Game Theory Resolution

[jay_shark]

The object of this game is to select a number from the closed interval [0,1] and to bet if you think your number is the highest . You only play one round so if you fold , the game is over .

a) A generous man decides to give you (hero) and your friend (villain) a free roll to enter this game . Hero posts the sb and villain posts the bb and you can raise to 3bb's or fold . Villain on the other hand can only call . What number should you raise with ??

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 .

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

We wish to maximize this function using calculus .

After simplifying you get

EV(x) = 3x-3x^2
EV'(x)= 3 -6x
Set this =0 so we get x=1/2 .

Finally we're done !!