The St Peterburg game.

[Peter van Velzen]

I watched another youtube movie by Professor Wildberger, in which he presented this game.

This game runs as follows: We flip an unbiased coin and if after n flips we get the first head, the player gets payed 2 dollars raised to the (n-1)th power. So if the first head shows up at the first flip you get $1, if it shows up at the second flip you get $2, if it comes at the third flip you get $4 etcetera. The person offering the most to play the game get’s to play it. How much can one offer to play such a game?

Using mere probability one would calculate the game was worth an infinite amount. Of course in real life this is not so. For we need to assess how much money there actually is!

For if the person paying out has no more than $30 then that is all you can expect to win plus – of course – the money you are offering to play the game! Let’s say you offer $2.

Then your possible outcomes are such:
Flip1: 0,5 x $1 = $0,50
Flip2: 0,25 X $2 = $0,50
Flip3: 0,125 x $2 = $0,50 etcetera

That is only up to and including flip 6 (which would yield $32)
After that all you are getting is $32 however long it takes.
That means that after flip 6 which is still worth $0,50 the probable wins are going down drastically
Flip 7 : $0,25
Flip 8 : $ 0,125 etcetera.

This means that all the flips beyond 6 together (even though there may be an infinite number, will but bring you no more than $0,50 all combined. So in this case the game is only worth $3,50!

But if you pay more , the payup will also increase. Should you offer $4 then?
If you do, that won’t change the first 6 flips, but from flip 7 on there is a difference. Flip 6 will get you not $0,25 but $0,265 so the game is worth $3,53. So NO you should not offer more than that.

Now if you play against Bill Gates the game is actually worth more, as he can pay you 30 billion.
Lets take the 2 log of 30 billion that will be a bit over 34. So on flip 34 he will pay you 17,2 billion and on flip 35 he will pay you 30 billion dollar (that’s all he has).
So the first 34 flips will be worth $17.
Flip 35 will be worth $0,43 and all flips up from flip 35 combined will be worth $0,87.
The game against Bill Gates is thus worth $17,87! Even if you play against 30 billion Bill Gates’s there would be no more in it than about $35!

And that is only true if you care for having more than $30.000.000.000

[Dat beloof ik]

It's getting late and I just finished several nightshifts, so please forgive for thinking not so clearly.
But.
It seems to me you are confusing exponentation and multiplication.

This part doesn't make any sense to me :

Flip1: 0,5 x $1 = $0,50
Flip2: 0,25 X $2 = $0,50
Flip3: 0,125 x $2 = $0,50 etcetera

Flip 3 should pay me 2 to the 2nd power.
That makes 4 to me, not 50 cents.

[Peter van Velzen]

It's getting late and I just finished several nightshifts, so please forgive for thinking not so clearly.
But.
It seems to me you are confusing exponentation and multiplication.

This part doesn't make any sense to me :

Flip1: 0,5 x $1 = $0,50
Flip2: 0,25 X $2 = $0,50
Flip3: 0,125 x $2 = $0,50 etcetera

Flip 3 should pay me 2 to the 2nd power.
That makes 4 to me, not 50 cents.

The chances of getting heads on the first flip are 1/2
The chances of not getting heads on the first flip but getting heads on the second flip are 1/4
The chances of not getting hears on the first or the second flop but getting on the third flip are 1/8
So if you multiply the probability of the event happening with the amount you get, the result is always $0,50
That is : Until that amount is maximized by the assets available,

I will explain again. Only the first head that comes op leads to payment. What would happen if you would flip again thereafter is irrelevant. So

H T T will give you $!
H T H will give you $1
H H T will give you $!
H H H will give you $1
T T T will give you nothing yet
T T H will give you $4
T H T will give you $2
T H H will give you $2
8 flps will sum up to $12
So on average you get $1,50 plus what happens on the follwing flips. (That is in the case of T T T).
Try this for 4 flips an you get $2 plus what happens on the follwing flips.
etcetera.