Rational Pirates Revisited

There are \(n\) pirates \(\left(P_{n}, P_{n-1}, \cdots P_1 \right) \) with the strict order of superiority \(\left(P_{n} > P_{n-1} > \cdots > P_1 \right) \).

They find 100 gold coins. They must distribute the coins among themselves according to the following rules:

1. The superior-most surviving pirate proposes a distribution.
2. The pirates, including the proposer, then vote on whether to accept this distribution.
3. In case of a tie vote the proposer has the casting vote.
4. If the distribution is accepted, the coins are disbursed and the game ends. If not, the proposer is thrown overboard from the pirate ship and dies (leaving behind \((n-1)\) pirates and their original ordering), and the next most superior pirate makes a new proposal to begin the game again.

Every pirate knows that every other pirates is perfectly rational, which means:

1. They will always be able to deduce something logically deducible.
2. Their first preference is that they want to survive.
3. Their second preference is that they want to maximize the number of coins they receive.
4. Other things remaining same, they prefer to kill a pirate rather than having him alive.

What is the minimum value of \(n\) such that the superior-most pirate can propose no distribution of coins that will let him survive?