\emph{Mediators} are third parties to whom the players in a game can
delegate the task of choosing a strategy; a mediator forms a
\emph{mediated equilibrium} if delegating is a best response for all
players. Mediated equilibria have more power to achieve outcomes
with high social welfare than Nash or correlated equilibria, but
less power than a fully centralized authority. Here we begin the
study of the power of mediation by using the mediation analogue of
the price of stability---the ratio of the social cost of the best
mediated equilibrium BME to that of the socially optimal outcome
OPT. We focus on load-balancing games with social cost measured by
weighted average latency. Even in this restricted class of games,
BME can range from as good as OPT to no better than the best
correlated equilibrium. In unweighted games BME achieves OPT; the
weighted case is more subtle. Our main results are (1) that the
worst-case ratio BME/OPT is at least $(1+\sqrt{2})/2\approx 1.2071$
(and at most $1+\phi\approx 2.618$ [Awerbuch Azar Epstein STOC'05])
for linear-latency weighted load-balancing games, and that the lower
bound is tight when there are two players; and (2) tight bounds on
the worst-case BME/OPT for general-latency weighted load-balancing
games. We also give similarly detailed results for other natural
social-cost functions.