Facility location (cooperative game)


The cooperative facility location game is a cooperative game of cost sharing. The goal is to share the cost of opening new facilities between the clients enjoying these facilities.
The game has the following components:
  • There are several consumers who need a certain service, e.g, electricity connection.
  • There are several locations where facilities can be built.
  • For every pair of consumer and location, there is a fixed cost of serving C from L. This cost is denoted Cost.
  • The cost of serving a group of consumers is lower than the sum of the cost of serving each consumer alone.
EXAMPLE:
  • There are two facilities, F1 which costs 2 and F2 which costs 2.
  • There are three consumers, Alice Bob and Carl.
  • Alice can be served only from F1, with cost 2. So the cost of serving her alone is 2+2=4.
  • Bob can be served from F1 with cost 2 or from F2 with cost 1. So the cost of serving him alone is 2+1=3.
  • Carl can be served only from F2, with cost 1. So the cost of serving him alone is 2+1=3.
  • The cost of serving Alice and Bob is 2+2+2=6.
  • The cost of serving Bob and Carl is 2+1+1=4.
  • The cost of serving Alice and Carl is 2+2+2+1=7.
  • The cost of serving all agents is 2+2+2+1+1=8.
The most socially-desirable outcome of the game is that all agents are served. The cost of this outcome can be shared among the agents. A cost-allocation is good if no sub-group of agents can deviate and get a lower cost for itself. In the above example:
  • The cost-vector is not in the core, since Alice can deviate and get a cost of only 4. Similarly, the vector is not in the core since Bob and Carl can deviate together and get a total cost of only 4.
  • The cost-vectors and are in the core.
A classic result in game-theory, the Bondareva–Shapley theorem, gives necessary and sufficient conditions for a game to have nonempty core.