Can such things be?

Thanks to all who replied to my earlier 'tossing and turning' query. As you may have gathered, I don't know very much about this stuff, and your answers were most illuminating. It appears that, in order to select fairly among n possibilities, you might (but presumably with probability 0) have to toss a coin an infinite number of times unless n is some power of 2.

Well, suppose we play the following game. I spin a fair coin; if it lands heads, I pay you £1. If it lands tails, I spin it again; this time, if it lands heads, I pay you £2. If it doesn't, I pay you £4 for heads on the next spin, else £8 on the next, else...

What stake should I ask you to put up for the privilege of playing this game against me?

David Burn London, England

the following game. I spin a fair coin; if it lands heads, I pay you £1. If it lands tails, I spin it again; this time, if it lands heads, I pay you £2. If it doesn't, I pay you £4 for heads on the next spin, else £8 on the next, else... What stake should I ask you to put up for the privilege of playing this game against me?'

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I wouldn't call you greedy if you asked me to put up an infinite sum to

If you are a millionaire, then the proper stake is about £11. You have a 1/2 chance of paying £1 (worth 50p), a 1/4 chance of paying £2 (worth 50p), ..., a 1/2^20 chance of paying £2^19=524,288 (worth 50p), and if the coin lands tails 20 or more times, you'll go bankrupt paying your full million (worth 1,000,000/2^20, just under £1).

I had to find out what 'tosser' might mean in the UK (I am in the US). I found it at http://www.dict.org/bin/Dict with a query string of tosser. The main interest here is that The DICT Development Group can be a good source for research.

The calculated expected return is, of course, infinite. This is mitigated, however, by the fact that you cannot pay my winnings beyond a certain amount. There is not infinite money to distribute. Eliminating the payouts above a given amount leaves this game with a finite expected value.

Accounting for my non-linear valuation of money and my risk tolerance, I could give you an exact stake that would make playing or not playing equivocal for me.

If we are just playing for theoretical numbers, I would bid any amount to play the game.

Cheers, Eric