A grant of a whopping \$100 million is available to the five districts making up a state. The districts have representation in the state assembly proportional to their population. District A has 35 representatives, District B has 25, District C has 16, District D has 14 and District E has 10.

If a coalition consisting of at least 51 members should form, then it can force the funds to come to their districts, where the money will be divided proportionally to the representation. For example, if the districts having 35 and 25 representatives get together, they can force all \$100 million to go to their two districts and that money will be divided such that \$(35/60)x100 million goes to District A and \$(25/60)x100 million goes to District B.

Each district has only its own self-interest in mind, so wants to receive as much money as possible and prefers a small alliance to a larger one when the money is the same.

Warm-Up:
Which coalition would District B most prefer?

Solution to Warm-Up:
B prefers the {B, C, E} coalition to any other winning coalition, because B would get the fraction 25/55 of the money, whereas if B allied itself with A, it would get only 25/60 of the money.

Questions for you:
1. Which coalitions might form if a simple majority is sufficient?

The representatives of District B have an ancient privilege. They can control the rules of when an allocation can take place. That is, they may require that an allocation require only a simple majority coalition (51 out of 100 representatives) as we've assumed up till now, but they can also demand that it require a 67 percent majority, or a 75 percent majority.

2. How strong a majority should District B prefer (51 percent, 67 percent or 75 percent) in order for B to receive as much money as possible?