## Pirate Game with New Priorities

A forum for good logic/math puzzles.

Moderators: jestingrabbit, Moderators General, Prelates

lordofthesnails
Posts: 19
Joined: Sat Jun 14, 2008 12:25 am UTC

### Pirate Game with New Priorities

Right, sorry if this has been posted but I couldn't find it. We all know the standard pirate game, but the rules have always taken for granted that the pirates prioritise in order:
1. (L)ife
2. (C)oins
3. (K)illing other pirates

(LCK for short)

The pirates I've read about don't necessarily value their own life or coins very highly, it seems possible they may be more interested in killing other pirates, so I propose we switch up the orders, as a salute to https://xkcd.com/1613/.

The original puzzle (thanks to Poker/Wardaft) is here:
Spoiler:
Five pirates find a treasure of gold and have to divide it among themselves. The most senior/fiercest/whatever pirate proposes a division, and if it does not get 50% of the vote, that pirate is killed and the next pirate makes a proposal.

There are five pirates (in order, A, B, C, D, and E) dividing a treasure of 1000 gold. As before, the pirates will vote on pirate A's proposal first, followed if it fails by pirate B's, and so on, with the majority deciding, and the current proposer breaking any ties. All pirates first and foremost want to survive, and given that, want to get as much gold as possible (on average, if probabilities are involved), and given that, want to kill as many pirates as possible, and will never go against these priorities. If there are still two or more best options after those three criteria are considered, pirates will decide in an effectively random manner, beyond the realm of any possible deal-making or predictability on the part of the other pirates. Oh, and to close off a potential loophole, all of the facts in this paragraph are common knowledge between all pirates - and I mean the kind of infinitely-layered common knowledge that is not present in the Blue Eyes problem.

How will the gold get divided?

It seems we have 6 scenarios:
1. LCK
2. LKC
3. CLK
4. CKL
5. KLC
6. KCL

I'm interested in which proposal gets accepted, and what the breakdown is. I believe I have an answer to each and would like to validate them, but most importantly I quite enjoyed thinking about the chaos perfect-logicians-who-don't-value-their-own-lives can create, and wanted to share it.

(I didn't want to include the occasionally cited 4th priority tiebreaker of "randomness" as being anywhere but 4th, but for a bonus point (or 24), feel free)

Poker
Posts: 60
Joined: Fri Jun 15, 2007 2:33 am UTC
Location: Multiple Places (only one now that you read this...)

### Re: Pirate Game with New Priorities

I was going to claim that we can rule out any case where C comes before L, because L needs to happen in order for C to go above 0, but that's not quite right - there is one change: now the pirates no longer care about living if they're not going to get any gold, at least not immediately. We can still rule out KCL and CLK, since in a situation where a pirate gets 0 coins, the next priority is living, but CKL cannot necessarily be eliminated like this.

For KLC:

Spoiler:
Every pirate wants to kill as many other pirates as possible, first and foremost, even beyond surviving. Therefore, every pirate will vote against every proposal, including his/her/pronoun own if allowed. So if it's forbidden to vote against your own proposal, then A, B, and C will be killed and D will get all the gold (because if the killing is going to stop there and D's making the proposal, D may as well get all the gold), while E gets nothing. On the other hand, if you are allowed to vote against your own proposal, D will also die, E will propose to get all the gold, reject it, and jump suicidally into the sea. Bad news is, this case isn't very interesting. Good news is, every pirate got their first desire - the maximum number of pirates died!

For LKC:

Spoiler:
Now the pirates still care about killing other pirates more than about getting coins, but not at the cost of their own lives. When it gets down to D and E, D will simply take all the gold (may as well at that point). C wants to live, but unlike in the traditional LCK case, C can't pay anybody off to survive, since gold is less important than killing pirates. So C dies. Now, when it gets to B, C knows that if B dies, C will also die, and since C wants to live the most, C will approve of any of B's proposals. With C's vote causing a tie, B's vote will break the tie, so any proposal of B's should pass, so B may as well take all the gold. Unfortunately for A, since everybody will live in the B case, everybody wants to kill more pirates, so A is doomed to die. This pattern continues for any number of pirates - the first one dies if there are and odd number of pirates and nobody dies otherwise, with the entire spoils going to the highest-ranking surviving pirate. More interesting than the KLC case, to be sure, but still less interesting than the traditional puzzle - the gold always goes to the same pirate.

