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:

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

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