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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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I think the answer is yes, there is a Nash equilibrium, and it can be attained just by letting each person in turn change his selection to the best possible alternative.

For more information, read this paper:

Weighted congestion games with separable preferences

by: Igal Milchtaich

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