Just wondering if anyone has done any math to figure out what the value of position is in a heads up pot.

For example, let's say I have a strategy to call a button raise from the BB every time my equity (with position) is 50%. When I plug my hand into an equity calculator, what % would I need to call? I'm guessing it's something like 52-54% but I actually have no idea and was wondering if anyone had done math on this.

It just equates to how much information you have. I guess it varies on the table but if you are in MP and every player including the button is a tight ass nit who plays only AA or KK you can disregard it every time when you are MP or later preflop because you know 95%+ of the time they're folding. Postflop I guess it depends on more factors but I don't know the math for it.

I agree with pocketacescracked. It depends on your hand and the table. Acting first is a big disadvantage when the other player has more information than you, a small disadvantage when you have more information than the other player.

If the other player is tight passive so you have a tight range on his hand, and you have a hand that will likely be strong or weak but not middle after the flop, say a pair or an A, position means little. If the other player is loose aggressive and you have something like KQo, position matters a lot.

yeah but there must be a way to calculate it and convert it into a % of equity

For example, let's say we had a big enough sample size of button vs BB hands with random players. The amount of money the button makes as a % of total pots would theoretically give us a value, would it not?

Does anyone have a guess at the top of their heads? I've been using 2-4% my entire poker career and wanted re-assurance.

No it can't be quantified, especially with "equity" because "equity" is not that important. The power of position is not mainly an equity advantage, it's an advantage of putting more money in when you have favourable equity, and putting less in when you are behind. It's about future betting, not current equity.

On a related note if you decided preflop base on your perceived equity then it's going to end badly. Future street equity distributions are what's more important, like ace-rag and a suited connector might have similar equity vs someone's range but the latter has an infinitely more favourable equity distribution.

Exactly solvable toy games provide some insight. For example, in the fixed-limit 1/2 street AKQ-game (where the OOP player must check), the IP player wins 1/18 bet when both are playing optimally. See Mathematics of Poker Example 13.1.

Part of that edge exists because the OOP player doesn't have the option to lead, though. But they solve for the full-street game later in the book and the IP player's win rate is then smaller (I don't remember it off the top of my head). The IP player's edge when both players have all strategic options available can be viewed as a measure of the value of position in the AKQ game.

So in a effort to talk about the real poker game we play try to see what happens when say CO raises 2.5bb with 20bb behind effective (covering say) with AdTh and BB calls with 6s5s and 20bb stack. Assume antes and small blind add up to 1.625bb extra so they go to the flop with 2.5*2+1.625=6.625bb and if they checked to the river it would be 58.4 vs 41.6%.

But of course you do have betting and so try to play this hand going over all flops and without knowing GTO of course try to see how most people would play it depending on how they connected , bets they face etc and see what happens by trying to play each side giving plausible ranges to the other. Yes the game tree gets high but most cases would end up to being resolved by flop or turn based on texture families (and having the 2 specific hands would simplify things) and so we dont need to be extra worried about the complexity that emerges as the 2 player poker game that is not very deep would tend to get resolved early on and higher complication tree branches have tiny probabilities anyway.

I may return and invite everyone to participate with more on this just for fun to see if we can estimate the EV of a specific duel like that over all flops on avg. The alternative of course is to simulate it vs a computer trying to play the hand objectively many times and not forcing on your approach the knowledge of his hand (yeah yeah you can do it you are training or trained to be scientists so you know what i mean). I bet without simulations we only need to examine a few cases to get a rough idea working with things like cbetting and what BB will do if he has air, how most people play when they connect, have draws etc. We only a need a rough idea.

Yes of course position finds its edge from acting later than opponent and at turn/river future bets and pot control, bluffing with more info etc but the reality is you cant avoid it all being a correction to the crude all in equity and the avg size a pot gets post flop. Just because you have position you wont be able to turn a trashy J3o vs KQo to profit.


I personally use a rough heuristic that splits the avg post flop pot size according to all in equity corrected by 5% on each side for position (x1.05 vs x0.95). I also use as avg pot 2.5 times the flop size. I asked people to help add statistics to improve that, but nobody offered anything to contribute to what to me appears to me a super massively important element in nearly solving a great deal of preflop strategy cases. (ie what to raise, call 3bet etc with and how to react depending on raise sizes ,antes etc because not everything is push or fold of course and calling is often a very important sector to want to quantify or approximate as much as possible)

My question then is, how do you translate that to % edge?

This is a tricky question, because certain hands may be better in a certain position than others. But I suppose we could do an overall assumption.

