A new "prisoner's paradox"
Imagine that you live in a medieval kingdom.
Its ruler, King Seyab, is known for his love
of mathematics, philosophy, and for cruelty.
As a very young king, 40 years ago,
he ordered a group of wise men to take an urn
and fill it with 1000 white and black balls.
The color of each ball was chosen by a coin flip,
independently of other balls.
There is no reason to doubt wise men's honesty
or accuracy in fulfilling king's order.
The king examined the contents of the urn
and filled another urn with 510 black
and 490 white balls. The contents of the two
urns is top secret and the subjects of King Seyab
never discuss it.
The laws of the kingdom are very harsh,
many ordinary crimes are punished by death,
and the courts are encouraged to met out the capital
punishment. On average, one person is sentenced
to death each day.
The people sentenced to death cannot apeal
for mercy but are given a chance to survive by the following strange
decree of the monarch.
The prisoner on the death row can sample
999 balls from the original urn.
He is told that the second urn contains
1000 balls, 490 of which are white.
Then he can either take the last ball
from the first urn or take a single random
ball from the second urn. The prisoner's
life will be spared if the ball turns out
to be white.
Now imagine that you have been falsely accused of squaring
a circle and sentenced to death. You have sampled
999 balls from the first urn. The sample contains
479 white balls. You have been told the contents
of the second urn. Will you take the last ball
from the first urn or sample a single ball from
the second one?
In view of how the balls were originally
chosen for the first urn, the probability of the last ball being white
is 0.50. The probability of sampling a white ball
from the second urn is only 0.49. It seems that
taking the last ball from the first urn is the optimal
However, you know that over 40 years, the survival
rate for those who took the last ball from the first
urn is either 48% or 47.9%. The survival rate
for those who sampled from the second urn is about 49%.
So what is your decision?
Should you accept a free gift?
The number of people sentenced to death in King Seyab's
kingdom is currently so large that the following system
had to be instituted. A prisoner samples 999 balls,
the prison guards note the color of the remaining ball
in the urn and then the prisoner returs to his or her
solitary cell to think about the decision. In this way,
the urn can be reused for the next prisoner while
earlier prisoners are still pondering their decisions.
Imagine that you are a Bayesian decision maker.
You have already seen your 999 ball sample (it contains
479 white balls)
and you have announced your decision to the prison
officials. Your decision was to take the remaining
ball from the first urn. You have not seen that ball yet.
A message from King Seyab has just arrived.
The king says the following.
"I have just learnt that there are 100 Bayesians
in the prison. Each one of them decided to take the
last ball from the first urn but none of them
was shown the ball yet.
I have a special offer for this group of 100 people.
I will spare the life of one randomly chosen prisoner
from this group if all members of the group agree to my terms.
In order to take advantage of this special offer,
the balls have to be reshuffled. In other words,
the prison guards will check the colors of the balls
waiting for each member of the group. If there are
N white balls among these 100 balls, the guards will
randomly (uniformly) assign 100 balls to the memebers
of the group. There will be N white balls among this
new set of balls. Then, I will uniformly choose one of the
black balls and replace it with a white ball.
Since you are a Bayesian decision maker, you will vote
against accepting king's offer. This is because
reshuffling of the balls decreases your chances of survival
by about 2% and king's offer will increase your chances
by about 1%. You know that all other members of the group
will also vote against the deal.
The king's special offer could have been a clear gain
for the group but every member of the group
would vote against it, thinking that he or she would
lose rather than gain.