14. Bargaining

Suppose that an individual is selected randomly to propose an alternative, all individuals vote for or against this alternative, and the proposal is implemented and the game ends if a majority is in favor, while otherwise the procedure is repeated until a proposal is accepted by a majority. Every alternative is the outcome of a subgame perfect equilibrium of the resulting extensive game, and, if the individuals' votes are observable, most alternatives are outcomes of subgame perfect equilibria in which every individual's vote is undominated. If only the outcomes, not the individuals' votes, are observable, and the individuals are sufficiently patient, then if the alternatives are distributions of a fixed amount of a good, every alternative is the outcome of a subgame perfect equilibrium with undominated voting, while if the set of alternatives is an interval of numbers and the individuals' preferences are single-peaked, the outcome of a subgame perfect equilibrium with undominated voting is close to the median of the individuals' favorite alternatives.

Now suppose that bargaining is on-going. An individual is selected randomly to propose a distribution of a fixed amount of a good, and all individuals vote for or against this proposal. If a majority votes in favor, the proposal is implemented in the current period, and otherwise the status quo is implemented. In both cases, the procedure is repeated in the next period, with the status quo in each period equal to the previous period's outcome. There are examples in which almost any distribution is the outcome of a stationary subgame perfect equilibrium of this model, including distributions in which some of the good is wasted.