Abstract
We consider a multisided assignment game with the following characteristics: (a) the agents are organized in m sectors that are connected by a graph that induces a weighted mpartite graph on the set of agents, (b) a basic coalition is formed by agents from different connected sectors, and (c) the worth of a basic coalition is the addition of the weights of all its pairs that belong to connected sectors. We provide a sufficient condition on the weights to guarantee balancedness of the related multisided assignment game. Moreover, when the graph on the sectors is cyclefree, we prove the game is strongly balanced and the core is fully described by means of the cores of the underlying twosided assignment games associated with the edges of this graph. As a consequence, the complexity of the computation of an optimal matching is reduced and existence of optimal core allocations for each sector of the market is guaranteed.
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1 Introduction
Twosided assignment games (Shapley and Shubik 1972) have been generalized to the multisided case. In this case, agents are distributed in m disjoint sectors. Usually it is assumed that these agents are linked by a hypergraph defined by the (basic) coalitions formed by exactly one agent from each sector (see for instance Kaneko and Wooders 1982; Quint 1991). A matching for a coalition S is a partition of the set of agents of S in basic coalitions and, since each basic coalition has a value attached, the worth of an arbitrary coalition of agents is obtained by maximizing, over all possible matchings, the addition of values of basic coalitions in a matching. When there are at least three sectors, the problem of finding an optimal matching in this multisided assignment market is known to be NPhard. See for instance Burkhard et al. (2009) for a survey on some special cases, together with its applications, algorithms, and asymptotic behavior.
If we do not require that each basic coalition has exactly one agent of each side but allow for coalitions of smaller size, as long as they do not contain two agents from the same sector, we obtain a larger class of games, see Atay et al. (2016) for the threesided case. But in both cases, the classical multisided assignment market and this enlarged model, the core of the corresponding coalitional game may be empty, and this is the main difference with the twosided assignment game of Shapley and Shubik (1972), where the core is always nonempty.
A twosided assignment game can also be looked at in another way. There is an underlying bipartite (weighted) graph, where the set of nodes corresponds to the set of agents and the weight of an edge is the value of the basic coalition formed by its adjacent nodes. From this point of view, the generalization to a market with \(m>2\) sectors can be defined by a weighted mpartite graph G. In an mpartite graph the set of nodes N is partitioned in m sets \(N_1,N_2,\ldots ,N_m\) in such a way that two nodes in a same set of the partition are never connected by an edge. Each node in G corresponds to an agent of our market and each set \(N_i\), for \(i\in \{1,2,\ldots ,m\}\), to a different sector. We do not assume that the graph is complete but we do assume that the subgraph determined by any two sectors \(N_i\) and \(N_j\), with \(i\ne j\), is either empty or complete. Because of that, the graph G determines a quotient graph \(\overline{G}\), the nodes of which are the sectors and two sectors are connected in \(\overline{G}\) whenever their corresponding subgraph in G is nonempty.
For each pair of sectors \(N_r\) and \(N_s\), \(r\ne s\), that are connected in \(\overline{G}\), we have a bilateral assignment market with valuation matrix \(A^{\{r,s\}}\). For each \(i\in N_r\) and \(j\in N_s\), entry \(a^{\{r,s\}}_{ij}\) is the weight in G of the edge \(\{i,j\}\), and represents the value created by the cooperation of i and j.
Given the mpartite graph G, a coalition of agents in N is basic if it does not contain two agents from the same sector and its members are connected in G. Then, the worth of a basic coalition is the addition of the weights of the edges in G that are determined by nodes in the coalition. An optimal matching in this market is a partition of N in basic coalitions such that the sum of values is maximum among all possible such partitions.
We show that if there exists an optimal matching for the multisided mpartite market that induces an optimal matching in each bilateral market determined by the connected sectors, then the core of the multisided market is nonempty. Moreover, some core elements can be obtained by merging of one core element from each of the underlying bilateral markets associated with the connected sectors.
Secondly, if the quotient graph \(\overline{G}\) is cyclefree, then the above sufficient condition for a nonempty core always holds and, moreover, the core of the multisided assignment game is fully described by “merging” or “composition” of the cores of the underlying bilateral games. A first consequence is that when \(\overline{G}\) is cyclefree, an optimal matching can be found in polynomial time. Secondly, for each sector there exists a core allocation where all agents in the sector simultaneously get their maximum core payoff. This means that, although agents in a same sector compete for the best partners in the other sectors, there is still some coincidence of interests among them.
This model of multisided assignment market on an mpartite graph G where the quotient graph \(\overline{G}\) is cyclefree can be related to the locallyadditive multisided assignment games of Stuart (1997), where the sectors are organized on a chain and the worth of a basic coalition is also the addition of the worths of pairs of consecutive sectors. However, in Stuart’s model all coalitions of size smaller than m have null worth. It can also be related with a model in Quint (1991) in which a value is attached to each pair of agents of different sectors and then the worth of an mtuple is the addition of the values of its pairs. Again, the difference with our model is that in Quint (1991) the worth of smaller coalitions is zero. In particular, the worth of a twoplayer coalition is taken to be zero instead of the value of this pair. Notice that in these models the cooperation of one agent from each side is needed to generate some profit. Compared to that, in our model, any set of connected agents from different sectors yields some worth that can be shared.
The assumption that one agent from each sector is needed to make any profit makes sense for instance in a supply chain network where some agents supply basic inputs for the industry, other agents purchase the final outputs and the rest are intermediaries who get their inputs from some agents in the industry, convert them into outputs at a cost and sell the outputs to some other agents (Ostrovsky 2008). In this setting, agents in excess in the large sectors of the market may not be able to find partners to complete a connected coalition between the suppliers of basic inputs and the final consumers, and hence get no reward in this market (see an example in Sect. 6). But there are other network situations in which the activity an agent carries out with one neighbour is independent of the activity this agent implements with other neighbours. Take as an example the network of European countries for road merchandise transport. A transport company can make a profit by its own by carrying goods to a neighbour country, but if it makes an agreement with a similar company in this second country, they both can reduce costs and hence make a larger profit, even if they are not part of a larger coalition that covers all the continent.
For arbitrary coalitional games, cooperation restricted by communication graphs was introduced by Myerson (1977) and some examples of more recent studies are Granot and Granot (1992), van Velzen et al. (2008), Grabisch and Skoda (2012), Grabisch (2013), and Khmelnitskaya and Talman (2014). The difference with our work is that in the multisided assignment game on an mpartite graph there exist wellstructured subgames, the twosided markets between connected sectors, that provide valuable information about the multisided market. This fact allows to find simple conditions for nonemptiness of the core, compared to other games defined on graphs (see for instance Deng et al. 1999).
Section 2 introduces the model. In Sect. 3, for an arbitrary mpartite graph, we provide a sufficient condition for the nonemptiness of the core. Section 4 focuses on the case in which the quotient graph is cyclefree. In that case, we completely characterize the nonempty core in terms of the cores of the twosided markets between connected sectors. From that fact, additional consequences on some particular core elements are derived in Sect. 5. Finally, Sect. 6 concludes with some remarks.
