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?
https://www.desmos.com/calculator/2e5w6rlzun
There's probably a better way, but you can see the maximum occurs when x=5.
Edit:
https://www.desmos.com/calculator/no1mnuld9f
Here's an animated version. It uses a floor function for stability and shows how carefully the individual probability needs to be.
Edited on October 27, 2020, 1:04 pm
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Posted by Jer
on 2020-10-27 12:59:09 |
Implementing by Desmos
