'Relevant profit' mode coming soon for free!

Okay, so it's an eight-handed table, you hold Deuces in the big blind and the button, who has been bullying everyone for the last three orbits because we're close to reaching the money, again shoves around 25 BB from the button. You're entirely certain he has a range not an inch tighter than the 31% range, and you must call you last 16.7 BB to go to showdown. Should you do it? 

Yes! Am I right?

Well, yes, but also...

What kind of player are you? How much is this bubble? Do you care about this particular mincash? Are you in a fighting mood, or do you really just want to guarantee a winning session?

Granted, in a perfect world where you always have thousands of buy-ins and no feelings regarding what happens, but instead you only focus on optimum play an the long run, the answer should be: fuck the bubble, I've read the situation well, I know what this guy is doing, calling is absolutely a positive-expectation play, not only immediately, but it may later stop people from messing with my blind, so I call.

But if those questions indeed bug your mind, it's no fault of yours, and you're not wrong to consider them - you are what is known in laymen's terms as a human. 

A human (by Flickr user vagueonthehow).

The number 17.1 BB is the exact point where calling the bet against the 31% range is break-even, meaning calling 17.1 BB has the exact same expectation of folding - that is, to have 17.1 BB by the end of the hand when you fold, and to end the hand with a total 171,000 BB over 10,000 hands when you call (averaging 17.1 BB). Any number greater than that is unprofitable to call (your expectation is lower than folding), and any number smaller than that is profitable. And 16.7 is smaller, so this is a profitable call. But...

What Perfect Preflop Play (get it here) is neglecting to tell you is, by how much? That is, if by calling my non-desperate stack of 16.7 BB I expect to end the hand with 16.8 BB but also expect to bust out before the bubble more than half the time, do I want to do it? I'm guessing most of you will answer "no."

By Flickr user lindenbaum

And that is why we're coming up with a free update that is going to tell you what you really want to know: When does calling become so profitable that it's a shame not to do it? 

It's really simple. We're not going to erase the information you currently find on PPP, but by messing around with the original formula just a little bit, we're going to bring you a whole new mode that you can choose to show you at what point your call starts yielding an extra 5% to your stack, or an extra 1.5 BB for stacks bigger than 30 BB. 

So now that we demand to increase our stack by at least 5% to warrant making the call, what is the cutoff? For this precise situation, it's 10.87 BB. That's because, when you make the all-in call with 10.87 BB, the total effective pot becomes 25.04 BB, and you have 45.6% equity in that, which amounts to 11.42 BB, which is bigger than 10.87 BB by 0.55 BB, which is 5% of 10.87 BB.

So if you have 10.87 BB or less against this exact player in this spot, then by calling you are going to increase your stack by at least 5%, effectively turning the call into too good to miss. So from a practical standpoint, this update (which also includes 6-max mode) is going to make PPP a whole lot better, and for free. 

The "relevant profit" concept will also be applied to Heads-up Monster, to make sure you're only shoving the small blind (especially with a decent stack) when there's enough to gain from it.