (C2) are satisfied. Now consider a feasible solution to K-MINCOALITION and assume it consists 3) Taxi-ride Sharing: There exists an instance with no 0 of ni sets of size i not from G, for i = 1, ..., K, and nK stable coalition structure even for K = 2, as illustrated in sets of size K from G. Then the total cost of the solution is Fig. 4. Participant ik can share a ride with participant ik−1 PK−1 0 PK−1 0 i=1 ni +nK +2nK . Subject to i=1 i·ni +K ·nK +K · or participant ik+1 (whereas participant is can share with 0 nK = n, this cost is uniquely minimized when nK = n/K, i or participant i ). Let the cost from vs to vd be s−1 1 is ik−1 that is, when there is a disjoint collection from G covering c(vs , vd ) c(vs , vd ) k is ik−1 . Assume that is ik−1 is identical for all , so N . Indeed, the minimum of linear relaxation of this integer c(vs , vs ) c(vd , vd ) c(vs , vd ) k are is ik+1 , is ik+1 and ik ik for all . Also, programming problem is determined by the minimum ratio we assume that test: s d 1 s d d d 1 2 1 1 c(vi , vi ) > c(vi , vi ) + c(vi , vi ) min min , , = (141) k k 2 s k−1 k−1 k i∈{1,...,K−1} i K K K s s 1 s d >c(v , v ) + c(v , v ) (139) On the other hand, if the answer to the instance G of EXACT- ik ik+1 2 ik+1 ik COVER-BY-K-SETS is NO, then this unique minimum cannot Hence, participant ik prefers to share with participant ik+1, be achieved by an integral solution, yielding a solution of cost n rather than with participant ik−1. This generates a cyclic strictly larger than K . preference (i1, ..., is). If there are odd number of participants arranged in a loop, then this can give no stable coalition structure. We remark that Eqn. (139) can be attained, when s is sufficiently large.