
The pirate game - dbalan
https://en.wikipedia.org/wiki/Pirate_game
======
cousin_it
Steven Landsburg has pointed out that the problem is a bit underdetermined,
and the usual solution is far from unique. For example, "the first pirate gets
all the coins and everyone else votes yes" is a subgame-perfect equilibrium
which is just as good as the usual solution, because no pirate has an
incentive to deviate unilaterally (one vote can't change the voting outcome).
There are many other equilibria like that. The moral of the story is that
games with more than two players seldom have unique solutions, unless you make
very strong and explicit assumptions.

Another problem which puzzles me even more is dividing a cake among three
people by majority vote. Let's say Alice and Bob make an agreement where each
of them gets half of the cake and Carol is left out. But that's unstable,
because Carol can offer Bob a different agreement where Bob gets 60%, Carol
gets 40%, and Alice is left out. Since Bob gets 10% more than before, he has
an incentive to switch. But that's unstable as well, because now Alice can
make a similar offer to Carol, etc. In the end there's no possible setup that
everyone will stick with. As far as I know, this problem is still mostly open
(though a few advances have been made).

To sum up, the idea of "perfectly rational" decision-making is surprisingly
difficult to nail down. It's been definitively solved only for the case of
two-player zero-sum games. When the game is not zero-sum, you get
complications like bargaining and equilibrium selection, and when you have
more than two players, you get an explosion of complexity without any clear-
cut answers.

~~~
lmm
> Steven Landsburg has pointed out that the problem is a bit underdetermined,
> and the usual solution is far from unique. For example, "the first pirate
> gets all the coins and everyone else votes yes" is a subgame-perfect
> equilibrium which is just as good as the usual solution, because no pirate
> has an incentive to deviate unilaterally (one vote can't change the voting
> outcome). There are many other equilibria like that. The moral of the story
> is that games with more than two players seldom have unique solutions,
> unless your assumptions are very strong and explicit.

I don't follow. If pirate A assigns themself all the coins, the other pirates
may as well vote against the plan - doing so risks nothing, and might result
in them getting more than 0 coins if others happen to also vote against.

~~~
cousin_it
Right, so you need a more restrictive equilibrium concept than subgame-perfect
equilibrium. It's a good math exercise to try fleshing out exactly which
concept is implied by your common sense reasoning (which I'm not arguing
against!) Some folks have suggested trembling hand equilibria and other ideas,
none of which are obvious from a naive reading of the problem.

Maybe it's more helpful to think about such problems in terms of "how would I
write a program to solve all problems in this class?", rather than "how would
I act using common sense in this particular problem?" That often makes things
clearer.

~~~
darkmighty
> "...no pirate has an incentive to deviate unilaterally (one vote can't
> change the voting outcome)..."

I would strongly challenge that assumption as having any realism (even for
perfectly rational agents and no communication): if an agent decided to vote,
he could assume all other perfectly rational agents (another usual assumption)
would also take the same decision (by symmetry), so he can safely vote yes.
This symmetry could only be broken if the agents have access to randomness.

~~~
cousin_it
Right, so we need to figure out what kinds of reasoning by symmetry are
allowed under perfect rationality (in all possible games, not just this one).
For example, should a rational person cooperate in the Prisoner's Dilemma
because the opponent will be forced to cooperate by symmetry? In that case,
won't irrational people eat rational people for breakfast, thus devaluing the
idea that rational behavior maximizes utility?

