<p>Condition 1: The pirates are logical and mathematically inclined.
Condition 2: Each pirate wants the most possible gold for himself. Everyone else be damned, if only it wasn’t for the fact that he needs a…
Condition 3: >50% agreement among pirates still left alive.
Condition 4: No planning beforehand. OP will have to check this, but based on my knowledge of other game theory problems, this is usually another condition to simplify things. Otherwise, we’re left with networks of potential agreements and backstabbers.</p>
<p>New answer! Given one more condition:</p>
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<p>I figured pirate 1 would want to offer pirate 5 more than what pirate 2 would offer him to absolutely ensure that pirate 5 votes for him, even though pirate 5 would be equally well-off with just 1 coin. Otherwise, there’s a chance that pirate 5 downvotes pirate 1 just for the hell of it. However, if we add a 5th condition stating that the pirates are not capricious and will take the largest sum offered on its first time, then a 98-0-1-0-1 would be the actual best answer.</p>
<p>Pirate 5 would not downvote this because he wouldn’t reach a scenario in which he could earn >1 coin (if he keeps downvoting with the intent of getting to scenario 1, he’d get to scenario 2, and be outvoted by pirates 3 and 4).</p>
<p>Pirate 4 downvotes this.</p>
<p>Pirate 3 does not downvote because he’ll only get to scenario 3 before being outvoted (he does not assume that pirate 5 wants to keep downvoting, because pirate 3’s best case scenario is pirate 5s worst case, and pirate 3 gets to establish the optimal route first).</p>
<p>Pirate 2 downvotes.</p>
<p>Pirate 1 doesn’t downvote.</p>