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Halloween Treats

The best way to approach this problem is to work backwards, starting with the simplest case, and expanding to the original problem, like was done in the solution to The Three Hats.

Case 1: Suppose Jar Jar gets to propose. Well, if he gets to propose, then he's the only one voting, and he gets all 100 pieces. No rocket science here.

Case 2: Suppose the goblin gets to propose. We can imagine that he might make any sort of proposition. But if Jar Jar votes "No", it reduces to Case 1. So Case 2's outcome is the same as Case 1's.

Case 3: Suppose the ghoul proposes. Now things get interesting- at least some "diplomacy" is involved. The ghoul points out the obvious, that if his vote doesn't pass, it falls to Case 2, and Jar Jar gets everything. So the ghoul offers to 1 piece to the goblin, nothing to Jar Jar, and the rest to himself. The goblin will have to accept or get nothing.

Case 4: The ghost needs two of the others' votes to pass, which means he has to improve on the value of Case 3, relative to each voter. He can't easily improve the ghoul's spoils, so no sense trying at all. He offers the ghoul nothing. He improves the goblin and Jar Jar's situation by increasing each of their shares by one. So 0 for the ghoul, 2 for the goblin, 1 for Jar Jar, and everything else for himself.

Case 5 (reality): The fair princess still only needs two of the others' votes to pass. What is the cheapest way she can buy their votes? She can buy the ghoul's vote for 1, and Jar Jar's vote for 2. The ghost and the goblin go home empty-handed.

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