If you take the time to think it though, you might well reinvent
superrationality, updateless decision theory and other fascinating things. In
fact, you'll quickly get to questions that I have no idea how to solve!

~~~
darkmighty
Yes I don't think this assumption makes a lot of sense in general if you don't
assume your perfectly rational players have access to randomness (I do think
it's the best assumption if the players can't communicate or have any
entropy).

In the case of Prisoner's Dilemma, if the cooperate/defect outcome had
sufficiently large cooperation bonus for fixed other payoffs (large enough
T>>R in wikipedia's notation), then with my assumptions it's clear that each
prisoner should flip a coin to decide, and each one gets ~T/4 expected payoff.

But the most glaring problem is of course assuming every player is perfectly
rational. Personally I'd only assume that if your players are all game
theorists with enough of time and paper :) I probably wouldn't even assume
myself as rational.

In conclusion, I believe equating maximin with optimal play/perfect
rationality is misguided, but maximin is a good safe bet.

------
aakilfernandes
I really hate questions like this. Since they mistake rationality for
computer-like adherence to a set of rules.

——————————————————————

Imagine a 100 rounds of ultimatum games. In each round, Alice proposes a split
of $1 and Bob can either accept or reject. If Bob rejects, the $1 is burned in
a fire. How much money will each end up with?

Lets work backwards, as we do in the pirate game.

In ROUND100, Alice knows Bob will take whatever non-zero offer she makes. She
can offer him 1 cent and Bob will agree.

In ROUND99, Alice knows that Bob can will accept whatever non-zero offer she
makes in ROUND100, so she can make whatever non-zero offer she wants this
round without fear of repercussion. She offers him 1 cent and knows he will
agree.

Continuing to work backwards, we find that in all 100 rounds, Alice takes 99
cents and offers Bob 1 cent. Alice ends up with $99 and Bob ends up with $1.

——————————————————————

This is obviously (and experimentally proven) not what would happen in real
life. In experimental outcomes, Bob rejects offers that are too low to
broadcast that he is “irrational”. As a result, Bob is able to negotiate a
much higher outcome (around 40%).

So who is more rational? The "rational" Bob who gets $1, or the “irrational”
Bob who gets $40?

~~~
robzyb
I really love questions like this.

The insight obtained from from comparing the computer-like rationality to what
would happen in the real world is truly fascinating to me.

Also, your example of Bob negotiating fits in very well with the theoretical
computer-like rationality. Alice's offer of 1c is equivalent to an offer of 0c
nominal in the hypothetical case.

The real-life 40% is more a commentary of utility of money as opposed to
decision making. I think that you're conflating concepts.

~~~
resu_nimda
What are some such insights to be gained from the pirate example? What did we
learn? How a set of "rational" computers might play out this scenario? How do
we apply that to our decision making? These questions might seem obtuse or
questioning the value of game theory, but I'm just having a hard time
understanding what to take from it.

 _The real-life 40% is more a commentary of utility of money as opposed to
decision making._

Only real-life humans make decisions though. If there is no implied utility of
money then Bob might as well reject all offers. And it does seem that the only
important question is "what would real humans do?" We can set up a simulation
that strictly adheres to a simple set of rules and watch how it plays out, but
what is that telling us?

~~~
the_af
> _What are some such insights to be gained from the pirate example? What did
> we learn?_

It's a logic puzzle. We learned that the intuitive solution of having the
first proposal be "I don't want any money, I just want to live" is too
conservative, and that there is a way better solution for the first pirate.

We didn't gain any deep insight in how to split money between real people,
since those cases seldom involve pure, unemotional logic.

------
ryandvm
This puzzle should be called "The VC game". Real pirates were actually far
more egalitarian. [http://www.newyorker.com/magazine/2007/07/09/the-pirates-
cod...](http://www.newyorker.com/magazine/2007/07/09/the-pirates-code)

~~~
kbutler
I initially thought the parent was being unfair to the VCs, but then
considered policies that exclude other investors, like preferred shares and
buyouts that fall just short of any payout to common shareholders, and so
forth, and realized it's pretty accurate.

------
personjerry
Isn't this our current economic status? The rich (A) become and stay rich by
scoring votes with key voters (C and E). These key voters are kept in line by
intimidation, because they know they could end up worse (like B and D).

~~~
h2077545
In addition B, C, D and E are usually the same people divided by A.

