Price of Anarchy

The worst-case ratio between the social cost of a Nash equilibrium and the social optimum.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

$\mathrm{PoA}(G)=\dfrac{\max_{s\in NE(G)} \mathrm{cost}(s)}{\min_{s}\mathrm{cost}(s)}$ for cost-minimization games.

Price of Stability uses $\min_{s\in NE(G)}$ instead.

Intuition

Quantifies how much efficiency is lost due to selfish (non-cooperative) behavior.

Bounded PoA justifies decentralized design choices.

Worked example

For nonatomic selfish routing with linear latencies, PoA $=4/3$ (Pigou network is tight).

For atomic congestion games with linear costs, PoA $=5/2$ (Awerbuch-Azar-Epstein; Christodoulou-Koutsoupias).

The math

Roughgarden's smoothness framework yields robust PoA bounds across many solution concepts (mixed, correlated, coarse-correlated).

Tight bounds often stem from Pigou-like example graphs.

Where it is used

Internet routing protocols like BGP let autonomous systems choose paths selfishly. The Price of Anarchy quantifies how much slower traffic becomes compared to an optimally coordinated routing scheme, guiding network design and traffic engineering.

In electricity markets, generators bid independently to maximize profit. The Price of Anarchy measures the efficiency loss from decentralized bidding relative to a centrally planned dispatch, informing market regulation and capacity planning.

More in Game theory

Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.