Friday Puzzler -- 5 Pirates

Hint

Alex Y

.sdrawkcab kroW

Re: Hint

Henry Higginbotham

I considered that, since it seems pretty obvious what would happen if it came down to the last two pirates, with the older one needing only his own support for his plan. I have a hunch that the three older pirates, who also know this, will find a way to keep it from getting that far by finding a way to keep everybody dry. But I can't yet tell you how that's going to happen.

Re: Hint

Larry Barrett

Let the pirates be named A, B, C, D, E in that age order.

If it came down to D and E, we know (at least, Henry knows) what will happen then.

So before it gets to that point, C will think this way: If I propose to split the treasure equally, D will vote against it for sure; E, however, realizing that if he votes against it I (C) will be thrown overboard and it will come down to D and E, and in that case he (E) will get nothing, will vote for it. Thus the treasure will be split equally. C goes on to think that E will vote for any proposal that gives him more than nothing, so C proposes to give a small part to E, most to himself (C), and nothing to D. E will vote for this, as well as C so the treasure would be split this way. This seems to be the best strategy for C.

Moving back another step will need to wait until tomorrow.

Re: Hint - continuing

Larry Barrett

So now consider B's strategy. B would realize that he only needs to get one 'yes' vote from either C, D, or E. B would also quickly grasp what D would propose (if only D and E remain) and what C would also propose (one unit of treasure to E, none to D, and the rest to C) in order to win a 'yes' vote from E.

So B would see that he could likely win over E by offering two units of treasure to E, none to D or C, and the rest to B. E would see that two units of treasuer was clearly better than what C would offer, and way better than what D would offer.

So now consider what A would propose, after thinking through what B, C, and D would likely offer. A would realize that he needs to win a 'yes' vote from two of the other pirates; E would be a likely candidate and A should be able to win over E by offfering three units of treasure. But what about B, C, or D? Each of them, if they in turn were the oldest survivors, would stand to get almost all the treasure. A would reason that D, being third in line and likely to get nothing from B or C, might be the most likely to vote yes if A offered something, say one unit of treasure. So A would propose that E get three units of treasure, D get one unit, B and C get nothing, and A get the rest (96 units of treasure).

In the rules Alex set out, nothing was said about side bets, collusion, secret arrangements, etc. Sounds like modern politics (and ancient politics and power struggles, too). But for example consider how D might think, before A made his proposal. D would have reasoned as above for what A, B, and C might propose, and seen that his best chance of getting more treasure was to convince E to not vote 'yes' to what A (or B or C) was about to propose. So D might have whispered to E that he, D, would promise E that he would give E substantially more treasure than what A, B, or C would offer, say four units of treasure, if E would not vote yes to whatever A, B, or C might propose. Just to make sure, D might even promise to split the treasure 50-50 with E! Of course, B and C might start thinking along these lines and come up with their own secret offers to E. E might even end up with all the treasure if A, B, C, and D conclude that staying dry (as Henry put it) was better than any amount of treasure.

Close

Alex Y

And your third paragraph introduces some interesting considerations if there is collusion [mot du jour ;-) ]. But considering that there is no talking other than by the eldest pirate still aboard, the answer is slightly different from what you posted.

Re: Hint - continuing

Alex Y

So now consider B's strategy. B would realize that he only needs to get one 'yes' vote from either C, D, or E. B would also quickly grasp what D would propose (if only D and E remain) and what C would also propose (one unit of treasure to E, none to D, and the rest to C) in order to win a 'yes' vote from E.

B would also realize (as would C, D, and E) that if he went over, the coins would be distributed as proposed by C -- 99,0,1 to C,D,E. If and when B were making this decision, everyone would know that it would never get to D&E alone, so that case can be ignored (once it is used to determine what C would do)

So B would see that he could likely win over E by offering two units of treasure to E, none to D or C, and the rest to B. E would see that two units of treasuer was clearly better than what C would offer, and way better than what D would offer.

Is E the best ne to seek a vote from? Can't B keep more than 98 for himself?

Re: Hint - continuing

Larry Barrett

So with this additional hint, I think that B could indeed do better by offering D 1 unit of the treasure if he will vote with B. D, otherwise, will get 0 from B and so should be glad to get 1 from B. So B proposes 0 to C and E, 1 to D, and the rest, 99 units, to himself.

Yes!

Alex Y

And only one more step.

Answer

Alex Y

The oldest Pirate will take 98 coins for himself, and offer 1 coin each to the third and fifth oldest, who will accept that offer, knowing that they will get nothing if they vote to throw the oldest overboard, as Larry showed in his latest post.

Here is the decision tree they all consider. In each case, the rational choice is made considering what each pirate would get on that line versus what they would get if they vote against the then-oldest pirate, and get what is offered on the line above.