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Pirates Dividing the Spoil
This is a classic logic problem. Briefly, the setup runs like this:
A band of cutthroats have skedaddled with some loot. Now comes the time to divide the take. The captain proposes a split of the treasure and the assembled pirates vote on it. If a majority of them approve of the split (or the vote is tied) the lucre is so divided. If a majority disapprove they make the captain walk the plank. Then the next highest ranked pirate takes over as captain and proposes his own split of the booty. How do they divide up the spoil?
Now, this discussion also comes with some basic assumptions:
- The pirates are greedy – every pirate will do his best to maximize his own share of the swag and deny it to others.
- The pirates are ruthless. There’s nary a one of them but would slit his captain prow to stern if it’d profit him a single guinea. Matter of fact these pirates are pretty well fed up with each other and would gladly heave each other over the side so long as it’s nice and legal according to the rules of this game.
- The pirates are shifty. Any single one of ’em would go back on his word quicker ‘n the wind shifts in a hurricane.
- The pirates are flawlessly logical. This might not be as sound an assumption as the previous three.
The standard version of the puzzle has five pirates in it (whom I’ll refer to as captain, 1st, 2nd, and 3rd mate, and the swabbie), with a hundred gold pieces. That’s just to make the logic easier to follow. The standard solution runs like this:
Suppose the first three ballots fail. That is, the captain, and the first and second mate are shark bait. That leaves a vote between the third mate and the swabbie. Because any vote that ends in a tie passes then the 3rd mate can force any split through he’d like. Since he’s greedy he’ll vote for 100 pieces for himself, and zero for the swabbie. Therefore any other spit benefits the swabbie more. All he needs is to get more than zero in one of the previous three ballots for him to vote in favor of one of them rather than end up with nothing at the hands of the 3rd mate.
Now suppose that only first two ballots fail instead. The 2nd mate is proposing a split for him, the 3rd mate, and the swabbie to vote on. The swabbie knows that if this ballot fails he gets the shaft, (and the 2nd mate understands this as well), so the second mate proposes the following split: 99 coins for him, none for the third mate, and one measly penny for the swabbie. The swabbie figures one pence is better than none, so he reluctantly allows this measure to pass. (He could offer one to the third mate and none to the swabbie, but then neither the swabbie nor the third mate will have a monetary incentive to agree, and both already harbor the opinion that the second mate would look better with a second smile.)
But that leaves the third mate in the lurch. Consequently, he can’t let the vote get to the third ballot. So we back up to where only the Captain is taking tea with Davy Jones, and the 1st mate is proposing the split. Knowing that an even vote favors his split he knows that he only has to keep one of his minions happy. Specifically the 3rd mate. The 2nd mate would just as soon have his guts for garters and take the majority of the loot itself. He could bribe the swabbie instead, but the swabbie will just as soon spill the first mate’s blood and take his miserable one coin from the 2nd mate instead. So the 1st mate proposes 99 for himself, one for the third mate, and the other two get boned.
Naturally, the other two don’t like this much. But the time to act isn’t in the second ballot, it’s in the first, when they can manage a slim majority to lock the other pirates out. Again, because exactly one coin is better than none, the captain can thus propose the following split: 98 coins for himself, zero for the 1st mate, 1 for the 2nd mate, 0 for the 3rd mate, and 1 for the swabbie. Because all five pirates are acting flawlessly logically the motion passes.
What’s Wrong With This Picture?
We all have an intuitive sense of how this would fail miserably should any real pirate captain try it. Shocked by the blatant unfairness the crew votes him off the ship and try again. Supposing it gets down to the swabbie he grudgingly allows the 3rd mate to keep all the gold, then stabs him while he’s sleeping. If the third mate doesn’t stab him first. Actually, a basic animal sense of fairness indicates that, if a perfectly logical split like this were proposed the whole endeavor would end in blood.
Okay, but assume the pirates are perfectly logical, and abiding by the rules of the game. Next time the ship makes port all but the captain are going to jump ship; presumably for other pirate ships which provide a more even split of the loot. Supposing every pirate captain and crew is this logical that won’t help them (getting either one or zero coins on every split), but maybe the transaction costs of hiring a new crew each voyage alone will cause the Captains to pay out a little bit better.
