# Congestion Game with Varying Price

I molded my problem as the following game (it is a congestion game with varying price):

$N$ players share resources $E$,

$S_i$ is the strategy space of player $i$ which is in $2^E$ (where $2^E$ is the power set of resources).

$P_e^i$ is the price of resource $e \in E$ considering player $i$. The price of resource $e$ is different for different users.

The goal of each player is to select a strategy $S_i$ which minimize its price $\sum_{e\in S_i}P_e^i$ .

My questions are:

1. Does this game have any Nash Equilibrium (NE)? If so under which conditions?
2. If it has any NE, what is a sample algorithm for achieving it?

I searched literature but could not find any appropriate information! Any solution is appreciated!

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For this to be a congestion game, presumably the prices $P_e^i$ have to depend on the number of players claiming resource $e$? Also, one would usually expect a price to be non-negative; in this case the obvious optimal strategy would be the empty strategy, with minimal price zero. So there seems to be something missing here. – joriki Feb 19 '13 at 18:45
@joriki: yes it could be depend on number of users using that service and consider they need some services but the total price should be minimized. – Reza Feb 19 '13 at 19:30