# Dividing the gold

There is a king and 3 women in a castle. One day, the king give 3 women a total of 100 golds. He said that each woman can propose an idea to share the gold among themselves. If the proposal passed with vote greater than half of the votes, the gold will be shared based on the proposal. Else, the woman who proposed the idea will be killed and the game continue. Assuming the propose sequence is from A to B to C, which one of them will have the greatest advantages in proposing the idea?

Info:

Only proposal with votes greater than half of the voters will be approved, proposal with votes equal to half of the voters is not approved,

King do not involved in voting

All of them are smart

All of them don't want to be killed (first priority)

All of them want the most gold (second priority)

All of them want to kill each other (third priority)

None of them can talk or communicate to each other during the whole process by any means

Hint:

The women with the greatest advantage can have all the gold

• I've reopened this. It's very similar to the earlier puzzle it was marked as a duplicate of, but the small difference (in tie-breaking rules) actually makes a big difference to the answer. The guiding principle for duplicates is: if it's obvious how to turn the posted solution to OLD_PUZZLE into a solution to NEW_PUZZLE, it's a dupe; otherwise, not. In this case it's not. Commented Apr 30, 2019 at 12:01

Since the order is strictly A, B, C, then, we can examine the naive approach:

B and C will just vote No to A's proposal regardless of what it is and effectively kill them (not only directly meets their third priority, but increases their second since there's less people to share between). Similarly, no matter what B proposes next, C will just vote No and it'll be a split vote, leaving B killed and only C remaining. At which point C proposes and votes Yes to taking all the money.

However, knowing this:

Since neither A nor B want to die (first priority), it's in B's best interest to go along with A's suggestion to out vote C, no matter what spilt they suggest, so of course A will propose she gets all.

In other words,

A has all the power and can guarantee survival and getting all the gold (though this means survival for everyone, so she still fails at priority three).

• In the duplicate question, the strategy (proposal) for $3$ players is different, any explanation? Commented Apr 30, 2019 at 4:22
• @athin - I think the difference is that in the other question, there's a tie breaking caveat (fiercest pirate's vote carries), where as here a tie is a failure. Which arguably makes this not a duplicate... Commented Apr 30, 2019 at 5:52