What if the Pirates Can Rely Even a Little Bit upon the Bloody Oaths Sworn by Their Crewmates?
Suppose the 2nd mate, tired of getting such a miserable rate of miles to the galleon, pulls aside0 the third mate and the swabbie. They agree each to vote against the first two proposals, keelhaul their officers, and split the remaining loot by thirds. Now, maybe the 2nd mate will betray them in the end and propose his traditional 99-0-1 split, but at the very least the 3rd mate and the swabbie are no worse off than before, and they still got to do some murderin’.
That last bit is interesting; simply by proposing an exchange of murders the final three pirates better their lot. The 2nd mate by most of the loot, the 3rd mate and the swabbie by getting to cut their officers to ribbons. If you relax the default problem’s prohibition from “No deals made” to “no deals made that involve promising even a single extra coin” and suddenly the whole calculation shifts.
First, the bloc executes everyone ranking just higher than the median pirate. Then, if there are enough pirates still available, a new bloc forms, and another round of executions begin. (Instead of our familiar crew of five, imagine a crew of nine. First, the lower five conspire against the upper four, then, when those four are pools of blood in the sea foam, the lower three conspire against the upper two.) In the end, no bloc smaller than three would be possible, since it’s never in the swabbie’s interest to kill the 3rd lowest ranking pirate and get zero coins instead of one.
Now Assume Pirates Aren’t Averse to a Little Bribery
Supposing the captain gets wind of this lower decks voting bloc. His life then hangs upon the question of whether he can bribe one of the members away. He only needs one — the 1st mate’s life is worth less than a bent penny in a Barbados barroom if the bloc has its way — and as such he’ll go along with whatever plan saves his yellow hide. The captain proposes a 98-0-0-0-2 split. Since he’s got the 1st mate’s vote in the bag, he’s relying on the swabbie’s greed to trump his bloodlust and betray the 2nd and 3rd mate’s voting bloc.
“Avast ye, ye scurvy bilge rat!” Exclaims the 2nd mate, by which I take him to mean “I can make a bid too.” He offers three doubloons to the swabbie, and a bidding war ensues. The final bids are, from the captain, a split of 0-0-0-0-100, and from the 2nd mate, first they murder the captain and the first officer, then they split the gold X-X-0-0-100, the “X”s denoting dead men. The swabbie ends up rich either way, but his cutlass is thirsty so as it turns out the captain can’t buy out the voting bloc that way.
What if he tries to bribe the 3rd mate as well? He proposes a split of 97-0-0-1-2, thus bettering both of the bottom pirate’s bottom line, at the expense of the first mutineer. On this bid he’d win, except a similar bidding war ensues. I’m not quite sure how this turns out. If the captain bribes one person and the mate bribes another (Captain: 0-0-0-100-0; Mate: X-X-0-0-100) the captain survives. It’s in the interests of the pirate the captain is bribing to vote with the captain and come away rich. On the other hand, if they bribe the same pirate (Captain: 0-0-0-100-0; Mate: X-X-0-100-0) then that pirate will take the option that first lets him shiver some timbers, and the captain dies. If you try some kind of even split in the bribes the captain dies as well.
He does have one option though; and that’s to bribe the 2nd mate. Suppose the captain offers a split of 0-0-100-0-0; the first mate (yellow as a jaundiced canary) votes along with, and the 2nd mate, greedy as ever, double-crosses his fellows for the gold. Three votes is enough to win the day and no pirates dance the jig o’ death.
Wait a Tick
“Avast and belay! Hoist ‘m by the yarrrdarrrm and ta Davy Jones with his pet goldfish!” By which I take the swabbie to mean “I say; when I agreed to this particular bit of mayhem I was promised that my lot would be bettered. But now, instead of the single gold piece I was promised if there were no collusion involved, I get none. What’s more, I lack even the joy of stretching the captain’s neck. If that’s the case then I, being a perfectly logical sort, will refuse to join this particular syndicate in the first place. What say you to that?”
By Jove, that intemperate blackguard may be on to something. If the ultimate logic of the voting bloc leaves two pirates high and dry and no pirates slaughtered then these ultimately logical pirates won’t enter into it. What if, though, the 1st mate organizes a voting bloc instead? (Bout time that cowardly sea dog bestirred himself.)