2 The multisided assignment problem on an mpartite graph and its related coalitional game
Let N be the finite set of agents in a market situation. The set N is partitioned in m sets \(N_1, N_2, \ldots , N_m\), each sector maybe representing a set of agents with a specific role in the market. There is a graph \(\overline{G}\) with set of nodes \(\{N_1,N_2,\ldots ,N_m\}\), that we simply denote \(\{1,2,\ldots ,m\}\) when no confusion arises, and we will identify the graph with its set of edges.^{Footnote 1} The graph \(\overline{G}\) induces another graph G on the set of agents N such that \(\{i,j\}\in G\) if and only if there exist \(r,s\in \{1,2,\ldots ,m\}\) such that \(r\ne s\), \(i\in N_r\), \(j\in N_s\) and \(\{r,s\}\in \overline{G}\). Notice that the graph G is an mpartite graph, that meaning that two agents on the same sector are not connected in G. We say that graph \(\overline{G}\) is the quotient graph of G.^{Footnote 2}
For any pair of connected sectors \(\{r,s\}\in \overline{G}\), there is a nonnegative valuation matrix \(A^{\{r,s\}}\) and for all \(i\in N_r\) and \(j\in N_s\), \(v(\{i,j\})=a_{ij}^{\{r,s\}}\) represents the value obtained by the cooperation of agents i and j. Notice that these valuation matrices, \(A=\{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}}\), determine a system of weights on the graph G, and for each pair of connected sectors \(\{r,s\}\in \overline{G}\), \((N_r, N_s, A^{\{r,s\}})\) defines a bilateral assignment market. Sometimes, to simplify notation, we will write \(A^{rs}\), with \(r<s\), instead of \(A^{\{r,s\}}\).
Then, \(\gamma =(N_1,N_2,\ldots ,N_m; G; \{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}})\) is a multisided assignment market on an mpartite graph. When necessary, we will write \(G^A\) to denote the weighted graph with the nodes and edges of G and the weights defined by the matrices \(\{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}}\). Given any such market \(\gamma \), a coalition \(S\subseteq N\) defines a submarket \(\gamma _{S}=(S\cap N_1,\ldots , S\cap N_m; G_{S}; A_{S})\) where \(G_{S}\) is the subgraph of G defined by the nodes in S and \(A_{S}\) consists of the values of A that correspond to edges \(\{i,j\}\) in the subgraph \(G_{S}\).
We now introduce a coalitional game related to the above market situation. To this end, we first define the worth of some coalitions that we name basic coalitions and then the worth of arbitrary coalitions will be obtained by just imposing superadditivity. A basic coalition E is a subset of agents belonging to sectors that are connected in the quotient graph \(\overline{G}\) and with no two agents of the same sector. That is, \(E=\{i_1,i_2,\ldots ,i_k\}\subseteq N\) is a basic coalition if \((i_1,i_2,\ldots ,i_k)\in N_{l_1}\times N_{l_2}\times \cdots \times N_{l_k}\) and the sectors \(\{l_1,l_2,\ldots ,l_k\}\) are all different and connected in \(\overline{G}\). Sometimes we will identify the basic coalition \(E=\{i_1,i_2,\ldots ,i_k\}\) with the ktuple \((i_1,i_2,\ldots ,i_k)\). To simplify notation, we denote by \(\mathcal {B}^N\) the set of basic coalitions of market \(\gamma \), though we should write \(\mathcal {B}^{N_1,\ldots ,N_m}\), since which coalitions are basic depends heavily on the partition in sectors of the set of agents. Notice that all edges of G belong to \(\mathcal {B}^N\). Moreover, if \(S\subseteq N\), we denote by \(\mathcal {B}^S\) the set of basic coalitions that have all their agents in S: \(\mathcal {B}^S=\{E\in \mathcal {B}^N\mid E\subseteq S\}\).
The valuation function, until now defined on the edges of G, is extended to all basic coalitions by additivity: the value of a basic coalition \(E\in \mathcal {B}^N\) is the addition of the weights of all edges in G with adjacent nodes in E. For all \(E\in \mathcal {B}^N\),
A matching\(\mu \) for the market \(\gamma \) is a partition of \(N=N_1\cup N_2\cup \cdots \cup N_m\) in basic coalitions in \(\mathcal {B}^N\). We denote by \(\mathcal {M}(N_1,N_2,\ldots ,N_m)\) the set of all matchings. Similarly, a matching for a submarket \(\gamma _{S}\) with \(S\subseteq N\) is a partition of S in basic coalitions in \(\mathcal {B}^S\).
A matching \(\mu \in \mathcal {M}(N_1,N_2,\ldots ,N_m)\) is an optimal matching for the market \(\gamma =(N_1,N_2,\ldots ,N_m; G; \{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}})\) if it holds \(\sum _{T\in \mu }v(T)\ge \sum _{T\in \mu '}v(T)\) for all other matching \(\mu '\in \mathcal {M}(N_1,N_2,\ldots , N_m)\). We denote by \(\mathcal {M}_{\gamma }(N_1,N_2,\ldots ,N_m)\) the set of optimal matchings for market \(\gamma \).
Then, the multisided assignment game associated with the market \(\gamma \) is the pair \((N, w_{\gamma })\), where the worth of an arbitrary coalition \(S\subseteq N\) is the addition of the values of the basic coalitions in an optimal matching for this coalition S:
with \(w_{\gamma }(\emptyset )=0\). Notice that if \(S\subseteq N\) is a basic coalition, \(w_{\gamma }(S)=v(S)\), since no partition of S in smaller basic coalitions can yield a higher value, because of its definition (1) and the nonnegativity of weights.^{Footnote 3} Trivially, the game \((N,w_{\gamma })\) is superadditive since it is a special type of partitioning game as introduced by Kaneko and Wooders (1982).
Multisided assignment games on mpartite graphs combine the idea of cooperation structures based on graphs (Myerson 1977) and also the notion of (multisided) matching that only allows for at most one agent of each sector in a basic coalition. It is clear that for \(m=2\), multisided assignment games on bipartite graphs coincide with the classical Shapley and Shubik (1972) assignment games. Notice also that for \(m=3\), multisided assignment games on 3partite graphs are a particular case of the generalized threesided assignment games in Atay et al. (2016), with the constraint that the value of a threeperson coalition is the addition of the values of all its pairs.
As for the related quotient graphs, for \(m=2\) the quotient graph \(\overline{G}\) consists of only one edge while, for \(m=3\), \(\overline{G}\) can be either a complete graph^{Footnote 4} or a chain. Figure 1 illustrates both the graph G and its quotient graph \(\overline{G}\) for the cases \(m=2\) and \(m=3\).
As in any coalitional game, the aim is to allocate the worth of the grand coalition in such a way that it preserves the cooperation among the agents. Given a multisided assignment market on an mpartite graph \(\gamma =(N_1,N_2,\ldots ,N_m; G; \{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}})\), a vector \(x\in \mathbb {R}^N\), where \(N=N_1\cup N_2\cup \cdots \cup N_m\), is a payoff vector. An imputation is a payoff vector \(x\in \mathbb {R}^N\) that is efficient, \(\sum _{i\in N}x_i=w_{\gamma }(N)\), and individually rational, \(x_i\ge w_{\gamma }(\{i\})=0\) for all \(i\in N\). Then, the core\(C(w_{\gamma })\) is the set of imputations that no coalition can object to, that is \(\sum _{i\in S}x_i\ge w_{\gamma }(S)\) for all \(S\subseteq N\). Because of the definition of the characteristic function \(w_{\gamma }\) in (2), given any optimal matching \(\mu \in \mathcal {M}_{\gamma }(N_1,\ldots ,N_m)\), the core is described by
A multisided assignment game on an mpartite graph is balanced if it has a nonempty core. Moreover, and following Le Breton et al. (1992), we will say an mpartite graph \((N_1,N_2,\ldots , N_m; G)\) is strongly balanced if for any set of nonnegative weights \(\{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}}\) the resulting multisided assignment game is balanced. Recall from Shapley and Shubik (1972) that bipartite graphs are strongly balanced. Our aim is to study whether this property extends to mpartite graphs or whether balancedness depends on properties of the weights or the structure of the graph.