For CKL:

Spoiler:
This is the original puzzle, except if pirates gets 0 gold they'll always prefer to kill other pirates, even if it means they die as well. However, this fact is never used in the original puzzle, so there is no change. The only case in which this may matter is if the puzzle were changed so that ties were a loss for the proposal: in this change, the original puzzle would see D die if it came to that, so D would support C's proposal even with no gold, whereas with this set of priorities D will still require 1 gold to not kamikaze on C.

lordofthesnails
Posts: 19
Joined: Sat Jun 14, 2008 12:25 am UTC

### Re: Pirate Game with New Priorities

I had somewhat intended K to be killing other pirates, but you did cover that. I agree with all the above, with the exception of:
Poker wrote:For LKC:

Spoiler:
Now the pirates still care about killing other pirates more than about getting coins, but not at the cost of their own lives. When it gets down to D and E, D will simply take all the gold (may as well at that point). C wants to live, but unlike in the traditional LCK case, C can't pay anybody off to survive, since gold is less important than killing pirates. So C dies. Now, when it gets to B, C knows that if B dies, C will also die, and since C wants to live the most, C will approve of any of B's proposals. With C's vote causing a tie, B's vote will break the tie, so any proposal of B's should pass, so B may as well take all the gold. Unfortunately for A, since everybody will live in the B case, everybody wants to kill more pirates, so A is doomed to die. This pattern continues for any number of pirates - the first one dies if there are and odd number of pirates and nobody dies otherwise, with the entire spoils going to the highest-ranking surviving pirate. More interesting than the KLC case, to be sure, but still less interesting than the traditional puzzle - the gold always goes to the same pirate.

I think:
Spoiler:
the fourth-lowest pirate reigns supreme

Poker
Posts: 60
Joined: Fri Jun 15, 2007 2:33 am UTC
Location: Multiple Places (only one now that you read this...)

### Re: Pirate Game with New Priorities

lordofthesnails wrote:I had somewhat intended K to be killing other pirates, but you did cover that. I agree with all the above, with the exception of:
Poker wrote:For LKC:

Spoiler:
Now the pirates still care about killing other pirates more than about getting coins, but not at the cost of their own lives. When it gets down to D and E, D will simply take all the gold (may as well at that point). C wants to live, but unlike in the traditional LCK case, C can't pay anybody off to survive, since gold is less important than killing pirates. So C dies. Now, when it gets to B, C knows that if B dies, C will also die, and since C wants to live the most, C will approve of any of B's proposals. With C's vote causing a tie, B's vote will break the tie, so any proposal of B's should pass, so B may as well take all the gold. Unfortunately for A, since everybody will live in the B case, everybody wants to kill more pirates, so A is doomed to die. This pattern continues for any number of pirates - the first one dies if there are and odd number of pirates and nobody dies otherwise, with the entire spoils going to the highest-ranking surviving pirate. More interesting than the KLC case, to be sure, but still less interesting than the traditional puzzle - the gold always goes to the same pirate.

I think:
Spoiler:
the fourth-lowest pirate reigns supreme

Yeah, good point, I jumped to conclusions:

Spoiler:
If we go up to six pirates, the second highest would support the highest, as before, but that's only two votes, which is not enough to force a tie. My bad.

measure
Posts: 127
Joined: Sat Apr 04, 2015 4:31 pm UTC
Location: Time-traveling kayak

### Re: Pirate Game with New Priorities

Poker wrote:
lordofthesnails wrote:[For LKC] I think:
Spoiler:
the fourth-lowest pirate reigns supreme

Yeah, good point, I jumped to conclusions:

Spoiler:
If we go up to six pirates, the second highest would support the highest, as before, but that's only two votes, which is not enough to force a tie. My bad.

I'm not so sure.
Spoiler:
At eight pirates, the top four would be incentivised to maintain the status quo lest they die when it collapses to four.

I think each power of two is stable, and other numbers reduce downward until they reach one. (8 is stable, 7,6,5 -> 4 is stable, 3 -> 2 is stable, 1 is stable)

lordofthesnails
Posts: 19
Joined: Sat Jun 14, 2008 12:25 am UTC

### Re: Pirate Game with New Priorities

oh SNAP

Gwydion
Posts: 336
Joined: Sat Jun 02, 2007 7:31 pm UTC
Location: Chicago, IL

### Re: Pirate Game with New Priorities

This is also what happens in the original game when there are many more pirates than coins. If we assume there are P pirates and C coins, where P>2C, then stable groups form when (P-2C) is a power of 2, with those highest-ranking pirates getting no coins but voting for their own survival, knowing that if the current proposal fails then so will theirs.

With too many pirates, none of the high-ranking ones are getting any coins anyway so the situation is essentially the same between LKC and LCK at the high end.