Here the odds are four to one against the captain, and whatever bribe he tries to make can be better than matched by the 1st mate, who can always add the sweetener “and you get to kill the captain.” What’s more, he can no longer bribe the top-ranking pirate in the voting bloc, since a prospective split of 0-100-0-0-0 still leaves three unhappy pirates and hence the captain’s blood running from the scuppers.
But then what happens? You’ve got the lower ranked pirates with every incentive to make a voting bloc again, only with the 1st mate in the position of the captain from before. He’s got to surrender all the ill-gotten wealth to keep his skin intact, and you end up with a final tally of X-0-100-0-0. Once again, the 2nd mate profits the entire amount, by the logic of bribing the only member of the voting bloc who’ll then have an incentive to keep you alive. But once again the swabbie is left with no incentive to join the original bloc of four against the captain.
Perhaps you could get by with a bloc of the three mates; 1st 2nd and third, leaving the captain and the swabbie on the outs. Then when the captain is gone the 1st mate is confronted by the traditional lower deck bloc of the 2nd mate, the 3rd mate, and the swabbie, and once again we’re left with a X-0-100-0-0 split so the 1st mate can delay his trip to the infernal regions.
One Final Wrinkle, I Promise
Suppose that the captain has one more line of motivation; if he’s going to go down then he’s going to take as many of the traitorous bilge rats with him. When a voting bloc arises he can structure his bribes so as to put paid to as many of the members as he can. It won’t save his dirty hide but he’ll at least have company with which to greet Davy Jones. When confronted by the bloc of three mates he offers to divide the loot fifty-fifty between the third mate and the swabbie. 0-0-0-50-50.
This is good news for the third mate; he’s been getting zero all along. He’d willingly betray his erstwhile companions for that much loot. “Belay that!” cries the 1st mate, and he proposes a X-0-0-50-50 split. This is even better for the latter two, so they assent to that. The captain goes over the side and they prepare to have a vote. But now that the 1st mate has promised to bribe the 3rd and the swabbie he’s got nothing with which to bribe the 2nd mate. The 2nd mate proposes a more bloodthirsty solution; x-x-0-50-50, which is still better for the last two. But when the 1st mate is overboard there’s nothing to stop the 3rd mate from proposing x-x-x-50-50, and so the captain has his revenge; two of the scurvy dogs who did him in will receive justice at the point of a cutlass.
However the perfectly logical mates can see where that train ends, and they’d prefer to get off at an earlier stop. The 2nd mate can choose to end it by supporting the 1st mate’s vote, preserving both their worthless lives or neither. This leads to the curious loot split of X-0-0-50-50, with the pirates voting X-yea-yea-nay-nay, those who are getting the entirety of the loot voting against, and those getting none of it voting for. The pirate captain is denied his ghostly revenge, however, this is as fair a split as is practicable in any of these scenarios.
To Sum Up
The final scorecard looks like this:
With no collusion: 98-0-1-0-1
With bloc votes but no bribery: X-X-99-0-1
With bloc votes and bribery: X-0-100-0-0
With bloc votes, bribery, and revenge: X-0-0-50-50
Conclusion:
If you’re the captains, sure as you value your hide keep your pirates from colluding!
If you’re the 1st mate, whatever you do make sure that bribery remains on the table.
If you’re the 2nd mate, steer your captain’s thoughts away from revenge
If you’re the third mate, whisper bloody thoughts of vengeance to your captain
If you’re the swabbie, sod the rules and knife ’em all whilst they sleep.
Question for all y’all: What ways of piratical murder have I failed to mention in here?
Published in Law
Matelotage.
Just going from the quoted setup, homo economicus will wind up with the two most junior splitting the spoil fifty-fifty.
EDIT: Wait, second to last gets everything, because tie vote defaults to proposal.
EDIT EDIT: Yeah, I can follow each step, but the whole of course makes no sense. This is a good way for two computers to negotiate a protocol or something, but it ain’t people. If murdering after the fact is allowed, why not before?
I’m just questioning how many men it takes to sail the ship. If they’re going to divide up the loot they won’t want to do it where anyone else might take an interest, which means they won’t be anywhere they can spend it. They’ll want to keep enough people alive to get back to port.
I just assumed this was off the Somali coast and it was a 26′ power boat with twin outboards.
First time learning this set of related logic problems. I’ve learned the solution to the first one–I started trying to solve it independently, but got impatient and read the answer.
