February 23, 2002

A Mailing List Thread...

2002-02-23: One Thread...

Rich Baker:

There is a room with a machinegun and a guy with two dice. A person is
taken into the room and the dice thrown. If they come up with two sixes
then the person is shot. Otherwise he or she is let out and a group of
ten people brought in. Again the dice are thrown and if they both come
up sixes then the people are shot. Otherwise a group ten times bigger
is brought into the room. This "game" goes on until a group is shot.
(There is an infinite supply of people. Nobody goes into the room
twice.) If you're taken into the room, what is the probability that you
get out alive?

Argument 1. You are killed if two sixes are thrown. This happens 1 in 36
times. Therefore your chance of getting out alive is 35/36 = 97%.

Argument 2. Most people who are taken into the room are killed,
therefore you are very likely to die. For example, suppose the third
batch are killed. Then 100 people who go into the room die and 11
survive. The chance of getting out alive is then 11/111 = 9.9%.
(Working out the true probability is left as an exercise.)

Which of these arguments is true? What's wrong with the other one? (You
can make the situation even more extreme by making the probability a
batch is shot even lower.)


Brad DeLong:

You do realize that the fact that people believe in argument (1) is
the key reason why the stock market goes insane from time to time?
And that the more people believe argument (1), the more likely we are
to have an insane stock market bubble followed by a catastrophic
crash?


Rich Baker:

No, I didn't realise either of those things. How does that work?

Rich, who believes argument (1).


Brad DeLong:

Everyone thinks that the chance that the stock market bubble will
collapse in the next month is small, therefore I should buy. No one
thinks that the fact that I am thinking of buying indicates that we
are near the end of the bubble.

As with so many things in statistics, probabilities differ because
they are based on different information sets--what the probability is
depends on what you know about the different alternate histories that
may flow forward from your particular situation.

If what you know is that you have been herded in the room and the guy
with the machine gun is rolling the dice, then in 35 of the 36
histories to follow you breath a sigh of relief and exit the room.

If what you know is that a bunch of people have been and will be
herded into this room, and that of those herded into the room about
90% die, then your view of the histories to follow looks very bleak.

If you know, in addition, that you are in the 5th round of this
"game"--that there have only been four rounds before you--then your
view of the histories to follow turns optimistic again.

If you don't know what round of this game you are in, but if someone
able to look in the future tells you "it's interesting: this play is
only going to last for five rounds" then you turn bleak again.

And if you know, in addition, that the causally-connected universe is
not infinite in either space or time, then you conclude that the game
cannot be played at all. After all, expected deaths in round 1 are
0.0278 of a sophont; expected deaths in round 2 are 0.27 sophonts;
expected deaths in round 3 are 2.63 sophonts; if you get to round 490
(and there is a 1/3 chance that you will) expected deaths are 9.26 x
10 ^ 36 sophonts--meaning that you have to have a room large enough
to hold 10 ^ 38 sophonts and a... rather large machine gun at your
disposal.

Indeed, such games can only be played by Devis that can create
universes infinite in duration and extent for their pure amusement,
and juggle infinities like the Flying Karamazov Brothers juggle
beanbags. A simple God who can make a square circle can't do it.

GCU Ritu?

GCU St. Petersburg Paradox


Brad DeLong:

Oooops! That was supposed to be round 40, not round 490.

In round 490, your expected deaths (as of round zero) are 2.89 x
10^481, and you need a room (and a machine gun) large enough for
10^489 sophonts...

GCU Red Faced!


Ritu Ko:

>GCU Ritu?

The arrangements are simple to make, but I need to find a reason to do it.
And, no, watching humans find new ways to kill themselves is *not* fun.

>GCU St. Petersburg Paradox

Which one is this?

Ritu
GCU The Devi who believes in argument [2]


Brad DeLong:

>>GCU St. Petersburg Paradox
>
>Which one is this?


A guy flips a coin repeatedly. If it comes up heads the first time,
he pays you 1 ruble and the game ends. If it comes up tails the first
time and heads the second, he pays you 2 rubles and the game ends. If
it comes up tails the first two times and heads the third, he pays
you 4 rubles and the game ends. Et Cetera. Et Cetera.

How much should you be willing to pay for the opportunity to play
this game? Your expected winnings are, after all, infinite.


GCU What Happens to a Vizier Who Persuades His Sultan to Give Him One
Grain of Wheat on the First Square of a Chessboard, Two Grains of
Wheat on the Second, Four on the Third, Eight on the Fourth, and So
On, Anyway?


Ritu Ko:

>How much should you be willing to pay for the opportunity to play
>this game? Your expected winnings are, after all, infinite.

I'd pay up to the cost of a book [somehow this has long been my limit for
deciding if I wish to indulge a non-essential whim.] If it cost more, I'd
pass.


>GCU What Happens to a Vizier Who Persuades His Sultan to Give Him One
>Grain of Wheat on the First Square of a Chessboard, Two Grains of
>Wheat on the Second, Four on the Third, Eight on the Fourth, and So
>On, Anyway?

Birbal never got the wheat. Akbar agreed but discovered to his extreme
mortification that there was not enough wheat in his entire empire.

Ritu
GCU Birbal was cool


Martin McGrane:

I think what we're overlooking here is that while there is an unlimited
supply of people, nothing has been said about the supply of bullets. Even
if there is an infinite supply of bullets and maintenance for the
machinegun is considered, there is still a finite rate of fire for the gun
which means that in larger groups people will be dead of old age before a
bullet reaches them.

Of course, if the size of the room is finite then at some point people
will start to die from the crush and from that point on the infinite
masses die with abandon.

If the game goes on for too long another problem arises. As the mass grows
too large in this finite space we start to form a black hole (or perhaps a
gravastar).We now have problems getting information past the event
horizon. How do we know if the game is finished if no information can
leave the room?

Should we keep sending people in? I think this is the more important
question. Remember that we have an infinite supply of people so whatever
amount we send in is inconsequential in comparison to what remains.

Martin
ROU Always Helpful


Brad DeLong:

>I think what we're overlooking here is that while there is an unlimited
>supply of people, nothing has been said about the supply of bullets. Even
>if there is an infinite supply of bullets and maintenance for the
>machinegun is considered, there is still a finite rate of fire for the gun
>which means that in larger groups people will be dead of old age before a
>bullet reaches them.

Ooooh! Well done!

>If the game goes on for too long another problem arises. As the mass grows
>too large in this finite space we start to form a black hole (or perhaps a
>gravastar).We now have problems getting information past the event
>horizon. How do we know if the game is finished if no information can
>leave the room?

Ooooh! Ooooh! Very well done indeed!

>
>Martin

Demonstrating once again that appropriate physical background
knowledge is essential for successful problem solving...


GCU Martin Wins!

Posted by DeLong at February 23, 2002 04:41 PM | TrackBack

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