= Price of anarchy
{title2=$\operatorname{PoA}$}
For a cost-minimization game, the price of anarchy is the ratio of the worst equilibrium social cost to the best feasible social cost:
$$
\operatorname{PoA}=\frac{\sup_{s\in\mathcal E}C(s)}{\inf_{s\in\mathcal S}C(s)}.
$$
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.
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