That one is really great! The other four problems you mentioned are interesting too, but I’ve not worked on them, yet.
I interpret your question as, “what other interesting variations (logic problems) could be made that you didn’t introduce?”
I don’t know yet, since I’ve not solved any of the four, nor learned your answer to them. But intuitively I would answer, “there must be more.”
What makes these refinements to the thought experiment interesting is that they incrementally add realistic aspects of actual human nature (praxeological knowledge).
As an empiricist, I say this requires an FDA phase 3 trial with volunteers.
Yes. Just like the ice hookers thread.
Parlay?
I was told there would be no math, or logic involve in, 🎶 A Pirate’s Life for Me 🎶…
Pirate recruiters lie.
Privateer recruiters, too.
I’m pretty sure you don’t need to preface that with pirate. Or alternately, are they recruiting for a pirate crew or are they stealing other people’s recruits?
Also:
Yeah, the whole problem relies on the pirates being perfectly logical and only murderin’ people when the rules allow. In essence, the problem assumes the pirates aren’t going to act like pirates.
Were the mutineers on the Bounty pirates? Only one survived. (And he was pardoned for becoming a Christian and leading the native population of Pitcairn Island to Christ and shepherding them.)
The first one I read about elsewhere. The remainder of this puzzle is stuff I thought up myself. I’m still not convinced I’ve got all the by-roads worked out. If you get a conclusion that I didn’t you might be right.
I think this:
For some people,
For purposes of discussion, let’s call those people “philosophers”.
Hank Rhody Freelance Philosopher is a philosopher.
Unfortunately, it is logically impossible for you to persuade Hank (or any philosopher) that this theorem (or any theorem) as a whole doesn’t make sense and each step does.
You use the phrase “of course” because to you it seems ridiculous to think that [A] is true in this case.
The problem with this logical exercise is that, as always, it excludes much more pertinent information.
The Captain has natural appeal, charm, and animal magnetism.
The 1st Mate is a sycophant, left-handed, and who harbors a deep unspoken passion for the Captain.
The 2nd Mate has OCD.
The 3rd Mate is young, impressionable, earnest and is merely happy he’s not the swabbie.
And the swabbie is the oldest of the crew, and a drunk who hasn’t successfully flemished a rope in years but who doesn’t eat, allowing the crew to divide his rations.
The psychological, the internal mechanisms, the grey matter, is what decides the final outcome of this distribution, and we must find what motivates each one to get the true picture.
If you are gently poking fun at some reactions to the puzzle, I appreciate it, even though I decided not to explore this branch of the conversation.
I would argue that for pirates murdering might be perfectly logical.
Congrats, then on the variations! All original work.
As to more variations: you implicitly assume that these four are selections from a logically closed set. I am not sure. I think each is adding an arbitrary new premise.
I was poking fun at the nature of the puzzle and, I suppose, the certainty, the faith, that logicians place upon their intellectual endeavors (and you are by no means the only one). The CFC debate is an example of this faith: chemically, CFC-mediated ozone depletion is logically unassailable, but it appears to be wrong. This would be due to unaccounted for variables. Logic is a tool, but it is more often used as a whip. If the accuracy of the predictions of relatively simple molecular models is so elusive, the predictions of the reactions of crew members are even more so.
Nevertheless it’s fun. And I was having fun.
The puzzle as I stated it has four officers and one seaman on the roster. The mechanics of the puzzle mean that, no matter what, the 3rd mate and the swabbie will survive. How much you wanna bet they were doing all the sailing to begin with?
I’m by no means up on 18th century maritime law, but my understanding is that both crimes warranted the death penalty. If you’re going to be hanged for mutiny and if every ship flying the Union Jack has her hand raised against you already, then I doubt it’s a large leap from there to piracy. As for the case of the Bounty specifically I don’t know.
Not at all. The proof works, but to get to that end point you have to carefully marshal your initial conditions such that no other outcome is possible. That means that it’s hard to derive any useful knowledge from the thought experiment; the analogy to anything in the real world is unlikely to share all those same assumptions.
Mostly I’m thinking about this one because logic puzzles are fun.
Well, I’m not 100% on what a logically closed set actually means, but yeah. I was iterating the problem. The iterated scenarios loosen the constraints on the pirates’ actions in orders to see what a slightly more realistic version of the original puzzle results in.
