Quantifying the Value of Position

Even though I’m too sloppy/lazy/uneducated to work out the details of complicated problems, I’m quite interested in the mathematics of poker. I feel like I do have a broad grasp of the game theory that underlies many situations and can use that to aid in my decision-making. Recently, I’ve been curious about how to quantify exactly the value of seemingly abstract concepts like position and implied odds. I think I may have come upon a sketch of how to work some of it out, though I doubt I’ll ever follow up on it.

We start with the “exploitability”, the idea that there is something about how you play that an opponent could potentially take advantage of. Conversely, “unexploitable” means that there is nothing an opponent could do to take advantage of how you play. Importantly, unexploitably is not always the most profitable way to play. Often, you will do something exploitable in order to exploit something exploitable an opponent is doing.

Suppose that you hold AQ in the big blind in a $1/$2 NLHE game. The action folds to the SB, who open shoves for $20. If you know that this opponent will only shove JJ+ and AK, you can fold your AQ. Though itself exploitable, this fold exploits your opponent’s excessively tight shoving range.

If you were to call with AQ, that would be less profitable than folding but it would also be unexploitable. In other words, there is nothing your opponent could do to take advantage of the fact that you are going to call with AQ, and in the long run you will make money, though not as much money, with a strategy that involves calling AQ. This is because the money you lose on the AQ call is more than made up for by all of the pots you win that your opponent could have picked up if he weren’t playing so tight.

It’s relatively simple to determine an optimal shoving range for the SB and then to determine how much you gain if the SB’s shoving strategy is tighter than optimal. Take it a step back and suppose that it is the player on the Button with just $20 and shoving an overly tight range. Now it should be clear that both the SB and BB benefit by having their blinds stolen less often than would be optimal. The money that the Button loses in missed steals is shared by the players in the blinds. The situation is actually more complex for both, but I want to back up again to accentuate a point.

Put the very tight short stack in the CO. It is still possible, and indeed relatively simple, to determine an optimal shoving range for him. If he shoves more tightly than optimal, we can calculate exactly how much Expected Value he loses as a result. This EV must be shared by the Button, SB, and BB. We already know that the blinds’ benefit primarily by having their live money stolen less often.

The Button, however, does not have any money in the pot, yet he too benefits from an overly tight CO. If we can calculate how much the blinds gain from the CO’s mistake, then the difference between what the CO loses relative to an optimal shoving strategy and how much the blinds collectively gain will be the value of the button.

I suppose it would be the value of the Button assuming the other players all play optimally, such that it would change if the Button were more or less skillful than the blinds. Still, it seems like this could be a strategy for someone more mathematically inclined than myself to approximate the value of position.

Am I on to something or just tired?

Edit: I was just tired. After sleeping on this, I realized that the BB’s EV is intrinsically dependent on that of the Button. In other words, the deeper the stacks are and the better Button plays, the more the Button is worth and the less the BB is worse. I was hoping to come at the Button backwards, treating the SB’s and BB’s EVs as roughly solvable quantities and using those to approximate the Button’s EV as a variable. But we can’t actually find the BB’s EV without knowing the Button’s EV, since most of the latter comes at the expense of the former. And the same problem exists for the SB.