Firstly, we need to assign specific hands a % edge before doing the position. All-in equity is not good enough. I think the best way would be to do a heads up game before going to a full table.

Find out how often you win with a particular hand in a given position and claim that is its equity.

So if A9o wins 74/100 hands on the button we can say it has 74% equity on the button. If it wins 53/100 hands in the BB we can say it has 53% equity in the BB. Do this with all the hands and take the average difference from the button vs. the BB and there you go. That average distance is the equity % of position.

Play against a computer that doesn't adapt, such as Snowie. This allows you to not skew the results (Only playing 72o versus an uber tight opponent making it more profitable than 87s).

Even this whole method is a gross simplification since certain hands win more money than others, but this will help you get started.

To clarify, I'm literally looking for how to mathematically convert this to percentage advantage. Like does this mean the in position player has a 2.682% edge?

Defining a poker players advantage as an edge is kind of a weird concept. I've seen it done before and it seems forced.

The first question is, edge as a percentage of *what*? The total amount bet/risked by the player?

That's where I get stuck. My guess in this toy game would be units per play vs amount risked. So 0.02682 / 1 (1 unit bet?) = 2.682%

I guess? You can make it mean whatever you want it to mean. Which leads me to my next question: why do you want to know what the number here is in terms of "the players edge?" What use do you plan to put it to (that you can't do by knowing that the player makes 0.02682 units/hand?)

This is from a comprehensive poker learning glossary I'm working on

"Edge (Ej) – the advantage you have over another player. One quantitative definition is Edge = EV / (Risk Money or Bet amount). This is equivalent to the % of a dollar you win per dollar wagered. For a bet decision, the edge can be written as Edge = Pot Odds *eq - (1- eq). Tournament edge might relate to your ROI compared to the average ROI of the opponent field. It is sometimes advised to hold off an all-in bet early in the tournament in a marginal situation if you have a tournament edge so you can take better advantage of it later on."

"Example: In roulette, if you bet on red, say, with a wheel having 0 and 00, your chance of winning is 18/38 and the total return for a $10 bet is $20 or you win $10. Therefore, the house EV is EV = 20/38 * 10 – 18/38 * 10 = 20/38. Therefore, house edge = EV/Risk = (20/38) / 10 = 5.26%."

"For a poker example, Assume that villain makes an all-in bet of 50 into a 100 pot making the pot 150 giving you pot odds of 3 to 1. If you call you have a 30% chance of winning. Your edge is 0.30*3 – 0.70 = 20%. The positive edge occurs because the 30% equity is greater than the required minimum equity of Bet/(Pot+Bet)=50/(150+50) = 25%. "

I'm basically looking to adjust betting frequency by some factor (edge?) for the out of position player, and use that as a baseline model

OK - why do you think that having this number in the form of an edge percentage will be useful?

I more or less agree with your definition. I'm not sure it's a useful number in poker, particularly because it's so hard to calculate outside of online poker.

When comparing the model I use to the GTO strategy, it's a bit too aggressive OOP. So I'm looking for a way to quantify the value of position to adjust betting frequency adequately

Are you programming a bot or something?

FWIW, the value of position will vary according to board texture and how the board interacts with ranges. e.g. on something like 874tt, in UTG vs BTN, the player on the button will benefit greatly because of his position, due to the board being so dynamic, and him acting last (so he can see how UTG acts on the turn, if a turn card is even seen). It contrasts with a board like AA2r, where there are no draws, and no possible overcards on the turn.

I'm basically just "solving" more advanced toy games. Texture is an important variable for sure, but I've found naive estimates to be surprisingly close

bump

You can do it with a solver, just plug in identical ranges and allow every strategic option. Remember a solver does not care about "initiative", the only difference will be that one player will be IP and another OOP. Doing this for IP open vs BB flat it turns out IP gets ~55% of the pot and BB ~45%.

We can say that for the average situation of 100bb stacks position is worth ~5% of the pot (without any range asymmetry considerations). As stacks get shorter it becomes less and less valuable. At around 5bb stacks position has no value.

Thanks but was looking for more on how to convert the solution presented in MOP above to "edge." In the example you gave the edge would be 10% if I'm understanding the concept correctly

birthday bump

If the model, or bot, you are working with is too aggro opp, why not just remove the weakest flop continuing combo or two from that range and let it stay aggressive with the stronger range? One or two check-folds or check-call instead of c-bet or check-raise.

Ultimately frequencies are the summation of the collective actions assigned to each combo contained within that range. If you are wanting to account for position, especially on the turn or river, it boils down to combos, imo.