5 Rational Pirates and 100 Gold Coins Backward Induction

Scenario

Five rational pirates ($P_1$ oldest down to $P_5$ youngest) must divide $100$ identical gold coins. The senior pirate proposes a distribution $(c_1, c_2, c_3, c_4, c_5)$. All remaining pirates vote. If at least $50\%$ of voters support the proposal, it passes. If rejected, the proposer is thrown overboard, and the process repeats with the next senior pirate. Pirates share strict lexicographic priorities:

1. Survival: Avoid being thrown overboard.
2. Greed: Maximize gold coins received.
3. Bloodthirst: Cast a 'no' vote if indifferent in coins.

Your Task

1. Using backward induction, analyze the subgame perfect equilibrium for $k = 1, 2, 3, 4, 5$ pirates.
2. State the exact optimal coin distribution proposed by $P_1$.
3. Identify which pirates vote in favor of $P_1$'s proposal.

Constraints

• Indivisible integer coins.
• Strict common knowledge of rationality.

Examples With 2 pirates, $P_4$ takes all 100 coins and gives 0 to $P_5$, casting his own vote ($1/2 = 50\%$) to pass.