Pricing in Resource Allocation Games Based on Lagrangean Duality and Convexification

We consider a basic resource allocation game, where the players' strategy spaces are subsets of $R^m$ and cost/utility functions are parameterized by some common vector $u\in R^m$ and, otherwise, only depend on the own strategy choice. A strategy of a player can be interpreted as a vector of resource consumption and a joint strategy profile naturally leads to an aggregate consumption vector. Resources can be priced, that is, the game is augmented by a price vector $\lambda\in R^m_+$ and players have quasi-linear overall costs/utilities meaning that in addition to the original costs/utilities, a player needs to pay the corresponding price per consumed unit. We investigate the following question: for which aggregated consumption vectors $u$ can we find prices $\lambda$ that induce an equilibrium realizing the targeted consumption profile? For answering this question, we revisit a well-known duality-based framework and derive several characterizations of the existence of such $u$ and $\lambda$ using convexification techniques. We show that for finite strategy spaces or certain concave games, the equilibrium existence problem reduces to solving a well-structured LP. We then consider a class of monotone aggregative games having the property that the cost/utility functions of players may depend on the induced load of a strategy profile. For this class, we show a sufficient condition of enforceability based on the previous characterizations. We demonstrate that this framework can help to unify parts of four largely independent streams in the literature: tolls in transportation systems, Walrasian market equilibria, trading networks and congestion control in communication networks.