------
afro88
There's a Korean gameshow called "The Genius" which plays on these kinds of
games. It's more socially oriented but the contestants and the "narrator" go
through the mathematics and strategies a bit. Highly recommended.

~~~
aninhumer
I second that recommendation.

The usual format is that they play a game each episode, and the winner and
another player of their choice are guaranteed to go to the next round. The
loser of the main game then plays a one-on-one game with another player of
their choice, and the loser of that game is eliminated. This creates an
interesting metagame surrounding the main games.

One thing that makes it work particularly well is that the participants all
seem to be relatively pleasant people. There were alliances and sudden-but-
inevitable betrayal, but no one was ever unpleasant about it. I'm not sure if
it's a cultural thing, or that they're all notionally "geniuses" (talented in
some field) rather than random people or celebrities, but that atmosphere
really made a good premise into an excellent show.

------
pierrebai
A tantalizing idea I've been trying to spell out properly that goes against
the usual reasoning behind similar problem is that of putting into question
the correctness of causality in hypotheticals.

Each such problem I've read about assumes that we can assume causality while
reasoning about hypothetical, but strangely, if we let go of that assumptions,
then we can arrive at different answers which hinge on otherwise surprising
behaviour. I relate this idea to the fact that in logical reasoning, all true
statements are, once proved, held to be simultaneously true. That is, given if
A then B, with A being true, we don't hold B to be true 'after' A, but to have
been always true, given A.

In the pirate problem, limiting ourselves to three pirates to shorten my
explanation, we end up with the split being: 99 to C, 0 to D, 1 to E. This is
because we assume that if only D and E were left, D would keep 100 to himself.
Now given that distribution (99,0,1), D should now change his hypothetical
proposal to (0,98,2). If we assume that C is not a 'non-causality' believer,
but both D and E to be, then E would vote no to (99,0,1) and D would do the
(0,98,2) split. You may argue that D could then do a (0,100,0) split, but
that's not how a non-causality believer MUST act, because he knows that to get
to that point, he must know thet E can logically know that he will do this
split. This can be justified by arguing that when a pirates survives, he will
enter such gold-splitting game later on. But my argument is subtler than this
and doesn't require it. It basically become this: all such pirates, posited to
be perfectly logical, are interchangeable. Thus they must all reason in the
same fashion. Thus, my argument is that _true_ pure logical minds see that the
true way to maximize their gold profit is to hold a world view that maximize
their profit, even if causality is discarded. Thus both D and E know that tehy
can maximize their profit by discarding their belief in causality. That is how
D ends up proposing (0,98,2) and E accepts it.

Of course, I ended up there by assuming C was not a believer, but given my
argument, C must also be ready to throw causality out of the window, otherwise
he will end up dead. I believe my argument ends up splitting (49,51,0), but
I'm not sure. Once causality is throw out, it's hard to tell, but intuitively,
with three pirates, those voting must be given almost equal gold and the
remaining pirates must not have a majority.

~~~
cousin_it
That's not completely right, but close enough to be dangerous. Indeed, quite a
few researchers around LessWrong, MIRI, FHI etc. agree that the right way to
formalize decision-making should use the timeless view that you advocate.
Decisions and precommitments are provable consequences of an agent's source
code, agents with the same source code can use that fact to coordinate, and so
on. I've done a lot of work on this, so feel free to ask :-)

------
asift
I love game theory, but I hate that the focus outside of academia is almost
always on simplistic single period games. These games can result in some
interesting conclusions and teach some valuable concepts, but they are
terrible at representing how people behave in the real world.

Anyone interested in a more complex and realistic examination of the economics
of pirating should consider Peter Leeson's book, _The Invisible Hook: The
Hidden Economics of Pirates_.

I would also recommend Leeson's _Anarchy Unbound: Why Self-Governance Works
Better Than You Think_ for some interesting applications of game theory in
more realistic historical contexts.

~~~
pavel_lishin
It's not meant to teach you about people or pirates; it's a simplified tool
used to teach concepts and methods of thinking.

Much like a two-body system is useful for explaining gravity, even though
there's virtually no place in the real world where that particular example
would be useful.

~~~
asift
I get that, which is why I said it's useful for teaching, but I disagree with
your notion that game theory is incapable of being anything other than an
educational tool.

When models utilize more realistic assumptions, they can be incredibly
powerful tools for understanding past behavior and predicting future behavior.
Historical evaluation is particularly powerful when game theory is utilized
with the analytical narrative form of analysis.

------
mrfusion
This kind of reminds me of the surprise execution paradox.

[https://en.m.wikipedia.org/wiki/Unexpected_hanging_paradox](https://en.m.wikipedia.org/wiki/Unexpected_hanging_paradox)

------
oilywater
I remember reading about this game on a private music torrent tracker what.cd
and was working on the solution for a couple of hours. I believe there was a
whole thread dedicated to these kinds of problems.

Quite an entertaining problem, skip the result and try it out yourself, the
sense of accomplishment is fulfilling, especially if your mood is shaky.

------
talmand
Let's see if my logic works. It seems some motivations and negotiations are
left out of the proposed solution.

