Q1
Sequentially choose random variable from uniform distribution , you can pause when you are shown the first number, or go on. If you go on, you can pause when you are shown the second number, or go on. If you skip twice, your chosen number is the third number anyway, otherwise, your number is the number when you chose to pause. If your number is the biggest one out of these three number, you win. Otherwise, you lose. What is your optimal strategy to win this game?
Solution:
assume the first number is , the second number is .
For 1st round:
If keep, win rate = , lose rate = . Thus, when should keep, else, go on.
For 2nd round:
- if
- if go on, win rate =
- if keep, win rate = 0
- if
- if go on, win rate =
- if keep, win rate =
Thus,
- if go on, win rate =
- If keep, win rate =
therefore, when should keep, else go on.
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