Small Blind Pre-Flop Opening Raise For +EV
We continue here with a third post on a small blind pre-flop opening. We noted in Post 29 that in such a situation the SB can assume the BB villain has a random hand. In that case we showed with tables and graphs the attainable EV for three showdown equities, 40%, 55% and 70% for various bet sizes and fold equities.
In Post 30, we provided tables that show critical bet sizes for villain indifference and for hero positive showdown EV for selected hands, mostly applicable for large stacks. For small stacks, the card equity needed for a positive EV shove was provided in a table and graph given bet size and big blind call %. That card equity value then defines the hands with which you can shove. A following table and graph display the associated shoving range for the same bet sizes and villain calling frequency assuming villain calling frequency defines his calling range. We also provided a link describing the Slansky-Chubukov theory on a small blind open.
Here we continue that discussion for more general cases by operating on the EV equation for a small blind open. We note that since future betting will likely take place if the raise is not a shove and the BB action is a call, that these results are only a first-cut look at the situation.
Expected Value Analysis
While the small blind may not get to open-raise pre-flop too often in a multi-player game, when he does, it can be a good opportunity to profit. The applicable EV equation in big blind units is as follows for SB raise and BB call:
EV = fe*1.5 + (1-fe)*(eq*(1.5 + R – 0.5) – (1-eq)*R)
= fe*1.5 + (1-fe)*(eq*(1 + 2R) – R)
where
fe = fold equity
eq = hero card equity
R = SB raise including the 0.5 blind completion. BB call = R – 0.5
The EV for various values of fe, eq and R are shown in the following table. We only show equity values less than 50%; EV is always positive for values greater than 50%.
It is seen that EV decreases with increasing raise size but increases with increasing equity and fold equity. For a relatively small raise like 3bb, the majority of cases examined show a profit (shaded cells). This is also true for large fold equity like 75%. So, if you think the big blind has a good chance of folding to an opening raise, then it is +EV to do so. Of course, the bigger the bet, the more likely it will cause a fold, but, as shown, big bets decrease EV. Here is where logic and psychology have to be considered along with the math for good decisions. What we can do with the math is to show the card equity or the fold equity needed to achieve +EV which we do in the next two sections.
Required Fold Equity
Here we manipulate the EV equation by setting it to 0 to provide the minimum fold equity for +EV. The equation is as follows:
Min fe = -(eq*(1+2R)- R))/(1.5-( eq*(1+2R)- R))
The results for various raise and card equity values are shown below:
It is seen that the minimum fold equity increases with raise size and decreases with increasing card equity. Blanks signify a plus EV situation for even a 100% call.
Required Card Equity
Here we manipulate the EV equation by setting it to 0 to provide the minimum card equity for +EV. The equation is as follows:
Min eq = ((1-fe)*R - fe*1.5 )/((1-fe)( 1+2R))
The results for various raise and fold equity values are shown below (another form of this table appeared in Post 30):
As to be expected increasing fold equity results in a decreasing minimum card equity. For raises of 3 and lower, if the big blind folds about 2/3 of the time, the small blind can play just about any two cards. The same is true for raises up to 5bb with fold equity at 75% or greater.
Summary
I believe it reasonable to say that the above results confirm that when the small blind has the chance to open raise in a multi-player cash game, he is in a good position to profit. On the other hand, it would be remiss not to remind readers that we did not consider a big blind raise nor did we consider likely future betting when equity realization issue can be a significant factor. Also, tournament factors such as antes and ICM were not considered. However, we do believe that one can benefit from studying the material presented here to develop a small blind opening strategy.