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Problem #17 HARD

Rivals on a Raft

Microsoft Meta Game Theory Logic

Problem Statement

Five crew members discover a chest with 100 gold coins. Crew rank: Captain (most senior) down to Sailor 5 (least). Tradition says the highest-ranked crew member alive proposes a split. The crew votes — if at least half vote yes (including the proposer), the split is adopted. Otherwise, that crew member is cast overboard and the next senior one proposes. All crew members are perfectly rational, self-interested, and prize their own survival above gold, but will choose more gold over less when survival is equal. What split does the Captain propose?

Answer & Quick Explanation

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Captain proposes 98 / 0 / 1 / 0 / 1 (Captain / Sailors 2–5). Passes 3–2. Sailors 3 and 5 rationally accept 1 coin each, since they'd receive 0 if the Captain were eliminated.

Detailed Editorial Solution

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Solve by backward induction from the two-person case up. Each crew member votes yes if and only if the current proposal gives them at least as much as the next scenario would. Step 1: 2 crew (Sailors 4 and 5): Sailor 4 proposes 100/0. Sailor 4 votes yes — that's 50%, so it passes. Sailor 5 gets 0. Step 2: 3 crew (Sailors 3, 4, 5): Sailor 3 needs 2 votes. Sailor 5 got 0 in the 2-person scenario, so Sailor 3 buys Sailor 5's vote with just 1 coin. Proposes 99/0/1. Passes 2–1. Step 3: 4 crew (Sailors 2, 3, 4, 5): Sailor 2 needs 2 votes. Sailor 4 got 0 in the 3-person scenario — cheapest to buy. Offers Sailor 4 one coin. Proposes 99/0/0/1. Passes 2–2. Step 4: 5 crew (Captain through Sailor 5): Captain needs 3 votes (including himself). Sailors 3 and 5 each received 0 in the 4-person scenario. Captain buys both votes with 1 coin each. Step 5: Captain's proposal: 98 / 0 / 1 / 0 / 1. Captain + Sailor 3 + Sailor 5 vote yes. Passes 3–2. Step 6: Sailor 2 and Sailor 4 vote no but are outvoted. Captain keeps 98 coins. Key Insight: The proposer is safe only while alive — death is the worst outcome. Crew members accept 1 coin over 0. The Captain exploits downstream scenarios to identify the two cheapest votes and pay the minimum bribe.