Suppose there's a game in which you have a 45% probability of winning. Let's make it a sedentary game, unlike tennis, so that you don't tire and the probability is always 45%.
Someone proposes a metagame: play a series of the original game, and if you win more games than your opponent, you win. But, there's a proviso: It must be an even number of games, and there's no tie-breaker, so again, it's not like tennis.
Your only strategy is to pick the actual, even, number of games. What's the choice of number of games that will minimize your probability of loss in the metagame?
(In reply to
Solution by tomarken)
This is not a surprising result.
If you play 2 games, you need to win 100% of them.
If play 4 games, you only need to win 75% of them.
If you play 6 games, you need to win 66.67% of them.
The more games you play, the closer you get to only needing to win over 50% of them. And initially, the % that you need to win drops pretty quickly. Thus, it is not surprising that initially your chance of winning the metagame improves as the the number of games increases.
On the other hand, your chance of winning an individual game is only 45%. So the more games you play, the less likely it becomes that you can win even 50% of them. Thus, it is not surprising that after some point, more games reduce your chances of winning the metagame.
Edited on October 27, 2020, 11:02 am
re: Solution