3 Balancedness conditions
The first question above is easily answered. For \(m\ge 3\), mpartite graphs are not strongly balanced. Take for instance a market with three agents on each sector. Sectors are connected by a complete graph: \(N_1=\{1,2,3\}\), \(N_2=\{1',2',3'\}\), \(N_3=\{1'',2'',3''\}\), and \(\overline{G}=\{(N_1,N_2), (N_1,N_3), (N_2,N_3)\}\). From Le Breton et al. (1992) we know that a graph is strongly balanced if any balanced collection^{Footnote 5} formed by basic coalitions contains a partition. In our example, the collection
is balanced (notice each agent belongs to exactly two coalitions in \(\mathcal {C}\)) but we cannot extract any partition. To better understand what causes the core to be empty we complete the above 3partite graph with a system of weights and analyse some core constraints.
Example 1
Let us consider the following valuations on the complete 3partite graph with three agents in each sector:
In boldface we show the optimal matching for each twosided assignment market. Now, applying (1), the reader can obtain the worth of all threeplayer basic coalitions and check that the optimal matching of the threesided market is
Notice that \(v(\{2,1',1''\})=9+0+0=9\), \(v(\{1,3',2''\})=0+5+0=5\) and \(v(\{3,2',3''\})=0+4+6=10\).
Take \(x=(u,v,w)\in \mathbb {R}^{N_1}\times \mathbb {R}^{N_2}\times \mathbb {R}^{N_3}\). If \(x=(u,v,w)\in C(w_{\gamma })\), from core constraints \(u_2+v_1+w_1=9\) and \(u_2+v_1\ge 9\) we obtain \(w_1=0\). Then, from \(v_3+w_1\ge 2\) we deduce \(v_3\ge 2\). Hence, \(u_1+v_3+w_2=5\) implies \(u_1+w_2\le 3\), which contradicts the core constraint \(u_1+w_2\ge 5\). Therefore, \(C(w_{\gamma })=\emptyset \).
We observe that the optimal matching \(\mu \) in the above example induces a matching \(\mu ^{23}=\{(1', 1''), (3', 2''), (2', 3'')\}\) for the market \((N_{2}, N_{3}, A^{\{2,3\}})\) which is not optimal. Let us relate more formally the matchings in a multisided assignment market on an mpartite graph with the matchings of the twosided markets associated with the edges of the quotient graph.
Definition 1
Given \(\gamma =(N_1,N_2,\ldots , N_m; G; \{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}})\), for each matching \(\mu \in \mathcal {M}(N_1,\ldots ,N_m)\) and each pair adjacent sector \(\{r,s\}\in \overline{G}\), we define a matching \(\mu ^{\{r,s\}}\in \mathcal {M}(N_r, N_s)\) by
We then say that \(\mu \) is the composition of \(\mu ^{\{r,s\}}\) for \(\{r,s\}\in \overline{G}\) and write
Conversely, given a set of matchings, one for each underlying twosided market, there may not exist a matching \(\mu \) of the multisided assignment market that is the composition of that given set of matchings. Take for instance matchings \(\mu ^{\{1,2\}}=\{(2,1'), (1,3'), (3,2')\}\), \(\mu ^{\{1,3\}}=\{(1,2''), (2,1''), (3,3'')\}\) and \(\mu ^{\{2,3\}}=\{(1', 2''), (2', 3''), (3', 1'')\}\) in Example 1. Since \((1',2'')\in \mu ^{\{2,3\}}\), \((2,1')\in \mu ^{\{1,2\}}\) and \((1,2'')\in \mu ^{\{1,3\}}\), there is no matching \(\mu =\mu ^{\{1,2\}}\oplus \mu ^{\{1,3\}}\oplus \mu ^{\{2,3\}}\) since both 1 and 2 should be in the same coalition of partition \(\mu \).
Next proposition states that whenever the composition of optimal matchings of the underlying twosided markets results in a matching of the multisided market on an mpartite graph, then that matching is optimal and the core of the multisided assignment market is nonempty. To show this second part we need to combine payoff vectors of each underlying twosided market \((N_r,N_s,A^{\{r,s\}})\), with \(\{r,s\}\in \overline{G}\), to produce a payoff vector \(x\in \mathbb {R}^N\) for the multisided market \(\gamma \). We write \(C(w_{A^{\{r,s\}}})\) to denote the core of these twosided assignment games.
Definition 2
Given \(\gamma =(N_1,N_2,\ldots , N_m; G; \{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}})\), let \(x^{\{r,s\}}\in \mathbb {R}^{N_r}\times \mathbb {R}^{N_s}\) for all \(\{r,s\}\in \overline{G}\). Then,
We then say that the payoff vector \(x=\bigoplus _{\{r,s\}\in \overline{G}}x^{\{r,s\}}\in \mathbb {R}^N\) is the composition of the payoff vectors \(x^{\{r,s\}}\in \mathbb {R}^{N_r}\times \mathbb {R}^{N_s}\). Similarly, we denote the set of payoff vectors in \(\mathbb {R}^N\) that result from the composition of core elements of the underlying twosided assignment markets by \(\bigoplus _{\{r,s\}\in \overline{G}}C(w_{A^{\{r,s\}}})\).
Proposition 1
Let \(\gamma =(N_1,N_2,\ldots , N_m; G; \{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}})\) be a multisided assignment market on an mpartite graph. If there exists \(\mu \in \mathcal {M}(N_1,\ldots ,N_m)\) such that \(\mu ^{\{r,s\}}\) is an optimal matching of \((N_r,N_s,A^{\{r,s\}})\) for all \(\{r,s\}\in \overline{G}\), then

1.
\(\mu \) is optimal for \(\gamma \) and

2.
\(\gamma \) is balanced and moreover \(\bigoplus _{\{r,s\}\in \overline{G}}C(w_{A^{\{r,s\}}})\subseteq C(w_{\gamma }).\)
Proof
To see that \(\mu =\bigoplus _{\{r,s\}\in \overline{G}}\mu ^{\{r,s\}}\) is optimal for \(\gamma \), take any other matching \(\tilde{\mu }\in \mathcal {M}(N_1,\ldots ,N_m)\) and let \(\tilde{\mu }^{\{r,s\}}\in \mathcal {M}(N_r,N_s)\), for \(\{r,s\}\in \overline{G}\), be the matching \(\tilde{\mu }\) induces in each underlying twosided market. Then, \(\tilde{\mu }=\bigoplus _{\{r,s\}\in \overline{G}}\tilde{\mu }^{\{r,s\}}\). Now, applying (1),
where the inequality follows from the assumption on the optimality of \(\mu ^{\{r,s\}}\) in each market \((N_r,N_s,A^{\{r,s\}})\), for \(\{r,s\}\in \overline{G}\). Hence, \(\mu \) is optimal for the multisided market \(\gamma \).
Take now, for each \(\{r,s\}\in \overline{G}\), \(x^{\{r,s\}}\in C(w_{A^{\{r,s\}}})\). Define the payoff vector \(x\in \mathbb {R}^N\) as in Definition 2, \(x_i=\sum _{\{r,s\}\in \overline{G}}x_i^{\{r,s\}}, \text{ for } \text{ all } i\in N_r,\,r\in \{1,2,\ldots ,m\}\). We will see that \(x\in C(w_{\gamma })\). Given any basic coalition \(E\in \mathcal {B}^N\),
where both inequalities follow from \(x^{\{r,s\}}\in C(w_{A^{\{r,s\}}})\) for all \(\{r,s\}\in \overline{G}\). Notice also that if \(E\in \mu \) the above inequalities cannot be strict and hence \(\sum _{i\in E}x_i=v(E)\). Indeed, if \(i\in E\cap N_r\), \(\{r,s\}\in \overline{G}\) and \(E\cap N_s=\emptyset \), then i is unmatched by \(\mu ^{\{r,s\}}\) and, because of the optimality of \(\mu ^{\{r,s\}}\), \(x_i^{\{r,s\}}=0\). Similarly, if \(i\in E\cap N_r\) and \(j\in E\cap N_s\), then \(\{i,j\}\in \mu ^{\{r,s\}}\) and hence \(x_i^{\{r,s\}}+x_j^{\{r,s\}}=v(\{i,j\})\). \(\square \)
The above proposition gives a sufficient condition for optimality of a matching and for balancedness of a multisided assignment game on an mpartite graph. However, this condition is not necessary. The matching \(\mu \) in Example 1 is optimal while \(\mu ^{\{2,3\}}\) is not. The core of the market in Example 1 is empty, but one can find similar examples with nonempty core (see Example 5).
