[jay_shark]

I will show that there is not a better strategy than the one I mentioned . Again , you can make the assumption that your grandmother gave you 50 cents as a gift so it's not coming from your pockets .

Instead of choosing numbers from 1-100 , lets pick a random number from the interval [0,1]. Note that the two questions are the same when n approaches infinity . In the example , I've used , n=100 .

Let a be your optimal pushing range ; a>=0
Let x be your opponent's optimal calling range . 1/3<=x<=1

We can write a in terms of x . Notice that (1-x)/(x-a) = 2
x=(2a+1)/3

The probability that your opponent wins given that he calls is (x-a) + (1-x)/2 ; x-(3x-1)/2 + (1-x)/2 = 1-x

EV(x): (3x-1)/2*[1.5x + 3.5*x -2.5*(1-x)]

We wish to maximize this function using derivatives . Ughhh

EV(x) = 1/2 *[22.5x^2 -15x +2.5] after simplifying

EV'(x) = 1/2*[45x -15] =0
22.5x = 7.5
x=1/3

Woohoo !!