Price of anarchy ($\mathrm{PoA}$)

For a cost-minimization game, the price of anarchy is the ratio of the worst equilibrium social cost to the best feasible social cost:Here $\mathcal{E}$ is the specified equilibrium set, $\mathcal{S}$ the feasible profiles, and $C$ the chosen nonnegative social cost. The ratio requires a positive denominator; zero optimum costs and empty equilibrium sets need separate conventions. A payoff-maximization formulation uses the reciprocal orientation to keep a larger number meaning worse efficiency.

In the two-edge routing game with one unit of flow and latencies $x$ and one, the equilibrium costs one while the optimum costs $3/4$, so the price of anarchy is $4/3$. The best equilibrium instead defines the price of stability. Bounds depend on the equilibrium notion and the allowed game class; one example does not establish a universal bound for all games.