Finally, even under the assumption of the proposition, that is, when the composition of optimal matchings of the twosided markets leads to a matching of the multisided market, the core may contain more elements than those produced by the composition of the cores of \((N_r,N_s,A^{\{r,s\}})\), for \(\{r,s\}\in \overline{G}\) (see Atay et al. 2016 for an example in the threesided case).
In the following section we see that the inclusion \(\bigoplus _{\{r,s\}\in \overline{G}}C(w_{A^{\{r,s\}}})\subseteq C(w_{\gamma })\) becomes an equality for some particular graphs.
4 When \(\overline{G}\) is cyclefree: strong balancedness
In this section we assume that the quotient graph \(\overline{G}\) of the mpartite graph G does not contain cycles. We will assume without loss of generality that it is connected, since the results in that case are easily extended to the case of a finite union of disjoint cyclefree graphs.
We select a node of \(\overline{G}\) as a source, that is, we select a spanning tree of \(\overline{G}\). Define the distance \(d=d(1,r)\) of any other node r as the number of edges in the unique path that connects this node to the source. Then, without loss of generality, we rename the nodes of \(\overline{G}\) in such a way that the source has label 1 and, given two other nodes r and s, if \(d(1,r)<d(1,s)\) then \(r<s\). Notice that the labels of nodes at the same distance to the source are assigned arbitrarily.
A partial order is defined on the set of nodes of a tree in the following way: given two nodes r and s, we say that sfollowsr, and write \(s\succeq r\), if given the unique path in the tree that connects s to the source, \(\{s_{1}=1,s_2,\ldots , s_q=s\}\), it holds \(r=s_p\) for some \(p\in \{1,\ldots ,q1\}\). If \(r=s_{q1}\) we say that s is an immediate follower of r. We denote by \(\mathcal {S}^{\overline{G}}_{r}\) the set of followers of \(r\in \{1,2,\ldots ,m\}\), we write \(\hat{\mathcal {S}}_r^{\overline{G}}=\{r\}\cup \mathcal {S}^{\overline{G}}_{r}\) when we need to include sector r, and we denote by \(\mathcal {I}^{\overline{G}}_r\) the set of immediate followers of \(r\in \{1,2,\ldots ,m\}\).
Our main result states that an mpartite graph G where the quotient graph \(\overline{G}\) is a tree is strongly balanced.
Theorem 1
Let \(\gamma =(N_1,N_2,\ldots , N_m; G; \{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}})\) be a multisided assignment market on an mpartite graph. If \(\overline{G}\) is cyclefree, then \((N,w_{\gamma })\) is balanced and
Proof
Notice first that when \(\overline{G}\) is a tree, there is a matching \(\mu \in \mathcal {M}(N_{1},\ldots ,N_{m})\) that is the composition of optimal matchings \(\mu ^{\{r,s\}}\) of each underlying twosided market \((N_r,N_s, A^{\{r,s\}})\), for \(\{r,s\}\in \overline{G}\). To see that, we define a binary relation on the set of agents \(N=N_1\cup N_2\cup \cdots \cup N_m\). Two agents \(i\in N_r\) and \(j\in N_s\), with \(r\le s\), are related if either \(i=j\) or there exist sectors \(\{r=s_1, s_2,\ldots ,s_t=s\}\subseteq \{1,2,\ldots ,m\}\) and agents \(i_k\in N_{s_k}\) for \(k\in \{1,2,\ldots ,t\}\) such that \(\{s_k,s_{k+1}\}\in \overline{G}\) and \(\{i_k,i_{k+1}\}\in \mu ^{\{s_k,s_{k+1}\}}\), for all \(k\in \{1,2,\ldots ,t1\}\). This is an equivalence relation and, because \(\overline{G}\) is a tree, in each equivalence class there are no two agents of the same sector. Hence, the set \(\mu \) of all equivalence classes is a matching and by its definition it is the composition of the matchings \(\mu ^{\{r,s\}}\) of the twosided markets: \(\mu =\bigoplus _{\{r,s\}\in \overline{G}}\mu ^{\{r,s\}}\). Now, by Proposition 1, \(\mu \) is an optimal matching for the multisided market \(\gamma \) and \(\bigoplus _{\{r,s\}\in \overline{G}}C(w_{A^{\{r,s\}}})\subseteq C(w_{\gamma })\), which guarantees balancedness. Since all twosided assignment games have a nonempty core (Shapley and Shubik 1972), the above inclusion guarantees balancedness of the multisided assignment market \(\gamma \).
We will now prove that the converse inclusion also holds.
Let \(u=(u^1,u^2,\ldots , u^m)\in C(w_{\gamma })\). We will define, for each \(\{r,s\}\in \overline{G}\), a payoff vector \((x^{\{r,s\}},y^{\{r,s\}})\in \mathbb {R}^{N_r}\times \mathbb {R}^{N_s}\). Take the optimal matching \(\mu =\bigoplus _{\{r,s\}\in \overline{G}}\mu ^{\{r,s\}}\) and \(E\in \mu \). Let us denote by \(\overline{E}=\overline{G}_{E}\) the subtree in \(\overline{G}\) determined by the sectors containing agents in E and take as the source of \(\overline{E}\) its sector \(s_1\) with the lowest label. Take any leaf^{Footnote 6}\(s_r\) of \(\overline{E}\) and let \(\{s_1,s_2\ldots ,s_q,s_{q+1},\ldots ,s_{r1},s_r\}\) be the unique path in \(\overline{E}\) connecting \(s_r\) to the source \(s_1\). Let \(s_q\) be the sector in this path with the highest label among those that have more than one immediate follower in \(\overline{E}\) (let us assume for simplicity that \(s_q\) has two immediate followers, \(s_{q+1}\) and \(s_{q'+1}\)). Figure 2 depicts such a subtree \(\overline{E}\).
For each sector \(s_t\) with \(t\in \{1,2,\ldots ,r\}\) we denote by \(i_t\) the unique agent in E that belongs to this sector. Then, we define
Iteratively, for all \(t\in \{q+1,\ldots ,r2\}\), we define
while for sector \(s_q\) we define \(x_{i_{q}}^{\{s_{q},s_{q+1}\}}=a_{i_{q}i_{q+1}}^{\{s_{q},s_{q+1}\}}y_{i_{q+1}}^{\{s_{q},s_{q+1}\}}\), and, assuming \(x_{i_q}^{\{s_q,s_{q'+1}\}}\) has been defined analogously from the branch \(\{s_{q'+1},s_{q'+2},\ldots , s_{r'1},s_{r'}\}\), we also define \(y_{i_q}^{\{s_{q1},s_q\}}=u_{i_q}^{s_q}\left( x_{i_q}^{\{s_q,s_{q+1}\}}+x_{i_q}^{\{s_q,s_{q'+1}\}}\right) \). More generally, if \(s_q\) has several immediate followers in \(\overline{E}\), then
We proceed backwards until we reach \(x_{i_1}^{\{s_1,s_l\}}\) for all \(\{s_1,s_l\}\in \overline{E}\) with \(s_1<s_l\).
