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A Simple Bargaining Procedure for the Myerson Value

Noemí Navarro EMAIL logo and Andres Perea

Abstract

We consider situations where the cooperation and negotiation possibilities between pairs of agents are given by an undirected graph. Every connected component of agents has a value, which is the total surplus the agents can generate by working together. We present a simple, sequential, bilateral bargaining procedure, in which at every stage the two agents in a link, (i,j) bargain about their share from cooperation in the connected component they are part of. We show that this procedure yields the Myerson value (Myerson, 1997) if the marginal value of any link in a connected component is increasing in the number of links in that connected component.

  1. 1

    See the book by Jackson (2008), Chapter 12, for a more formal definition.

  2. 2
  3. 3
  4. 4

    For a definition of games with potential, see Monderer and Shapley (1996).

  5. 5
  6. 6

    In the case of the literature implementing the Shapley value, the value function is defined as a function of coalitions of players. In the case of the Myerson value, the value function is mainly a function of the links in a network.

  7. 7

    Currarini and Morelli (2000) and Slikker and van den Nouweland (2001) adapt the DCG to network settings. In Currarini and Morelli (2000) players’ demands are not contingent on network structures, while Slikker and van den Nouweland (2001) consider a simultaneous version of the DCG where demands are link-contingent, i.e., players send a claim or demand per link in which they would be willing to participate.

  8. 8

    The BFS procedure was first introduced by Pérez-Castrillo and Wettstein (2001) and consists of a two-stage negotiation process in which all currently active players first simultaneously bid against all the other players for becoming a proposer. After net bids are paid and received, the proposer, who is the player with the highest net bid, makes a feasible proposal to all of the active players, who sequentially accept or reject the proposal. If they all accept, then the set of active players cooperate together and distribute the value as proposed. If one player rejects then the proposer leaves the game and the mechanism starts all over again. In this case, the set of active players coincides with the previous set of active players excluding the player whose proposal has just been rejected. An alternative mechanism that also implements the Shapley value is provided by Hart and Mas-Colell (1996)

  9. 9

    For the sake of completeness, Vidal-Puga and Bergantiños (2003) apply the BFS mechanism to implement the Owen value, which is an extension of the Shapley value to cooperative situations where players are organized in a-priori unions.

  10. 10

    In addition to these examples, van den Nouweland and Borm (1991) find, in a context of TU games with communication structures, necessary and sufficient conditions for the resulting game to be link-convex when the underlying TU game is convex.

  11. 11

    Optimal quantity for firm i in a connected component S is given by

    and therefore its profit is given by

  12. 12

    Alternatively, if we assume that there is perfect competition in each node i, then the consumer surplus in each market is a quadratic function of the number of links that each firm i has, namely . If w is the sum of all market surpluses minus an increasing, concave function C on the number of links inside a component, we would also obtain a link-convex value function.

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Published Online: 2013-5-1
Published in Print: 2013-1-1

©2013 by Walter de Gruyter Berlin / Boston

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