Cascading to Equilibrium: Hydraulic Computation of Equilibria in Resource Selection Games

Drawing intuition from a (physical) hydraulic system, we present a novel framework, constructively showing the existence of a strong Nash equilibrium in resource selection games with nonatomic players, the coincidence of strong equilibria and Nash equilibria in such games, and the invariance of the cost of each given resource across all Nash equilibria. Our proofs allow for explicit calculation of Nash equilibrium and for explicit and direct calculation of the resulting (invariant) costs of resources, and do not hinge on any fixed-point theorem, on the Minimax theorem or any equivalent result, on the existence of a potential, or on linear programming. A generalization of resource selection games, called resource selection games with I.D.-dependent weighting, is defined, and the results are extended to this family, showing that while resource costs are no longer invariant across Nash equilibria in games of this family, they are nonetheless invariant across all strong Nash equilibria, drawing a novel fundamental connection between group deviation and I.D.-congestion. A natural application of the resulting machinery to a large class of constraint-satisfaction problems is also described.