Suppose there are 2 potential policies that could be passed: policy A and policy B, and there are 7 voters in the legislature who will vote and the winning policy will be determined by majority rule. All 7 voters initially prefer policy B over policy A by 1.

However, suppose there are 2 coalition-building groups: Pro-A and Pro-B.

• Pro-A would like policy A to win, Pro-B would like policy B to win.
• Both have some capital (e.g., campaign contributions, favors, etc.) that they can use to bribe legislators and influence their votes. Everyone knows that Pro-B has an overall bribing budget of 3. Pro-A is not budget constrained and has a huge budget, but they still would like to efficiently use their capital.
• But both want to bribe in the most cost-efficient way.

First, Pro-A buys votes, i.e., it can choose which legislator to give how much to.

• After observing how much Pro-A gives to each legislator, Pro-B then buys votes, i.e., likewise chooses which legislator to give how much to.
• The 7 legislators get to keep all the bribes they receive from Pro-A and Pro-B and they vote accordingly.
  • For example: if say legislator #3 is given 5 by Pro-A and 2 by Pro-B, overall, he would be -1+5-2 = 2 in favor of policy A and hence would vote for policy A.

Suppose Pro-A has already bribed, and now Pro-B faces 7 legislators who favor policy A to B by 1.5, 5, 0.75, 2, 3, 1, -1 respectively (i.e., 1^st prefers A to B by 1.5, 2^nd prefers A to B by 5, etc.).

(a) is it safe to say that given its budget constraints, Pro-B cannot successfully provide bribes that could get it majority support for Policy B?

- it has only one legislator in favor of Policy B, and needs 3 more

- the lowest amount of bribe it can extend to make 3 more legislators indifferent is 0.75 + 1 + 1.5 = 3.25, which is larger than its budget. Therefore, it would not have sufficient cash to execute a bribery that would get 3 more legislators to favor Policy B.

(b) is it safe to say that Pro-A has bribed the legislators as follows: 2.5, 6, 1.75, 3, 4, 2, 0 (totaling 19.25)?

(c) can Pro-A still win given the following bribe structure: 2.25, 2.25, 1.75, 2.25, 2.25, 2, 0 (totaling 12.75)?

- Pro-B needs at least 0.75 to bribe legislator who favors Policy A by 0.75 to make it indifferent

- Pro-B also needs at least 1 to bribe legislator who favors Policy A by 1 to make it indifferent

- after bribing the 2 legislators listed above, Pro-B is left with only 1.25 to bribe a 4th legislator to ensure Policy A doesn't get majority

- therefore, Pro-A needs at least 2.25 (-1 + x = 1.25 wherein x = 2.25) to ensure that not only are legislators indifferent between A and B, and that Policy A gets the majority vote

(d) Is there any other bribe structure that Pro-A could have done to save money yet still get the majority?