More Thoughts on Button Straddling
Earlier today, Zach posted an article about the expected value of button straddling. That inspired me to do the following analysis. Hopefully, in conjunction with Zach’s analysis, we can offer some insight into the situation, or at least provide some inspiration for others to take this work further.
So let’s start with some basics. Let’s call our button EV x and our button straddle EV y. Basically, if y>x, then we want to straddle. The next step taken in the article posted earlier today was to try and define y as a function of x, or y=f(x). Basically, the desire here is to use x to get y. Therefore, based solely on x, we can decide whether or not to button straddle.
Before continuing, it is important to mention that y cannot possibly be a function of x. This is for several reasons. The act of straddling changes the preflop action (the button acts last), causes the button straddler to play a strategy that takes into account the money invested preflop, and will often result in changes to opponents’ strategies . However, if we can create a reasonable estimate of y given x, then we can use that estimate to more accurately guess if the actual value of y is greater than x.
So, what goes into f(x). The following factors would all likely need to be components of a function to estimate y from x. In truth, these are inseparable as they all occur together in the act of straddling. However, considering their individual effects and their relative magnitudes can help determine whether straddling is right in a certain situation.
Doubling the stakes - This will likely have a positive effect on x. If we doubled the stakes and doubled each player’s stack, we will almost certainly increase our winrate.
Halving stack sizes (in big blinds) - This will likely have a negative effect on x. The value of position is magnified when stack depth increases. Therefore, decreasing stack depth reduces the advantage of position along with the winrate (in big blinds).
Acting last preflop - Acting last is always an advantage. Therefore, this should have some positive effect on winrate.
Putting in a blind bet of 2bb - This component is difficult for me to model. It’s clearly not separable from the other effects. However, consider a blind bet of 2 bb on the button. This almost certainly has an expectation that is less than not blind betting. Therefore, separated from the other effects, this likely has some sort of negative impact on our winrate.
Opponent adjustments - This is much less easily predicted. However, opponents adjusting poorly to a straddle will definitely increase our winrate. For example, if we knew button straddling would cause the whole table to move all-in with any two cards, then we would straddle.
Rather than trying to come up with a function that brings all of these factors together, I am taking the approach of modeling toy scenarios that seem to be clearly better or worse than button straddling and deriving a function from the toy scenario that will serve as a bound on the actual value of y.
Consider the following function as a lower bound on y, f(x)=1.5x-2. This function is basically modeling a situation where we pay 2bb as a fee that is not included in the pot in order to increase our winrate by 1.5. Let’s imagine that the mechanism for increasing our winrate is to simply double the blinds for the hand. Some may argue that a corresponding increase of 1.5 to winrate is too optimistic, but it seems reasonable to me as a lower bound. Straddling on the button is almost certainly better since we will be last to act and our 2 bb will be in the form of a bet rather than a fee. This means that if our EV for the the button is 4bb/hand, the we should almost certainly be button straddling.
An upper bound on y is a more difficult number to reach. However, I think f(x)=2x-.5 is a reasonable upper bound for ordinary low stakes circumstances. In this bound, I am basically claiming that one is effectively paying a fee of .5bb to double their winrate. I am pretty confident that this scenario is more favorable than button straddling independent of wonky opponent adjustments. Therefore, by this bound one should never button straddle with a winrate of less than .5bb/hand on the button.
From this analysis, I’m concluding that it is probably good to straddle from the button if one feels they have a significant edge over their opponents. Increased average stack depth and poor opponent adjustment are two factors that could make button straddling more favorable.
This analysis is begging for improvement. Please feel free to post any thoughts in the comment section.
