Consider the following two-player game. The players simultaneously draw one sample each from a continuous random variable X, which follows $Uniform\ [0, 100]$. After observing the value of her own sample, which is private information (that is, opponent does not observe it), players simultaneously and independently choose one of the following: $SWAP$, $RETAIN$. If both the players choose $SWAP$ then they exchange their initially drawn numbers. Otherwise, if at least one person chooses $RETAIN$, both of them retain their numbers. A player earns as many Rupees as the number she is holding at the end of the game. what is the probability that the players will exchange their initially drawn numbers?
this is a question from one of master's entrance exams, now i think that the question is incomplete because given the information in the question if the it is up to player to draw any number from the given interval of $ [0,100]$, why would any player choose any number less than $100$. all the players will choose to maximize their payoff and hence they must draw $100$. and hence the required probability should be equal to 0
But if there is a missing point from the question, for instance the players might draw from 101 balls numbered 0 to 100 in that case the question seems like Monty Hall Problem.
If i am wrong then can anyone please help me with this question?