The following is an interview question.
Question: Given $2$ players, you are player $1$ and another player (player 2).
$(i)$ Both players select a number between 1 and 100 inclusively and the one who tells the bigger number has to pay the lower number to the other player. If both numbers are the same, then player $1$ needs to pay the number to player $2.$ The player $2$ says a random number between 1 and 100. What is the optimal strategy ?
$(ii)$ Now, we assume that player $2$ is smart. Same rules above apply. Which strategy to beat him ?
Since player $2$ selects number randomly, player $2$ has expected value of choosing $50.5.$ So, player $1$ should choose $50$.
For $(ii),$ I have no idea how to start at all.
player $1$ can always pick number $1$ to win. But that is not optimal.