Nash equilibria with minimum potential in undirected broadcast games

In this paper, we consider undirected network design games with fair cost allocation. We introduce two concepts Potential-Optimal Price of Anarchy (POPoA) and Potential-Optimal Price of Stability (POPoS), where POPoA is the ratio between the worst cost of Nash equilibrium with optimal potential and the minimum social cost, and POPoS is the ratio between the best cost of Nash equilibrium with optimal potential and the minimum social cost, and show the following.
• The POPoA and POPoS for undirected broadcast games with n players are .
• The POPoA and POPoS for undirected broadcast games with |V| vertices are .
• There exists an undirected broadcast game with n players such that , .
• There exists an undirected broadcast game with |V| vertices such that .