Question
You and friend play a game where you both select an integer $1-100$. The winner receives $\$1$ from the loser. The winner is the player who selects the strictly higher number. If there is a tie, then nothing happens. However, a player can also win by selecting a value exactly $2$ below the larger integer. For example, if you select $80$ and your friend selects $82$, you are the winner in this case. Assume both you and your friend play optimally. The optimal strategy here is a mixed strategy, where you select a random value $X$ from some appropriately determined distribution. Find $\text{Var}(X)$.