In addition, if \(i\in N_r\) and for some \(\{r,s\}\in \overline{G}\), \(r<s\), i is unmatched by \(\mu ^{\{r,s\}}\), define \(x_i^{\{r,s\}}=0\). Similarly, if \(i\in N_r\) and for all \(\{s,r\}\in \overline{G}\), \(s<r\), i is unmatched by \(\mu ^{\{s,r\}}\), define \(y_i^{\{s,r\}}=0\).
We will first check that the payoff vectors \((x^{\{r,s\}},y^{\{r,s\}})\) we have defined are nonnegative for all \(\{r,s\}\in \overline{G}\). From (4) to (9) above, it follows that, for all maximal path in \(\overline{E}\) starting at \(s_1\), \(\{s_1,s_2,\ldots , s_r\}\), and all \(t\in \{ 1,2,\ldots , r1\}\), we can express \(x_{i_t}^{\{s_t,s_{t+1}\}}\) in terms of the payoffs in u to agents in following sectors in \(\overline{E}\):
where the first equality follows from (7) and the second from (9).
Hence, if \(T=\{i_t\}\cup \{i\in E\mid i\in N_r,\,r\in \hat{\mathcal {S}}^{\overline{E}}_{s_{t+1}}\}\), we have
Notice that for \(t=1\), because of efficiency of \(u\in C(w_{\gamma })\), we obtain
Equation (10), together with (9) gives, for all \(t\in \{2,\ldots ,r\}\),
where the inequality follows from the core constraint satisfied by \(u\in C(w_{\gamma })\) for coalition \(T=\{i_t\}\cup \{i\in E\mid i\in N_r, r\in \mathcal {S}^{\overline{E}}_{s_{t}}\}\), that is,
Now, again making use of (4) to (12), we express \(x_{i_t}^{\{s_t,s_{t+1}\}}\) in terms of the payoffs in u to agents in sectors that do not follow \(s_t\) in \(\overline{E}\):
where the first equality follows from (9), the second from the definition of \(x^{\{s_{t},s_{t+1}\}}_{i_{t1}}\), the last equality from (11), and \(T_l=\{i_t\}\cup \{i\in E\mid i\in N_r,\,r\in \hat{\mathcal {S}}_l^{\overline{E}}\}.\) Recursively applying the same argument (in first place to \(x_{i_{t1}}^{\{s_{t1},s_t\}}\)), we eventually obtain
with \(T'=\{i_1\}\cup \{i\in E\mid i\in N_r, r\in \mathcal {S}^{\overline{E}}_{s_{1}}\}\), T as defined above, and where the inequality also follows from \(u\in C(w_{\gamma })\).
Once proved that for all \(\{r,s\}\in \overline{G}\), \((x^{\{r,s\}},y^{\{r,s\}})\) is a nonnegative payoff vector, let us check it is in \(C(w_{A^{\{r,s\}}})\). If \(\{i,j\}\in \mu ^{\{r,s\}}\) for some \(\{r,s\}\in \overline{G}\), then i and j belong to the same basic coalition E of \(\mu \) and \(x_i^{\{r,s\}}+y_j^{\{r,s\}}=a_{ij}^{\{r,s\}}\) follows by definition from Eqs. (5) and (7).
Since by construction of \((x^{\{r,s\}},y^{\{r,s\}})\), see (5), this vector satisfies \(x^{\{r,s\}}_{i}+y^{\{r,s\}}_{j}=a_{ij}^{\{r,s\}}\) for all \(i\in N_{s}\), \(j\in N_{s}\) and \(\{r,s\}\in \overline{G}\), it only remains to prove that if \(i\in N_r\), \(j\in N_s\), with \(\{r,s\}\in \overline{G}\), \(r<s\), and \(\{i,j\}\not \in \mu ^{\{r,s\}}\), then \(x_i^{\{r,s\}}+y_j^{\{r,s\}}\ge a_{ij}^{\{r,s\}}\). Since i and j are not matched in \((N_r,N_s,A^{\{r,s\}})\), they belong to different basic coalitions in \(\mu \). Let E and \(E'\) be the basic coalitions containing i and j respectively. Let us consider a maximal path \(\{s_1,s_2,\ldots , s_t,\ldots ,s_p\}\) in \(\overline{E}\) with origin in the node in \(\overline{E}\) with the lowest label (that we will name the source of the subtree \(\overline{E}\)) and such that there exists \(t\in \{1,\ldots ,q\}\) with \(r=s_t\). We write \(i_1\in E\cap N_{s_1}\). Similarly, let \(\{s_1',s_2',\ldots ,s_l',\ldots ,s_p'\}\) be the maximal path in \(\overline{E'}\) with origin in the node in \(\overline{E'}\) with the lowest label (the source) and such that there exists \(l\in \{1,\ldots ,p\}\) with \(s=s_l'\).
Recall from (13) that \(y_{j}^{\{r,s\}}=u(R)v(R)\), where \(R=\{j\}\cup \{b\in E'\mid b\in N_k, k\in \mathcal {S}^{\overline{E'}}_{s'_{l}}\}\), and from (14) that \(x_i^{\{r,s\}}=u((T'{\setminus } T)\cup \{i\})v((T'{\setminus } T)\cup \{i\})\), where \(T=\{i\}\cup \{b\in E\mid b\in N_k, k\in \mathcal {S}^{\overline{E}}_{s_{t}}\}\) and \(T'=\{i_1\}\cup \{b\in E\mid b\in N_k, k\in \mathcal {S}^{\overline{E}}_{s_{1}}\}\). Since \(E\cap E'=\emptyset \), \((T'{\setminus } T)\cup \{i\}\) and R are also disjoint. Then,
since \(v((T'{\setminus } T)\cup \{i\}\cup R)= v((T'{\setminus } T)\cup \{i\})+v(R)+a_{ij}^{\{r,s\}}\) and \(u\in C(w_{\gamma })\). This completes the proof of \(C(w_{\gamma })=\bigoplus _{\{r,s\}\in \overline{G}}C(w_{A^{\{r,s\}}}).\)\(\square \)
A first remark on the computation of an optimal matching for multisided assignment markets is appropriate. Although the solution of the twosided assignment problem is solvable in polynomial time, the solution of its multisided extension is NPhard (see Garey and Johnson 1979). However, for a multisided assignment market on an mpartite graph where the quotient graph that connects the sectors is cyclefree, an optimal matching is computed in polynomial time. Indeed, from Theorem 1 it follows that the composition of optimal matchings of each underlying twosided market yields an optimal matching of the multisided assignment market. Since in a market with m sectors any tree connecting the sectors has \(m1\) edges, we have \(m1\) underlying twosided markets and we only need to solve \(m1\) bilateral assignment problems to build an optimal matching for the multisided market.
Now, we ask whether the class of mpartite graphs with cyclefree quotient graph is a maximal domain for strong balancedness.
A multisided market on a 3partite graph with two agents in each sector is strongly balanced, that is, it has a nonempty core given any possible system of weights, even if the quotient graph contains a cycle. To prove this we only need to check that balancedness conditions in Lucas (1995) for \(2\times 2\times 2\) assignment games are satisfied.
But if an mpartite graph has a cycle in its quotient graph and all sectors in the cycle contain at least three agents, then we are always able to find a system of weights such that the corresponding multisided assignment game has an empty core.
Corollary 1
An mpartite graph, with at least three nodes on each sector, is strongly balanced if and only if the quotient graph has no cycles.
