?? EXERCISE 224.1 (Cohesion in legislatures) The following pair of games is designed
2 to study the implications of different legislative procedures for the cohesion of a governing coalition. In both games a legislature consists of three members. Ini- tially a governing coalition, consisting of two of the legislators, is given. There are two periods. At the start of each period a member of the governing coalition is randomly chosen (i.e. each legislator is chosen with probability 1 ) to propose a
bill, which is a partition of one unit of payoff between the three legislators. Then the legislators simultaneously cast votes; each legislator votes either for or against the bill. If two or more legislators vote for the bill, it is accepted. Otherwise the course of events differs between the two games. In a game that models the cur- rent US legislature, rejection of a bill in period t leads to a given partition dt of the
pie, where 0 dt 1 for i = 1, 2, 3; the governing coalition (the set from which i 2
the proposer of a bill is drawn) remains the same in period 2 following a rejection
in period 1. In a game that models the current UK legislature, rejection of a bill brings down the government; a new governing coalition is determined randomly, and no legislator receives any payoff in that period. Specify each game precisely and find its subgame perfect equilibrium outcomes. Study the degree to which the governing coalition is cohesive (i.e. all its members vote in the same way).