On the Nash dynamics of congestion games with player-specific utility

Strategic issues in multi-agent routing in communication networks can be studied by modeling the routing problem as a congestion game. Such a game has a finite number of players and a finite set of resources; each player chooses a nonempty subset of the resources and the payoff to a player depends only on the number of players using each resource. Rosenthal proved that congestion games with symmetric utilities have pure strategy Nash equilibria. This is because the Nash dynamics give rise to an acyclic digraph. An explicit potential function on pure strategy profiles for the Nash dynamics is known for such games; this helps in understanding the Nash dynamics and verifies that the corresponding digraph is acyclic. Milchtaich considered congestion games with player-specific utilities where each player selects exactly one resource. lie proved that the Nash dynamics give rise to an acyclic digraph when there are two resources. This implies the existence of a potential function on pure strategy profiles for the Nash dynamics, but no explicit potential function appears to be known as yet. In this paper we define an acyclic digraph on the integer lattice of each dimension and show that the Nash dynamics for any such congestion game can be embedded into one of these digraphs. We find an explicit potential function on each of these lattice digraphs. This potential function can then be restricted to the embedded digraph of the Nash dynamics of such a congestion game. Apart from giving an alternate proof of the acyclicity of the digraph of Nash dynamics for such games, such an explicit function helps in understanding these dynamics.