Proof
The “if” part follows from Theorem 1. To prove the “only if” part take an mpartite graph G such that the quotient graph \(\overline{G}\) has a cycle: \((N_1,N_2,N_3,\ldots ,N_p)\). Define the weights \(a^{12}_{ij}\), \(a^{13}_{ik}\) and \(a^{23}_{jk}\), for \(i,j,k\in \{1,2,3\}\) as in Example 1, and \(a^{rs}_{pq}=0\) elsewhere. It is straightforward to see that if \(x\in C(w_{\gamma })\), where \(\gamma =(N_1,\ldots ,N_m; G; \{A^{\{r,s\}}\}_{(r,s)\in \overline{G}})\), then the restriction of x to the coalition formed by the three first agents of \(N_1\), \(N_2\) and \(N_3\) should be in the core of the game in Example 1, which is a contradiction since it has an empty core. \(\square \)
The supply chain networks in Ostrovsky (2008) constitute a more general model that can be inscribed in the theory of matching with contracts, where utility may not be fully transferable among agents. These networks are somehow directed (vertical networks): each agent needs to buy some input from a preceding agent to transform it in some kind of output that serves as an input for the activity of a following agent, until the final consumer is reached. Hence, by definition, the network contains no cycles. In a generalized model in Hatfield and Kominers (2012), the network is determined by the set of feasible bilateral contracts between agents, and cycles are allowed. Nevertheless, acyclicity is needed to guarantee existence of stable allocations. Compared to that, in our model the graph that connects the agents is undirected and may contain cycles. But the main difference is that the set of agents is partitioned in sectors and it is the abscence of cycles in the quotient graph that connects the sectors what characterizes the existence of (core) stable allocations.
5 When \(\overline{G}\) is cyclefree: optimal core allocations
In markets where agents are organized in sectors, it has been observed that agents may present conflict or coincidence of interests depending on whether they belong to the same sector or to different sectors. The first example is the twosided assignment market in Shapley and Shubik (1972). Agents are partitioned in a set of buyers and a set of sellers and although one could think that there is a competition between buyers to be matched to the best sellers it turns out that there is a core allocation where all buyers get their maximum core payoff, which shows some coincidence of interests among buyers (and the same can be said for the sellers). Moreover, there is opposition of interests between the two sectors, since in this buyersoptimal core allocation, all sellers get their minimum core payoff. In the twosided market of Shapley and Shubik (1972), the existence of the two optimal core allocations is a consequence of the lattice structure of the core. Nevertheless, Roth (1985) points out generalized bilateral markets where the same phenomena holds without an underlying lattice structure.^{Footnote 7}
The fact that, when \(\overline{G}\) is cyclefree, the core of the multisided assignment game on an mpartite graph is completely described by the cores of all underlying twosided markets allows us to deduce some properties of \(C(w_{\gamma })\) from the known properties of \(C(w_{A^{\{r,s\}}})\), with \(\{r,s\}\in \overline{G}\). One of these consequences is that, for each sector \(r\in \{1,2,\ldots ,m\}\), there is a core element \(u\in C(w_{\gamma })\) where all agents in sector r simultaneously receive their maximum core payoff, which is their marginal contribution to the grand coalition. This is one property of twosided assignment markets that does not extend to arbitrary multisided market, but it is preserved when sectors are connected by a tree and the value of basic coalitions is defined additively as in (1).
Proposition 2
Let \(\gamma =(N_1,N_2,\ldots ,N_m; G; \{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}})\) be a multisided assignment market on an mpartite graph. If \(\overline{G}\) is cyclefree, then for each sector \(k\in \{1,2,\ldots ,m\}\) there exists \(u\in C(w_{\gamma })\) such that

1.
\(u_i\) is the maximum core payoff for all \(i\in N_k\) and moreover

2.
\(u_i=w_{\gamma }(N)w_{\gamma }(N{\setminus }\{i\})\) for all \(i\in N_k\).
Proof
Let us assume without loss of generality that \(\overline{G}\) is a tree. Take any \(k\in \{1,2,\ldots ,m\}\). For all \(s\in \{1,2,\ldots ,m\}\) with \(\{k,s\}\in \overline{G}\),^{Footnote 8} take \((x^{\{k,s\}},y^{\{k,s\}})=(\overline{x}^{\{k,s\}},\underline{y}^{\{k,s\}})\) the element of \(C(w_{A^{\{k,s\}}})\) that is optimal for all agents in \(N_k\). Similarly, for all \(r\in \{1,2,\ldots ,m\}\) such that \(\{r,k\}\in \overline{G}\), take the element \((x^{\{r,k\}},y^{\{r,k\}})=(\underline{x}^{\{r,k\}},\overline{y}^{\{r,k\}})\) of \(C(w_{A^{\{r,k\}}})\) that is optimal for the agents in \(N_k\). These optimal core elements exist in any bilateral assignment market (see Shapley and Shubik 1972). Moreover, by Demange (1982) and Leonard (1983), it is known that for all \(i\in N_k\), \(\overline{x}_i^{\{k,s\}}=w_{A^{\{k,s\}}}(N_k\cup N_s)w_{A^{\{k,s\}}}(N_k\cup N_s{\setminus }\{i\})\) and \(\overline{y}_i^{\{r,k\}}=w_{A^{\{r,k\}}}(N_r\cup N_k)w_{A^{\{r,k\}}}(N_r\cup N_k{\setminus }\{i\})\). Finally, for all \(\{r,s\}\in \overline{G}\) with \(r\ne k\) and \(s\ne k\), take an arbitrary core element \((x^{\{r,s\}},y^{\{r,s\}})\in C(w_{A^{\{r,s\}}})\).
Now, if we consider the composition of the core elements defined above, we get, given \(k\in \{1,2,\ldots ,m\}\), \(\overline{u}^k=\bigoplus _{\{r,s\}\in \overline{G}}(x^{\{r,s\}},y^{\{r,s\}})\).
Then, for all \(i\in N_k\), if \(\{r,k\}\in \overline{G}\) with \(r<k\),
for all other \(u\in C(w_{\gamma })\), as a consequence of Theorem 1.
Moreover, if \(k\in \{1,2,\ldots ,m\}\) is such that there exists \(r\in \{1,2,\ldots ,m\}\) with \(\{r,k\}\in \overline{G}\) and \(r<k\), and there exists \(s\in \{1,2,\ldots ,m\}\) with \(\{k,s\}\in \overline{G}\) and \(k<s\), then
for all \(i\in N_k\).
Similarly, if k is a leaf of \(\overline{G}\), then
for the only \(r\in \{1,2,\ldots ,m\}\) such that \(\{r,k\}\in \overline{G}\) and for all \(i\in N_k\). Also, if k is the source of the tree \(\overline{G}\), then
for all \(i\in N_k\).
Then, for all \(k\in \{ 1, 2, \ldots , m\}\) we have \(\overline{u}_{i}^{k}=w_{\gamma }(N)w_{\gamma }(N{\setminus } i)\) for all \(i\in N_{k}\). \(\square \)
In multisided assignment games on an mpartite graph with \(\overline{G}\) cyclefree, unlike the case of Shapley and Shubik (1972) twosided markets, optimal core allocations for a sector k may not be unique. Indeed, from the proof of Proposition 2, the reader will realize there is a lot of freedom of choice of core allocations for those bilateral markets in which sector k is not involved.
Notice that, for each sector \(k\in \{1,2,\ldots ,m\}\) there is also a core allocation where all agents in this sector get their minimum core payoff. The proof is analogous to the one in Proposition 2: we only need to choose, for each twosided market in which sector k takes part, the kminimum core allocation, and for each twosided market not involving sector k, an arbitrary core allocation. Their composition produces a minimum core allocation for sector k in the multisided market on an mpartite graph, as long as the quotient graph is cyclefree. When \(\overline{G}\) contains cycles, optimal core allocations for all sectors may not exist (see Example 5 in the “Appendix”).