This, by the way, doesn’t work. Our pirates are smarter than Stalinist stooges. If some members of that group of five will succumb to skullduggery once their own murderous work is done then they’re not going to join in the initial bloc to begin with. They’ll have logically computed their own bloody ends and chosen a different path.
Try it again, assuming that the bottom three ranking pirates will always form a bloc in favor of killing everyone higher on the list. That part at least holds from the five pirate version of the game. A three vote lower decks bloc of the three lowest-ranking pirates will have enough votes between them to strategically murder each higher ranking pirate, unless those higher ranking pirates alter their behavior in order to not get murdered. Which they will, because nobody wants to get murdered.
I haven’t worked through how they’d most effectively respond, according to the rules of the game.
I’m guessing you could get it down to where all the sailing is done by a single Australian man.
More broadly, I think I need to be a little more careful on how I consider the promises pirates make, one to another. Let me break that down a bit.
The offer: I will take some action now if you will promise to take some action later.
The promise: I will take the later action if you do your thing now.
The problem here is that pirates, being pirates, aren’t the most truthful bunch out there. (You heard it first! Here on Ricochet!) A pirate will gleefully make a promise and break it as soon as it’s in his best interest to do so. Jumping back to the blood oaths section and the original collusion cartel, let me go into a bit more detail.
The 2nd mate, knowing the captain is going to make the traditional 98-0-1-0-1 proposal, decides that he can do one better. He makes an offer to his lower deck hearties, to wit that they vote against their temporary best interest in order to profit. As we’ve seen, the outcome of X-X-99-0-1 is strictly better for all involved in the deal, particularly the 2nd mate.
The offer here is “I will vote against the original proposal and hang the captain, if you, the 3rd mate, will vote against the 1st mate’s proposal, and hang the 1st mate.” The promise is the 3rd mate’s agreement to vote along party lines come the second vote. But what happens when that second vote comes?
The captain proposes 98-0-1-0-1, and gets his skull duggered. The first mate proposes X-0-1-0, and the 3rd mate promptly double crosses his fellows; by voting yea on this proposal he ends with one gold coin where following through on his promise, he ends on zero gold coins. Furthermore, by instigating this whole deal the 2nd mate is worse off than he started; instead of the captain’s one gold coin he gets zero gold coins. The swabbie too; though his vote isn’t strictly necessary for the logic to play out.
What therefore happens? If we assume that pirates will always break their word the moment it becomes even slightly advantageous to do so we’re reducing the puzzle to the original no-deal-making configuration. If we assume that pirates will always hold to their sworn oaths then Mr. Balzer will complain they’re not being sufficiently piratical. If we assume that they’ll sometimes break their oaths, when the reward is great enough, then we no longer have deterministic puzzles, and I’d have to express outcomes as “this seems likely.”
Jump to the last, the Captain’s attempt at revenge. Against the assembled three mates bloc he proposes a split of 0-0-0-50-50. This is immediately greeted with the first mate making a counter offer of X-0-0-50-50, which his block agrees sounds mighty fine, so they slit the captain’s throat and make him walk the plank. Somehow. Everything is in the best tradition of piracy so far. But now that the captain’s dead, what’s to stop the 1st mate from proposing X-50-0-0-50, which suits him better than keeping his campaign promise?
Outraged, his fellow mates will vote against, but the swabbie is in the same position as he was before, so why shouldn’t he vote to save the 1st mate’s life? Oh, right, he’s a pirate. He’d cheerfully vote against if he could be assured that he’d get at least 50 in whatever other settlement comes up. If the 2nd mate promises X-X-50-0-50 he’s got a deal. Maybe; that murder is better than what the 3rd mate was getting in this iteration, but worse than the 50 he was initially promised.
If at every point of the chain pirates accept a promise, are subsequently betrayed, and out of outrage demand a new and better promise, then we reduce the scenario to the X-X-X-100-0 split from the swabbie’s original calculation, and the ghost of the captain has his gruesome revenge. This seems like a pretty miserable outcome, but it’s a darn sight better than what the voters get.
This is tennis without a net. Or paint. Or a court.
And?
It’s enjoyable as far as exercise, but I can’t figure out the rules, so I stopped playing. Play ball!