The [0, 1] Half-Street Fixed-Limit Game - Balanced Play vs. Exploitative Play

This is my official 5000th post, and I feel like what this post illustrates is pretty important, so I hope some people get a lot out of it.

Given x_1, the optimal exploitative value betting range is going to be [0, y_1] such that y_1 = x_1 / 2. With this value betting range, how often do we have to bluff so that we're unexploitable?

We'd like the EV of Villain calling with x_1 to be the same as the EV of folding x_1. Earlier we did this and found that the resulting equation was when y_1 = (1 - y_0)/a. We already know what y_1 will be, so let's change this equation around to make it more convenient for finding y_0 given y_1:

y_1 = (1 - y_0)/a
ay_1 = (1 - y_0)
y_0 = 1 - ay_1

So given x_1, we should be value betting [0, x_1 / 2], and bluffing [1 - (a/2)x_1, 1] if we want Villain's call with x_1 to be break even.

Earlier we found that the optimal exploitative bluffing frequency was set by y_0 = x_1/(1-a). Now we're looking at the balanced bluffing frequency being set by y_0 = 1 - (a/2)x_1. The relationship between these two values is very important in the discussion of balance vs. exploitative play.

Let's take a look at an example scenario. Suppose P = 1.5 and Villain calls with [0, 0.3], making x_1 = 0.3. Note that a = 0.4. Then the optimal exploitative value betting frequency is set by y_1 = 0.15, the optimal exploitative bluffing frequency is set by y_0 = 0.3/(1-0.4) = 0.5 and the balanced bluffing frequency is set by y_0 = 1 - (0.4/2)(0.3) = 0.94. If we plot these values on a number line, we can find some interesting results.


Note: Don't fall into the trap of thinking that all of this is a lot of theoretical stuff that doesn't apply to "real" poker. In no-limit hold'em, this situation would basically be like we had 2/3 pot left behind on the river in position and Villain had us covered, and after Villain checks to us, we have similar ranges. Additionally, this game can be slightly altered so that we don't have the same ranges.

I've colored the 5 important segments of the [0, 1] distribution in the above graphic. In the bright red, this is the part of Hero's range that he will value bet with, all of which will show a profit. In the dark red, this is the part of Villain's range that he calls with, but that does not beat the worst hand that Hero value bets. This section exists because Hero bluffs a non-zero percentage of the time, and because of the effect of pot odds.

The blue section is the range of hands that Hero can bluff with if he wants to exploit Villain with an increased bluffing frequency. By bluffing hands from this section, he increases his EV at the risk of being exploited by Villain calling more.

The green section is the range of hands that Hero should always bluff. If he doesn't bluff these hands, he loses EV. The problem is that there is no risk to bluffing these hands because Villain cannot exploit Hero by calling more. Moreover, if you don't bluff these hands, you become exploitable by Villain calling less. So basically, you have to bluff these hands because 1) it's +EV compared to checking, and 2) if you don't then you become exploitable.