Once proved the existence of sectoroptimal core allocations for an assignment market on an mpartite graph with a cyclefree quotient graph \(\overline{G}\), the question arises whether some other extreme core points or some singlevalued solutions of the coalitional game can be obtained in the same way by composition of the corresponding solutions in the underlying twosided markets. Next proposition shows that indeed all extreme core allocations of the multisided assignment game are obtained as the composition of extreme core allocations of the underlying twosided markets.
Proposition 3
Let \(\gamma =(N_1,N_2,\ldots ,N_m;G;\{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}})\) be a multisided assignment market on an mpartite graph. If \(\overline{G}\) is cyclefree, then any extreme core allocation \(x\in Ext(C(w_{\gamma }))\) is the composition of extreme core allocations of the underlying twosided markets, \(x=\bigoplus _{\{r,s\}\in \overline{G}}x^{\{r,s\}}\), where \(x^{\{r,s\}}\in Ext(C(w_{A^{\{r,s\}}}))\) for all \(\{ r,s\}\in \overline{G}\).
Proof
From Theorem 1, it is straightforward to see that \(x\in Ext(C(w_{\gamma }))\) satisfies \(x=\bigoplus _{\{r,s\}\in \overline{G}}x^{\{r,s\}}\) for some \(x^{\{r,s\}}\in C(w_{A^{\{r,s\}}})\). Assume now that \(x^{\{r',s'\}}\not \in Ext(C(w_{A^{\{r',s'\}}}))\) for some \(\{r',s'\}\in \overline{G}\). Then, there exist two different elements, \(y^{\{r',s'\}}\) and \(z^{\{r',s'\}}\), in \(C(w_{A^{\{r',s'\}}})\) such that \(x^{\{r',s'\}}=\frac{1}{2}y^{\{r',s'\}}+\frac{1}{2}z^{\{r',s'\}}\).
We now consider two different elements in \(C(w_{\gamma })\) by composing \(\bigoplus \limits _{\begin{array}{c} \{r,s\}\in \overline{G}\\ {\{r,s\}\ne \{r',s'\}} \end{array}}x^{\{r,s\}}\) either with \(y^{\{r',s'\}}\) or \(z^{\{r',s'\}}\),
It is then straightforward to check that \(x=\frac{1}{2} x^y+\frac{1}{2} x^z\), which contradicts the assumption \(x\in Ext(C(w_{\gamma }))\). \(\square \)
However, the converse implication does not hold, that is, the composition of extreme core allocations of the underlying twosided markets provides an element in \(C(w_{\gamma })\) which may not be an extreme point (see Example 4 in the “Appendix”).
We now consider singlevalued core selections that are not extreme points but usually interior core points. As a consequence of Theorem 1, the composition \(\eta ^{\oplus }(w_{\gamma })=\oplus _{\{r,s\}\in \overline{G}}\eta (w_{A^{\{r,s\}}})\) of the nucleolus^{Footnote 9} of the twosided markets between connected sectors belongs to \(C(w_{\gamma })\). Moreover, wellknown algorithms to compute the nucleolus of a twosided assignment game (Solymosi and Raghavan 1994; Martínez de Albéniz et al. 2014) can be used to obtain \(\eta ^{\oplus }(w_{\gamma })\). However, this composition does not coincide with the nucleolus of the initial msided market \(\gamma =(N_1,N_2,\ldots ,N_m; G; \{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}})\), as Example 3 in the “Appendix” shows.
If we select the \(\tau \)value or fairdivision point^{Footnote 10} as the cooperative solution concept to distribute the profits in each bilateral market, we can propose the composition of the \(\tau \)values of all connected twosided markets, \(\tau ^{\oplus }(w_{\gamma })=\oplus _{\{r,s\}\in \overline{G}}\tau (w_{A^{\{r,s\}}})\) as an allocation of the profit of the multisided assignment market with a tree quotient graph. Because of Theorem 1, this composition belongs to \(C(w_{\gamma })\) and can be considered as a fair division solution for the msided market. However, different to the twosided case, it may not coincide with the \(\tau \)value of the initial msided market \(\gamma =(N_1,N_2,\ldots ,N_m; G; \{A^{\{r,s\}}\}_{\{r,s\}\in \overline{G}})\). In fact, the \(\tau \)value of a multisided assignment market on an mpartite graph may lie outside the core (see Example 2 in the “Appendix”), even when the quotient graph \(\overline{G}\) is cyclefree.
6 Concluding remarks
We have considered multisided markets where agents are on an mpartite graph that induces a cyclefree network among the sectors. Basic coalitions do not need to have agents from all sectors, it is enough not to have two agents from the same sector. Moreover, the worth of a basic coalition is the addition of the worths of all its pairs that are an edge of the mpartite graph.
A similar situation is considered in Stuart (1997), although restricted to the case in which the network that connects the sectors is a chain. There, the worth of a basic coalition is also defined additively, but, as in the classical multisided assignment games in Kaneko and Wooders (1982) and Quint (1991), the set of basic coalitions is smaller since it is required that a basic coalition contains exactly one agent of each side. Although the core of Stuart’s multisided game is also nonempty, it does not contain the composition of all core elements of the underlying two sided markets.
Indeed, take \(N_{1}=\{1,2,3\}\), \(N_{2}=\{1',2',3'\}\) and \(N_{3}=\{1'',2''\}\), and consider the chain \(\overline{G}=\{\{N_{1},N_{2}\},\{N_{2},N_{3}\}\}\). Assume also that \(a^{\{r,s\}}_{ij}=1\) for all \((i,j)\in N_{r}\times N_{s}\) such that \(\{N_{r},N_{s}\}\in \overline{G}\), but, unlike the model we present in this paper, only triplets may have a positive value. It is easy to see that \((0.5,0.5,0.5;0.5,0.5,0.5)\in C(w_{A^{\{1,2\}}})\) and \((0,0,0;1,1)\in C(w_{A^{\{2,3\}}})\). However,
since an optimal matching consists of two triplets and hence the unassigned agents in sectors \(N_{1}\) and \(N_{2}\) can only receive zero payoff in the core.
Our paper answers the question about what conditions suffice so that multisided markets inherit the properties of twosided markets. The answer is a cyclefree network structure with nonnegative weights.
Notes
A graph consists of a (finite) set of nodes and a set of edges, where an edge is a subset formed by two different nodes. If \(\{r,s\}\) is an edge of a given graph, we say that the nodes r and s belong to this edge or are adjacent to this edge.
Equivalently, we could introduce the model by first imposing a (weighted) mpartite graph on \(N=N_{1}\cup N_{2}\cup \ldots \cup N_{m}\) with the condition that its restriction to \(N_{r}\cup N_{s}\), for all \(r,s\in \{1,\ldots ,m\}\) and different, is either empty or a bipartite complete graph. Then, the quotient graph \(\overline{G}\) is easily defined.
If we allow for negative weights, then the valuation fuction might not be superadditive. Consider for instance \(\overline{G}=\{(N_1,N_2),(N_2,N_3)\}\); take \(S=\{1,2',2''\}\), \(T=\{1,2'\}\) and assume the weights are \(a_{12'}=6\) and \(a_{2'2''}=2\). Then, according to our definition, S is a basic coalition and \(v(S)=62=4\). But \(S=T\cup \{2''\}\) and \(v(S)=4<v(T)+v(\{2''\})=6\). Moreover, \(v(S)\ne w_{\gamma }(S)\). In this case, we should keep T and \(\{2''\}\) as basic coalitions, and obtain \(w_{\gamma }(S)\) by superadditivity. But then, which are the basic coalitions would depend on the weights, not only on the network.
A graph is complete if any two of its nodes are connected by an edge. Hence, an mpartite graph with more than one node in some of the sectors is never complete in this sense. A complete mpartite graph is an mpartite graph such that any two nodes from different sectors are connected by an edge.
