[jay_shark]

Here is one possible solution . I will follow with the same set up as I did for my Game Theory Resolution problem .

Solution: 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 2/3 which is verified since 1/3 + 2/3*1/2 = 2/3

So the probability that you win given that he calls is 1/3 .

Lets compute your optimal EV when you bet .

EV = 3*(1-x)/2*[2x + 4*1/3*(1-x) -2*2/3*(1-x)]
EV= 3x*(1-x) after simplifying .

Take the derivative of the EV function

EV' = -6x+3 which means that x>=1/2 if we set the derivative =0 .

Therefore a=1/4 .

So we should only check and call or possibly check and fold if our number is less than 1/4 and that we will always bluff with hands [1/4,1/2]. Note that there are variations to one's bluffing interval .

If we check , then our opponent knows that our number is less than 1/4 .

I haven't worked out the check and call or check and fold scenario but I'll come up with an answer shortly .