Coalitions in Nonatomic Network Congestion Games

Abstract

This work shows that the formation of a finite number of coalitions in a nonatomic network congestion game benefits everyone. At the equilibrium of the composite game played by coalitions and individuals, the average cost to each coalition and the individuals' common cost are all lower than in the corresponding nonatomic game (without coalitions). The individuals' cost is lower than the average cost to any coalition. Similarly, the average cost to a coalition is lower than that to any larger coalition. Whenever some members of a coalition become individuals, the individuals' payoff is increased. In the case of a unique coalition, both the average cost to the coalition and the individuals' cost are decreasing with respect to the size of the coalition. In a sequence of composite games, if a finite number of coalitions are fixed, while the size of the remaining coalitions goes to zero, the equilibria of these games converge to the equilibrium of a composite game played by the same fixed coalitions and the remaining individuals.1. At the CE, the average social cost, the individuals' cost and the average cost to each coalition are lower than the equilibrium cost at the WE of the corresponding nonatomic game.2. At the CE, the average cost to a coalition is lower than that to any other larger coalition. If a coalition sends flow on a certain arc, then any other larger coalition sends more on it. R

Routing Game on Parallel Networks: The Convergence of Atomic to Nonatomic

“…Their proof is based on the convergence of variational inequalities corresponding to the sequence of Nash equilibria, a technique similar to the one used in this paper. Wan [19] generalizes this result to composite games where nonatomic players and atomic splittable players coexist, by allowing the atomic players to replace themselves by players with heterogeneous sizes.…”

Finite composite games: Equilibria and dynamics

“…This result shows intrinsic link between the different frameworks discussed in this paper. The reader is referred to Haurie and Marcotte [10] or Wan [33] for details.…”

Delegation equilibrium payoffs in integer-splitting games

“…The same concepts can be defined in games where each player has a finitely divisible stock, even in the presence of a set of nonatomic players who hold each an infinitesimal stock, using the notion of composite equilibria [3,8]. However, the general case where each player n can divide arbitrarily her stock m n composed of finitely (or countably) many atoms m n i , i ≥ 1 and a nonatomic part m n 0 with +∞ j=0 m n j = m n deserves further study, since the definition of delegation equilibrium by backward induction is no longer available.…”

“…Several papers (for example, [1,2,4]) deal with variation of the structure of players in congestion games, in particular, comparing the social optimum cost to the cost at the Wardrop equilibrium [10]. Other approaches (for example, [8]) evaluate the "advantage of coalitions". In all such frameworks, a procedure has to be defined that allows for such a cooperative behavior.…”