The authors introduce and analyze 'multistage situations, ' which generalize 'multistage games' (which, in turn, generalize 'repeated games'). One reason for this generalization is to avoid the perhaps unrealistic constraints--inherent to noncooperative games--that the set of strategy tuples must be a Cartesian product of the strategy sets of the players. Another reason is that, in most economic and social activities (e.g., in sequential bargaining without a rigid protocol), the 'rules of the game' are rather amorphous; the procedures are rarely pinned down. Such social environments can, however, be represented as multistage situations and be effectively analyzed through the theory of social situations.

Consider an environment with a finite number of alternatives, and agents with private values and quasilinear utility functions. A domain of valuation functions for an agent is a monotonicity domain if every finite-valued monotone randomized allocation rule defined on it is implementable in dominant strategies. We fully characterize the set of all monotonicity domains.