I study games with incomplete markets where people must sink their investments before they can join a match. I focus on competitive matching markets where there is a public price to join any match. Despite the First Welfare Theorem, coordination failures can arise because of market incompleteness. But are coordination failures stable? I introduce a trembling-hand and prove that—in a general class of models with general heterogeneity of types, costs of investments, and matching surpluses—coordination failures are not stable. My main theorem is a modified First Welfare Theorem: even with endogenous and incomplete markets, every perfect, competitive equilibrium is efficient..