If I were C I would have voted no on A's proposal and try to make a deal with
D and E. The idea is that knowing if A were thrown overboard then B might be
more apt to provide a more beneficial coin sharing program in an effort to
save his life. After all, he just watched the previous guy get tossed
overboard and die. The most I could lose would be 1 coin if B decided to try
A's method on D and E and they went along with it.

The key is that I would have already negotiated a deal with D and E, if A or B
doesn't share the wealth fairly then we vote to toss them. Once I'm in charge,
I'll share the coins equally as possible.

~~~
sokoloff
That fails because one of the explicit stipulations in the problem is: "The
pirates do not trust each other, and will neither make nor honor any promises
between pirates apart from a proposed distribution plan that gives a whole
number of gold coins to each pirate."

I read that as "an actual proposal, by the leader" not as a "theoretical
future proposal, iff that negotiator becomes the leader" and I believe that's
the intended reading.

Thus, D and E cannot trust that you'll hold your word to give them 33 coins
each provided you get to be in charge.

~~~
talmand
If that were true then the whole thing falls apart as soon as the first
proposal is made. They all have to trust that A will follow through with the
proposal, of which they apparently do not.

But yes, I glossed over that part apparently.

~~~
the_af
They do trust A (the first proposal). If all of them accept A's proposal, they
are all committed to honoring it. What they cannot trust is any hypothetical
proposals made by a pirate whose turn is not to make the current proposal.

"I have the turn and I propose we split this way": trusted.

"I don't have the turn yet, but when it's my turn, I promise I'll propose this
/ kill him / won't kill you": not trusted.

------
andyraskin
I was asked to solve this puzzle once during a project management interview.
The solution here says "work backwards," but I found that "work recursively"
was a more helpful (and instructive) way to think about it.

~~~
tamana
The are very similar. Recursion is expressing a problem in terms of a smaller
case of the same problem. Working backwards/induction is building up from a
base case.

~~~
andyraskin
Recursion is exactly the same (as induction) because it also requires
specifying what to do in a base case. I guess I think recursion is a more
instructive term than "working backwards" because it speaks to a rigorous
programmatic approach.

~~~
hisham_hm
Wikipedia is not written for programmers only.

------
blobbers
The results of the pirate game seem a lot like the stock allocation at my
company. A billion dollar acquisition and Pirate A figuring out how to
distribute the coins... guess where they end up!

Note to self: don't get thrown overboard.

------
pratyushag
I think this is a great game and a portrayal of reality where a small number
of people get most of the wealth, a 50% number are middle class with enough to
be just okay (or at 1) and with a still a very large number who can barely
have a comfortable life (say $4k/month household income).

------
Pxtl
Shouldn't it be E: 2 in the solution?

If pirates, all things being equal, prefer violence to peace (which is
implied) then E getting 1 from A is less preferable than killing A and B and
getting 1 from C.

