Abstract
We consider a multi-sided assignment game with the following characteristics: (a) the agents are organized in m sectors that are connected by a graph that induces a weighted m-partite 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 multi-sided assignment game. Moreover, when the graph on the sectors is cycle-free, we prove the game is strongly balanced and the core is described by means of the cores of the underlying two-sided assignment games associated with the edges of this graph. Moreover, once selected a spanning tree of the cycle-free graph on the sectors, the equivalence between core and competitive equilibria is established.
Highlights
Two-sided assignment games (Shapley and Shubik 1972) have been generalized to the multisided case
We consider a multi-sided assignment game with the following characteristics: (a) the agents are organized in m sectors that are connected by a graph that induces a weighted m-partite 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 show that if there exists an optimal matching for the multi-sided m-partite market that induces an optimal matching in each bilateral market determined by the connected sectors, the core of the multi-sided market is non-empty
Summary
Two-sided assignment games (Shapley and Shubik 1972) have been generalized to the multisided case. For any pair of connected sectors {r , s} ∈ G, there is a non-negative valuation matrix A{r,s} and for all i ∈ Nr and j ∈ Ns , v({i , j }) = ai{rj ,s} represents the value obtained by the cooperation of agents i and j Notice that these valuation matrices, A = { A{r,s}}{r,s}∈G , determine a system of weights on the graph G, and for each pair of connected sectors {r , s} ∈ G, (Nr , Ns , A{r,s}) defines a bilateral assignment market. The multi-sided assignment game associated with the market γ is the pair (N , wγ ), where the worth of an arbitrary coalition S ⊆ N is the addition of the values of the basic coalitions in an optimal matching for this coalition S:. Multi-sided assignment games on m-partite graphs combine the idea of cooperation structures based on graphs (Myerson 1977) and the notion of (multi-sided) matching that only allows for at most one agent of each sector in a basic coalition. Our aim is to study whether this property extends to m-partite graphs or whether balancedness depends on properties of the weights or the structure of the graph
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