When You Get Sidetracked

16 February 2017

You know that advice blog piece I clicked on yesterday? Because I found it so enlightening . . . amusing too . . . I made the mistake of clicking on the "Home" page which led me to a post they did about pirates. And who can't resist pirate stories . . . well, you have to admit, the title is intriguing . . . in light of the pirate treasure maps I've received:

"The Puzzle of the Pirate Booty" was the title . . . since I seriously doubt the link will even work when you're of age, I've copied and pasted here for you . . . because I really found it hysterical. Yeah, I know it's king of long . . . and you'll likely just roll your eyes and skip to my next post . . . but just in case you find this stuff as intriguing as I do . . . 
 

The Puzzle Of The Pirate Booty

By Tim Urban

Ten Perfectly Rational Pirate Logicians (PRPLs) find 10 (indivisible) gold pieces and wish to distribute the booty among themselves. The pirates each have a unique rank, from the captain on down. The captain puts forth the first plan to divide up the gold, whereupon the pirates (including the captain) vote. If at least half the pirates vote for the plan, it is enacted, and the gold is distributed accordingly. If the plan gets fewer than half the votes, however, the captain is killed, the second-in-command is promoted, and the process starts over. (They’re mutinous, these PRPLs.)

Pirates always vote by the following rules, with the earliest rule taking precedence in a conflict:

Self-preservation: A pirate values his life above all else.

Greed: A pirate seeks as much gold as possible.

Bloodthirst: Failing a threat to his life or bounty, a pirate always votes to kill.

Under this system, how do the PRPLs divide up their gold?

SOLUTION:

Think of it from the view of the lowest-ranked pirate and work up from there.

The lowest ranked pirate (we’ll call him P10 and work our way up to the captain, who’s P1) would love to have every other pirate die so he can take all 10 gold pieces for himself. He also has no threat to his life, because by the time he’s promoted to captain, it means everyone else is dead.

P9 would also love to have all pirates above him killed, because if it gets down to just himself and P10, he can allot all 10 gold pieces to himself and none to P10. The vote will end up 1-1 (P9 will vote in favor and P10 will vote against), and since half of the members voted for the plan, it’ll be enacted and P9 will not be killed. So he, too, has no threat to his life.

P8 knows all this. He knows that if it gets down to the final two pirates, it’ll end up as:

P9: 10 pieces             P10: 0 pieces

And he knows that P10 and P9 know this, so if it ends up getting to the point where there only three pirates left and he’s captain, he’ll have that in mind. He could allot the gold pieces like this:

P8: 10             P9: 0               P10: 0

That’s much worse for P9 than the outcome if P8 is killed, so P9 will vote against the plan. But P10 is in the same position as he’ll be if P8 is killed: zero gold pieces. This is when rule #3 kicks in: the pirates are bloodthirsty. If it won’t put either a pirate’s life or bounty at stake, “a pirate always votes to kill.” In this case, P10 will vote to kill because it won’t affect his life or bounty either way. So if P8 wants to win P10’s vote, he needs to give him one gold piece. Then rule #2 kicks in—greed. P10 will understand that he should vote in favor of the plan, because if he doesn’t, he’ll get nothing after P8 is killed, so he’ll vote for the plan and it’ll be enacted. P8’s life will be spared and he’ll make off with an awesome nine gold pieces. So P8 will allot the pieces like this:

P8: 9               P9: 0               P10: 1

In the case where there are four pirates left and P7 is captain, P7 knows all of the above and he knows that the other three pirates all know all of the above—and he only needs one of the three other pirates to vote in favor of his plan for it to be enacted in order to spare his life. So he’ll allot them like this:

P7: 9               P8: 0              P9: 1   P10: 0

He knows P8 and P10 will vote against it, because if he dies, they’ll be in the above situation where P8 gets nine pieces and P10 gets one, and his plan has them both getting zero. But he’ll get P9’s vote, because P9 knows that if he votes against it, P8 will enact his plan and P9 will get nothing. One piece is better than zero pieces, so P7’s plan will be enacted.

You can continue this logic upwards through the ranks. In each case, if the highest-ranked pirate allots one piece to all pirates who will be getting zero pieces should he be killed, his plan will be enacted, so it’ll go like this:

If it gets down to the final five pirates, the plan will be:

P6: 8   P7: 0   P8: 1   P9: 0   P10: 1

If it gets down to the final six pirates, the plan will be:

P5: 8   P6: 0   P7: 1    P8: 0  P9: 1   P10: 0

If it gets down to the final seven pirates, the plan will be:

P4: 7   P5: 0   P6: 1   P7: 0   P8: 1   P9: 0   P10: 1

If it gets down to the final eight pirates, the plan will be:

P3: 7   P4: 0   P5: 1    P6: 0   P7: 1    P8: 0  P9: 1   P10: 0

If it gets down to the final nine pirates, the plan will be:

P2: 6   P3: 0   P4: 1   P5: 0   P6: 1   P7: 0   P8: 1   P9: 0   P10: 1

But no one will die at all, because the original captain will allot them like this:

P1: 6   P2: 0   P3: 1   P4: 0   P5: 1    P6: 0   P7: 1    P8: 0  P9: 1   P10: 0

He’ll get votes from P3, P5, P7, P9, and himself to create a 5-5 split—and his plan will be enacted.

So the solution is: 6, 0, 1, 0, 1, 0, 1, 0, 1, 0

Note about collusion: Collusion at first seems like it could provide an interesting wrinkle. For example, P9 and P10 could collude and decide that instead of P9 voting to enact P1’s plan (which would give him one gold piece and P10 zero gold pieces), they could both vote no, and then continue to vote no again and again, preventing a majority vote until they’re the only two left. This would be great for P9, because he’d get ten gold pieces instead of one, and also good for P10 because, despite ending up with zero pieces in both cases, he’d have killed more people in the latter case, and as per rule #3, pirates are bloodthirsty, so he’d rather do that than P1’s plan. But this fails, because P10 would rather have one gold piece than kill those other pirates (since rule #2, greed, trumps rule #3, bloodthirst). So if they went with this collusion, once they got to the final three, P10 would break the agreement and vote yes to P8’s plan to snatch his last chance at one gold piece. Knowing this would happen, P9 would vote yes to P7’s plan. This would continue upwards until the top, so P9 would ultimately vote yes to P1’s plan, as described in the original solution.

P9 could try to thwart this problem by promising P10 two gold pieces if he agrees to vote no each time until they’re the only two left—but that fails too, since once there are only the two of them left, P9 has no incentive to honor the plan and will take all ten for himself. This is why all collusion inevitably fails.

Oddly, I do know who this guy is . . . your mom almost did a speech in one of her classes based on one of his "Ted Talks" about procrastination. It's a pretty funny piece . . . involves a monkey and panic monster!

The brain explained by Tim Urban . . . for procrastinators

If you read through that whole plan, then you’re probably wondering what it has to do with anything. It doesn’t have anything to do with anything . . . sometimes I just think things are funny. Besides, I thought this post might help me with my own “Pirate Puzzle Mystery!”