Or is it that E knows that B will be offering next, and B will offer him 0, so
it makes sense for him to accept A's offer of 1?

~~~
vog
_> Or is it that E knows that B will be offering next, and B will offer him 0,
so it makes sense for him to accept A's offer of 1?_

Yes. That's why it is sufficient to offer E: 1.

------
archibaldJ
Reminded me of the game theory course by Stanford on Coursera. It was in the
exercise on [backward
induction]([https://en.wikipedia.org/wiki/Backward_induction](https://en.wikipedia.org/wiki/Backward_induction)),
if I remember correctly.

------
mcv
It's a nice demonstration of how the lack of trust ultimate screws everybody
except the leader.

------
drelihan
I was asked this question in an interview 10+ years ago. I thought it was a
really good question to promote logical discussions. Lots of branches and
extensions you can add to see how people think. Really fun.

------
dsego
If you like this type of puzzle, I can recommend the book: “How Would You Move
Mount Fuji?: Microsoft's Cult of the Puzzle”.

------
quickpost
This game reminds me of some employee stock option negotiations I've been on
the lessor end of...

------
efnx
These outcomes assume you can live rich (having ~98 coins) in the same space
as poor pirates without them stabbing you and taking your coins! Real pirates
would distribute more evenly for fear of death after distribution.

------
ALee
Well now I have to choose a new interview question.

------
logfromblammo
Now that you have that down, allow the pirates to secretly bribe each other
out of their own savings before the vote. If multiple votes occur, previous
bribes are added to a pirate's savings. If a pirate is thrown overboard, his
remaining personal savings are looted and added to the pool of coins to be
divided. The personal savings of each pirate are secret from all the other
pirates, but all pirates always have more in savings than the next pirate in
order of decreasing seniority.

According to the pirate's code, a bribe is a binding contract, but only to the
extent that breaching the terms of the contract results in returning the
amount of the bribe _after_ the breach occurs. Otherwise, the bribed pirate is
thrown overboard. A pirate may therefore spend the bribe money before
reneging, if reneging will still yield enough money to pay back the bribe
afterward.

Pirate P[x] has savings s[x], and s[x] > s[x+1].

The degenerate case where n=1 is easy to deduce. P[1] proposes {pool}, votes
yes, and wins.

The case where n=2 is also easy. P[1] proposes {pool, 0}, votes yes, and wins.

When n=3, it gets more complicated. P[1] needs one more vote to win. If P[2]
is able to propose a split, he will be proposing {0, pool + s[1], 0}. So the
cost of P[2]'s vote is at least pool + s[1] + 1, which is normally impossible
to achieve for P[1]. But since P[2] does not know how much s[1] is, other than
that it is more than s[2], P[1] might be able to risk it. But since P[1]
doesn't know s[2], bluffing a lower amount for s[1] is risky, as if the bluff
amount is lower than s[2] + 1, then P[2] will immediately know to vote no. The
cost of P[3]'s vote is at least 1. P[1] and P[2] will therefore be bidding
competitively for P[3]'s vote. P[1] has the choice of offering value to P[3]
publicly in the proposal, or secretly as a bribe. Either way, P[1] is likely
to propose 0 for P[2].

So if P[1] proposes { pool, 0, 0 }, and bribes P[3] to vote yes with s[1],
P[2] might try bribing P[3] with any amount from 1 to s[2] to vote no, with no
effect. The net effect is { +pool -s[1], 0, +s[1] }.

Remember also that any bribe P[1] pays to P[2] _could_ be added to a bribe
from P[2] to P[3]. If P[1] bribes 1 to P[2] to vote yes, and s[1]-1 to P[3] to
vote yes, and P[2] bribes s[2]+1 to P[3] to vote no, then if P[2] and P[3]
both vote no, P[1] is thrown overboard for losing the vote, and P[2] is thrown
overboard for reneging on a bribe and not having the coin to pay it back, so
P[3] gets everything. If P[3] suspects that P[1] may have bribed P[2] to vote
yes, and knows that P[2] bribed him to vote no--which would be useless unless
P[2] himself intended to vote no--then P[3] may vote no on the possibility of
getting pool + s[1] + s[2], which would otherwise be impossible for him. So
knowing this, P[1] may be confident that any bribe to P[2] to vote yes _would
not_ reasonably be re-bribed to P[3] to vote no.

So P[1] could bribe 1 to P[2] to vote yes, tell P[3] that he had bribed P[2],
tell P[2] that he had told P[3], and then propose {pool - 1, 0, 1}. It would
be useless for P[2] to bribe P[3] unless he intended to reneg, he can't add
P[1]'s bribe to his own, and he knows that s[1] - 1 >= s[2]. With the 1 from
the pool, he could not match such a theoretical bribe anyway. So the net
effect is now { +pool -2, 1, 1 }.

I'm not exactly sure if that is a stable solution or not.

I think maybe that P[3] might be able to get more by bribing P[1] to vote no,
or P[2] to vote yes.

------
vacri
If the pirates were that strictly rational and interested in their own
survival, they wouldn't be pirates. They'd have a nice, safe job, free from
threat of death or injury.

~~~
mikeash
I don't normally go for pop culture quotes here, but this one seems rather
appropriate:

Oh, get a job? Just get a job? Why don't I strap on my job helmet and squeeze
down into a job cannon and fire off into job land, where jobs grow on little
jobbies?!

~~~
vacri
Yeah, and while you're doing that, why not have a chat to the pirates so
bloodthirsty to kill each other, but with such a code of honour that they're
satisfied with the booty split which has 98% of the money going to only 20% of
the crew.

Why are you so disturbed about the ridiculousness of my comment, yet quite
happy to take the ridiculousness of the situation's premise?

~~~
mikeash
What gives you the impression that I'm "disturbed"?

~~~
vacri
Probably an artifact of seeing a bizarre response in general to the comment,
really. The tone of the quote also doesn't help, as it sounds pissed-off.

