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Algorithmic game theory studies the interaction of algorithms with strategic behavior. It asks both how hard equilibria are to compute and how mechanisms should be designed when participants choose actions to improve their own utility.

For example, a shortest-path algorithm optimizes travel time for one specified traveler, while a routing game asks what happens when many travelers each choose a route. A Nash equilibrium can have a larger total delay than a centrally chosen allocation. Quantifying that loss by a price of anarchy links incentives with computational optimization. The input representation, equilibrium concept and allowed running time must all be specified.

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.