A few days ago Dave Radcliffe (@daveinstpaul ) tweeted an interesting problem:
Consider this game. Start with $100 and toss a coin repeatedly. If heads, the pot is increased by 50%, otherwise it is decreased by 40%. Would you be willing to play this game and if so for how many rounds?
It’s a tempting proposition. We stand to gain more than we stand to lose and there are equal odds of winning and losing.
I thought at first that the answer is “Yes and for as many rounds as possible!”. Curiously it doesn’t quite work out that way.
Recently in Australia there has been a debate about mandatory pre-commitment. Should gamblers be forced to determine a limit on the amount of money they will gamble before they start gambling? Will this help problem gamblers?
Dave Radcliffe’s problem implies a type of pre-commitment. We pre-commit to a certain number of rounds of the game.
Suppose we pre-commit to two rounds. There are four equally possible outcomes:
- Two heads, in which case we finish with $225
- One head, then one tail, in which case we finish with $90
- One tail then one head, in which case we also finish with $90
- Two tails, in which case we finish with $36
The expected value is the average of these which is $110.25. So the expected return is positive ($10.25) but notice that the odds of a positive return are 1 in 4.
No matter how many rounds we pre-commit to the expected return will be positive. In fact, the more rounds we pre-commit to, the greater the expected value (100*1.05^n, where n is the number of rounds we pre-commit to).
But if we are only allowed one game it would be unwise to pre-commit to too many rounds.
Here’s another way of looking at it. If the coin shows a head, followed by a tail, we multiply the money we start with by 1.5*0.6. In other words we multiply by 0.9 .
If we play many rounds, then we might multiply the money we start with by:
depending on the order of heads and tails.
The more rounds we play of the game the more likely it is that there will be a roughly equal number of heads and tails. So when we rearrange the terms in this long product we are likely to end up with a large number of pairs like this:
There may be a few extra 1.5′s at the end or a few extra 0.6′s but in a long game with many rounds we would expect there to be a roughly equal number of heads and tails.
So now we have
0.9 * 0.9 * 0.9 ……..
… with a few extra factors at the end. As we play more and more rounds there are probably quite a lot of these 0.9′s which multiply to give quite a low factor so we very probably end up losing most of our money.
So the more rounds we play, the greater the expected return AND the lower the probability of a positive return.