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Consider a three-member legislature that operates under majority rule. There are a status quo y and an alternative x. A lobbyist L1 wants to have x enacted and a lobbyist L2 wants to have y retained. Each lobbyist can “bribe” a legislator to vote for his preferred policy. For simplicity assume that only bribes of exactly p can be made (with p > 0 a fixed number). If a legislator receives a bribe from only L1 she votes for x; if she receives a bribe from only L2 she votes for y. If a legislator receives a bribe from both lobbyists or no lobbyists she votes for y. The preferences of the two lobbyists are given by the utility functions:
Utility of L1(x enacted)=u*-pBL1
Utility of L1(y enacted)=-pBL1
Utility of L2(x enacted)=-pBL2
Utility of L2(y enacted)=u*-pBL2
where BL1 is the number of legislators L1 bribes (either 0, 1, 2, or 3), and BL2 is the number of legislators L2 bribes (either 0, 1, 2, or 3). Assume that u* ≥ 3 p. A pure strategy for each lobbyist indicates which legislators (if any) she bribes. For example, (b, 0, b) indicates that bribes were made to legislators 1 and 3 but not 2. A mixed strategy is then a probability distribution (lottery) over the eight possible pure strategies. Find the Nash Equilibria to this game. Hint: First delete strictly dominated strategies