Paper provided by Econometric Society in its series Econometric Society World Congress 2000 Contributed Papers with number 0704.For the assignment game, we analyze the following mechanism: sellers, simultaneously, fix their prices first; then buyers, sequentially, decide which object to buy, if any, among the remaining objects. The first phase of the game determines the potential prices, while the second phase determines the actual matching. We prove that the set of subgame perfect equilibria in pure strategies in the strong sense of the mechanism coincides with the set of sellers' optimal stable outcomes when buyers use maximal strategies. That is, the mechanism leads to the maximum equilibrium prices and to an optimal matching.Pérez-Castrillo acknowledges the financial support from projects DGES PB 92-0590, DGES PB 96-1192, and SGR 98-62. Sotomayor acknowledges J.S. Guggenheim Foundation, FAPESP, and FIPE - Sao Paulo - Brazil.Peer reviewed