# Myerson–Satterthwaite theorem

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The Myerson–Satterthwaite theorem is an important result in mechanism design and the economics of asymmetric information, due to Roger Myerson and Mark Satterthwaite. Informally, the result says that there is no efficient way for two parties to trade a good when they each have secret and probabilistically varying valuations for it, without the risk of forcing one party to trade at a loss.

Formally, the theorem applies if a prospective buyer A has a valuation $v_A \in [x_a,y_a]$, and the prospective seller B has an independent valuation $v_B \in [x_b,y_b]$, such that the intervals $[x_a,y_a]$ and $[x_b,y_b]$ overlap, and the probability densities for the valuations are strictly positive on those intervals. Under those conditions, there is no Bayesian incentive compatible social choice function that is guaranteed in advance to produce efficient outcomes and guarantees buyers and sellers non-negative returns regardless of $v_a$ and $v_b$.

The Myerson–Satterthwaite theorem is among the most remarkable and universally applicable negative results in economics — a kind of negative mirror to the fundamental theorems of welfare economics. It is, however, much less famous than those results or Arrow's earlier result on the impossibility of satisfactory electoral systems.