Given a player set N, a collection of coalitions \(\mathcal {C}=\{S_1,S_2,\ldots , S_k\}\) with \(S_l\subseteq N\) for all \(l\in \{1,2,\ldots ,k\}\), is balanced if there exist positive numbers \(\delta _{S_l}>0\) such that, for all \(i\in N\), it holds \(\sum _{i\in S_l \subseteq \mathcal {C}}\delta _{S_l}=1\).
Given a tree, a leaf is a node with no followers.
The idea that the core models competition dates back to Edgeworth (1881) and is also explained by Shubik (1959). For a given player, the minimum core payoff can be interpreted as the amount of value guaranteed to this player due to competition and the difference between the minimum and maximum core payoffs can be interpreted as a residual bargaining problem.
Recall that, because of the labeling of the nodes at the beginning of Sect. 4, \(\{k,s\}\in \overline{G}\) implies \(k<s\).
The nucleolus of a coalitional game (N, v) is the payoff vector \(\eta (v)\in \mathbb {R}^N\) that lexicographically minimizes the vector of decreasingly ordered excesses of coalitions among all possible imputations (Schmeidler 1969). An imputation for the game (N, v) is a payoff vector \(x\in \mathbb {R}^N\) that satisfies \(\sum _{i\in N}x_i=v(N)\) and \(x_i\ge v(\{i\})\) for all \(i\in N\). The excess of coalition \(S\subseteq N\) at \(x\in \mathbb {R}^{N}\) is \(v(S)\sum _{i\in S}x_{i}\).
The fairdivision point of a twosided assignment market is the midpoint of the buyersoptimal and the sellersoptimal core allocations Thompson (1981). The \(\tau \)value is a singlevalued solution for coalitional games introduced in Tijs (1981). It is known that for twosided assignment games the \(\tau \)value and the fairdivision point coincide (Núñez and Rafels 2002).
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Acknowledgements
Open access funding provided by MTA Centre for Economic and Regional Studies (MTA KRTK). We thank to participants of CMUUPitt Theory Seminar, the 3rd Pennsylvania Economic Theory Conference, the 13th Meeting of the Society for Social Choice and Welfare, SING12, and Games 2016 for their comments and suggestions. We also are grateful to Javier Martínez de Albéniz for his helpful comments and suggestions. The authors thank an anonymous referee for his/her useful comments and suggestions. A. Atay acknowledges support from the Hungarian National Research, Development and Innovation Office via the grant PD128348, and the Hungarian Academy of Sciences via the Cooperation of Excellences Grant (KEP6/2018). M. Núñez acknowledges the support from research grant ECO201786481P (Agencia Estatal de Investigación (AEI) y Fondo Europeo de Desarrollo Regional (FEDER)) and 2017SGR778 (Generalitat de Catalunya).
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Appendix
Appendix
We first consign to this “Appendix” two examples that show that for a multisided assignment game on a cyclefree quotient graph, the composition of the \(\tau \)values (or the nucleolus) of each underlying twosided market may not coincide with the \(\tau \)value or the nucleolus of the initial multisided market. Similarly, the third example shows that by composition of arbitrary extreme core allocations of each twosided market we may not obtain an extreme core allocation of the multisided market.
Example 2
Let us consider an assignment market \(\gamma \) on a 3partite graph such that the quotient graph is \(\overline{G}=\{\{1,2\},\{2,3\}\}\) which is cyclefree. The sectors are \(N_1=\{1,2\}\), \(N_2=\{1',2'\}\), and \(N_3=\{1'',2''\}\). The valuation matrices of the two underlying twosided markets are
and the value of triplets is given by the following threedimensional matrix
The \(\tau \)value of this multisided market game is \(\tau (\gamma )=(\frac{5}{9},\frac{24}{9}; \frac{29}{9}, \frac{15}{9}; \frac{15}{9},\frac{20}{9})\) which is not in the core since \(\tau _{2}+\tau _{1'}+\tau _{2''}=\frac{24}{9}+\frac{29}{9}+\frac{20}{9}=\frac{73}{9}<9=v(\{2,1',2''\})\). Hence, \(\tau (\gamma )\) cannot coincide with \(\tau (w_{A^{\{1,2\}}})\oplus \tau (w_{A^{\{2,3\}}})\).
Example 3
Let us consider an assignment market \(\gamma \) on the following 4partite graph related to the the quotient graph \(\overline{G}=\{\{1,2\},\{2,3\},\{2,4\}\}\) which is cycle free. The sectors are \(N_1=\{1,2\}\), \(N_2=\{1',2'\}\), \(N_3=\{1'',2''\}\), \(N_4=\{1''',2'''\}\), and the valuation matrices of the twosided markets are
The nucleolus of the three underlying twosided markets are
and their composition is
while the nucleolus of the sixplayer game \((N, w_{\gamma })\) can be computed and is
Example 4
Let us consider an assignment market \(\gamma \) on a 4partite graph related to the quotient graph \(\overline{G}=\{\{1,2\},\{2,3\},\{2,4\}\}\) which is cyclefree. The sectors are \(N_{1}=\{1,2\}\), \(N_{2}=\{1',2'\}\), \(N_{3}=\{1'',2''\}\), and \(N_{4}=\{1''',2'''\}\). The valuation matrices of the three underlying twosided markets are
Take respective extreme core allocations of the three underlying twosided markets \(A^{\{1,2\}}\), \(A^{\{2,3\}}\), and \(A^{\{2,4\}}\): (2, 1; 0, 1), (2, 0; 0, 2), and (1, 0; 0, 1). Then, by composition we get a core allocation for the multisided assignment market, \(x^{\oplus }=(2,1;3,1;0,2;0,1)\in C(w_{\gamma })\). But, there exist two core elements
and
such that \(x^{\oplus }=\frac{1}{2}y+\frac{1}{2}z\). Hence, \(x^{\oplus }\notin Ext(C(w_{\gamma }))\).
The last example shows that assumptions of Proposition 1 are not necessary for the nonemptiness of the core. In this example, the core of the multisided assignment game is nonempty and the matching induced on one twosided market is not optimal. The same example shows that when \(\overline{G}\) is not cyclefree, optimal core allocations for each sector may not exist.
Example 5
Let us consider an assignment market \(\gamma \) on a complete 3partite graph G where \(M_1=\{1,2\}\), \(M_2=\{1',2'\}\) and \(M_3=\{1'',2''\}\). The valuation matrices of the three underlying twosided markets are
and the value of triplets is given by the following threedimensional matrix
Notice first that this market does not satisfy the sufficient condition in Proposition 1. Indeed the optimal matching is \(\mu =\{(1,1',1''),(2,2',2'')\}\) but \(\mu ^{\{1,2\}}=\{(1,1'), (2,2')\}\) is not optimal for \(A^{\{1,2\}}\). Nevertheless, the core is nonempty. For instance, \(x=(4,5;0,0;6,9)\) and \(y=(4,4;1,0;5,10)\) belong to \(C(w_{\gamma })\). Moreover \(x_{1''}=6\) and \(y_{2''}=10\) are respectively the marginal contributions of agents \(1''\) and \(2''\) and hence these are their maximum core payoffs. However, there is no core element where agents \(1''\) and \(2''\) simultaneously receive 6 and 10. Indeed, if \((x,y,4x,4y,6,10)\) were a core allocation, then core constraints would imply \(x+(4y)\ge 4\) and \(y+(4x)\ge 5\), which lead to the contradiction \(y\le x\le y1\). As a consequence, in this market there is no optimal core allocation for the third sector.
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Atay, A., Núñez, M. Multisided assignment games on mpartite graphs. Ann Oper Res 279, 271–290 (2019). https://doi.org/10.1007/s10479019032565
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DOI: https://doi.org/10.1007/s10479019032565