Abstract | We propose a simple and intuitive cost mechanism which assigns costs for the competitive
usage of m resources by n selfish agents. Each agent has an individual demand; demands are
drawn according to some probability distribution. The cost paid by an agent for a resource
it chooses is the total demand put on the resource divided by the number of agents who
chose that same resource. So, resources charge costs in an equitable, fair way, while each
resource makes no profit out of the agents.
We call our model the Fair Pricing model. Its fair cost mechanism induces a noncooperative
game among the agents. To evaluate the Nash equilibria of this game, we
introduce the Diffuse Price of Anarchy, as an extension of the Price of Anarchy that takes
into account the probability distribution on the demands. We prove:
² Pure Nash equilibria may not exist, unless all chosen demands are identical.
² A fully mixed Nash equilibrium exists for all possible choices of the demands. Further
on, the fully mixed Nash equilibrium is the unique Nash equilibrium in case there are
only two agents.
² In the worst-case choice of demands, the Price of Anarchy is £(n); for the special case
of two agents, the Price of Anarchy is less than 2 ¡ 1
m .
² Assume now that demands are drawn from a bounded, independent probability distribution,
where all demands are identically distributed, and each demand may not exceed
some (universal for the class) constant times its expectation. It happens that the constant
is just 2 when each demand is distributed symmetrically around its expectation.
We prove that, for asymptotically large games where the number of agents tends to
infinity, the Diffuse Price of Anarchy is at most that universal constant. This implies
the first separation between Price of Anarchy and Diffuse Price of Anarchy.
Towards the end, we consider two closely related cost sharing models, namely the Average
Cost Pricing and the Serial Cost Sharing models, inspired by Economic Theory. In contrast
to the Fair Pricing model, we prove that pure Nash equilibria do exist for both these models. |