Re: The Pirate Game Theory Problem and the Trouble with Voter Apathy
Re: The Pirate Game Theory Problem and the
Trouble with Voter Apathy
Source: http://sci.tech−archive.net/Archive/sci.logic/2006−08/msg00242.html
• From: "Rupert"
• Date: 4 Aug 2006 03:39:06 −0700
Rupert wrote:
Rupert wrote:
Matthew Wampler−Doty wrote:
There is a well known problem in game theory regarding 10
"Voting
Pirates" and a split of 100 gold pieces of treasure. The
problem may be
presented a variety of ways; I will give one for the purposes
of
demonstration.
Each pirate has a unique rank from 1 to 10. Each round the
highest
ranking (say with rank R) pirate offers a split to the other
pirates,
and they all vote to approve or not (all R of them). The head
pirate
needs at least a 1/2 majority for his split to go through, and if
it
fails to go through, he is killed, and then game is replayed
with one
fewer pirates.
A unit of Gold may not be split. A pirate prefers N+1 units of
gold
over N units, always. A pirate prefers to keep his life over
any amount
of gold.
For a presentation of the "classical" solution to this problem,
I refer
to Wikipedia: http://en.wikipedia.org/wiki/Pirate_game
But another, less palatable solution to this problem goes as
Re: The Pirate Game Theory Problem and the Trouble with Voter Apathy 1
Re: The Pirate Game Theory Problem and the Trouble with Voter Apathy
follows:
All 10 pirates survive, the pirate with rank 10 gives himself
all of the
gold and the others nothing, and they all vote to approve of
this.
Why is this a solution? Because for any one pirate, if they are
voting
to approve, they cannot change their vote unilaterally and get
change
the outcome that the head pirate wins the majority. This
makes this
horrible solution a Nash Equilibrium, technically.
Another solution is the "naive" everyone gets 10 gold pieces
solution,
and no pirate dies. This would happen if the pirates all held
the
contingent strategy that if the pirate with rank 10 didn't split
the
gold evenly, they'd all vote to kill him. Again, no pirate has
any
unilateral power, and the first pirate certainly wants to keep
his life,
so he provides and they all vote conform to the voting
strategy.
One can place more restrictions on the pirates. One could
demand that
they all have complete contingent strategies, for instance. I
will
contend (and prove, if necessary) that having complete
contingent
strategies, and subgame perfect equilibria don't "cure" the
pirates of
their apathy.
In fact, I have yet to find a truly acceptable, formal criteria to
pick
out the classical solution to this problem. Any ideas?
On a side note, the riddle and the problem of apathy came up
in a class
I had under an Economics professor. His solution was to try
to ban
apathy, and demand that agents "vote their preferences." I
was later
surprised to discover that he had written this article back in
2004:http://www.slate.com/id/2107240/
Go figure.
Re: The Pirate Game Theory Problem and the Trouble with Voter Apathy 2
Re: The Pirate Game Theory Problem and the Trouble with Voter Apathy
Matthew P. Wampler−Doty
Suppose a pirate never accepts an outcome when by following a certain
strategy he can enforce a better outcome for himself. Call such a
pirate rational of order 1.
Suppose a pirate never accepts an outcome when by following a certain
strategy he can enforce a better outcome for himself, given that all
the pirates are rational of order 1. Call such a pirate rational of
order 2.
And so on.
Assume the pirates are rational of order n for every positive integer
n.
Then the outcome is the one presented in the Wikipedia article.
I should be a bit clearer about what I mean. When I say a pirate never
accepts an outcome when he can enforce a better one, I mean he never
accepts the *possibility* of an outcome when he can enforce a better
one. Thus the pirates are maximiners.
The pirates are maximiners of order 1. That is, of all the available
strategies, they pick the one such that the worst possible outcome is
as good as possible.
And the pirates are maximiners of order 2. That is, of all the
available strategies which are consistent with being a maximiner of
order 1, they pick the one such that the worst possible outcome is as
good as possible, given that all the other pirates are maximiners of
order 1.
And so on.
This forces the solution in the Wikipedia article.
I don't think we need to assume that the pirates are full maximiners. I
think we just need to assume that death has infinite disvalue (the
pirates will never allow the possibility of death when it can be
avoided) and the pirate will never select a strategy when there is
another strategy that strongly dominates it (is guaranteed to produce a
better outcome).
I shouldn't have said "strongly dominates". I should have said "weakly
dominates". A strategy weakly dominates another if it is guaranteed to
produce an outcome at least as good, and there is a possibility that it
Re: The Pirate Game Theory Problem and the Trouble with Voter Apathy 3
Re: The Pirate Game Theory Problem and the Trouble with Voter Apathy
will produce a better outcome.
.
Re: The Pirate Game Theory Problem and the Trouble with Voter Apathy 4