The whole point of picking a value for R is that it essentially transforms the game from "guess whether A is smaller or larger than B" to "guess a number betwee

The whole point of picking a value for R is that it essentially transforms the game from "guess whether A is smaller or larger than B" to "guess a number between A and B." Regardless of how A and B are chosen, if your R is not between them, then you break even; if your R is between them, then you win. So your advantage is entirely based on your probability of picking between A and B. As long as that probability is nonzero, then you have an edge. The way you guarantee having nonzero probability is by choosing with a distribution that is positive everywhere (e.g the normal distribution). No matter where the interval [A,B] is, you have a chance of choosing within it. Depending on the opponent's distribution vs yours, that chance may be very small, but it's still strictly greater than 0.