Resource Buying Games

TL;DR

This paper investigates the existence and complexity of pure Nash equilibria in resource buying games, revealing that equilibrium existence depends on cost functions and underlying structures like matroids.

Contribution

It characterizes conditions under which pure Nash equilibria exist in resource buying games, especially relating to cost functions and matroid structures.

Findings

Pure Nash equilibria exist for marginally non-decreasing cost functions.

Matroids are the key structure for equilibria with marginally non-increasing costs.

Equilibrium existence varies with cost function behavior and game structure.

Abstract

In resource buying games a set of players jointly buys a subset of a finite resource set E (e.g., machines, edges, or nodes in a digraph). The cost of a resource e depends on the number (or load) of players using e, and has to be paid completely by the players before it becomes available. Each player i needs at least one set of a predefined family S_i in 2^E to be available. Thus, resource buying games can be seen as a variant of congestion games in which the load-dependent costs of the resources can be shared arbitrarily among the players. A strategy of player i in resource buying games is a tuple consisting of one of i's desired configurations S_i together with a payment vector p_i in R^E_+ indicating how much i is willing to contribute towards the purchase of the chosen resources. In this paper, we study the existence and computational complexity of pure Nash equilibria (PNE, for short) of resource buying games. In contrast to classical congestion games for which equilibria are guaranteed to exist, the existence of equilibria in resource buying games strongly depends on the underlying structure of the S_i's and the behavior of the cost functions. We show that for marginally non-increasing cost functions, matroids are exactly the right structure to consider, and that resource buying games with marginally non-decreasing cost functions always admit a PNE.