In the simplest full-street AKQ game, with limit betting of 1 unit and
pot size of P units, the equation for the second player's value in the
game was derived as a function of alpha, and thus as a function of both
bet size and pot size.



For bet size 's', pot size 'P', Alpha as 'a', and a betting frequency
'x' of the first player's Aces, the value of the game to the second
player was derived as:



F(alpha) = F(s,P) = ((x)*(3a-1)+(1-2a))*(1/6)





At P=2, or equivalently, for a half-pot sized bet, we have:



Alpha = 1/(1+2) = 1/3



==> 3a - 1 = 0

==> F(s,P) remains constant for all x (at these specific s and P values)

==> Player 1 is indifferent to betting an Ace for half-pot or checking an Ace



Specifically, at s = P/2 we have that a=1/3 :



F(s,P) = (1-2a)*1/6 = (1- 2/3)*(1/6) = (1/3) * (1/6) = 1/18



On the second paragraph of Pg. 165, the authors make a very vague claim
about how this is normalized for a bet size of 1 and pot size of 2, and
how the value of F(s,P) should scale upwards in proportion to the new
bet size. For example, they claim that:



F(1,2) = 1/18 ==> F(2,4) = 1/9



I don't see how this is the case, other than some verbiage they present as "proof". If anything the relationship seems to be:



F(P, P/2) = P/36



I don't know how to show this, however. How do I show what they are claiming?