Consider ·the following game form representation of worker selection from a menu of contracts. The number of firms is finite. (fwo will be enough.) Firms simultaneously and independently name contracts (pairs (w, e)); with each firm limited to a finite number of contracts. (If you wish, assume each firm can name no more than four contracts.) The con tracts tentatively offered are announced, and then firms simultaneously and independently decide whether they wish to withdraw some of the contracts they offered initially. The new set of tentatively offered contracts is announced, and- if any contracts were removed, firms are again given the opportunity to remove contracts. And so on, until either no contracts are left or, in a given round, no contracts are removed. (Since there are finitely many firms and finitely many contracts, this process must termi nate.) Workers then self-select from among the contracts (if any) that are left. Otherwise, things are just as in the models of this chapter.
What can you say about the subgame perfect equilibria of this game? Can you give a precise characterization? If not, can you give partial charac terizations? Can you describe at least one equilibrium? (There are many proper subgames here, and you may find things easier if you look for Nash equilibria where workers play sequentially rational strategies. That is, don't worry about subgame perfection for the firms.)