Price of anarchy beyond symmetry and homogeneous costs in network games

Determine bounds or exact characterizations for the price of anarchy in the quadratic network game where each agent i chooses x_i ≥ 0 and has payoff u_i(x) = - (1/2) γ_i x_i^2 + (β_i + ∑_{j ≠ i} g_{ij} x_j) x_i, without imposing that the cost coefficients γ_i are equal across i and without assuming that the induced spillover matrix M with entries m_{ij} = g_{ij}/γ_i is symmetric. The goal is to quantify inefficiency of Nash equilibria relative to social optima under these general conditions.

Background

In the symmetric and homogeneous-cost case (all γ_i equal and M symmetric), the paper proves Proposition 1 that the price of anarchy equals ((1 - ρ(M))/(1 - 2ρ(M)))^2 under 2ρ(M) < 1, where ρ(M) is the spectral radius. This relies on diagonalizing M and comparing Nash equilibrium to the social optimum in the spectral basis.

The author highlights that these symmetry assumptions are strong and not realistic in many applications, noting the difficulty of varying γ_i while maintaining a fixed social welfare function. The open question seeks a welfare/PoA characterization when these assumptions are removed.