Recently noted Pot Limit Omaha authority and noted CardPlayer Magazine columnist Jeff Hwang wrote an excellent essay on the topic of All-In EV. With Jeff's permission, we have reprinted his essay which originally was published in Cardplayer Magazine on September 12, 2011. We hope that this article will be helpful in eliminating the common misconceptions about the All-In EV stat which is found in PokerTracker 3, and in turn encourage discussion of this stat's new name in PokerTracker 4 - All In Equity. Take it away Jeff!





Due to popular demand from online poker players, Hold’em Manager and PokerTracker 3 both include versions of a controversial and poorly understood statistic in their poker-tracking software called “All-In EV.” The idea behind this stat is to enable the user to determine whether he is running above, below, or around expectation.

There are a couple of problems with the All-In EV stat in either form:

It’s not EV. It should be obvious given some thought, but EV by definition is forward-looking. EV stands for expected value, and as such is a forward-looking concept. The All-In EV stat, on the other hand, is results oriented, adjusting results based on actual hands shown. (Technically speaking, the term “results-adjusted win” would be more accurate!) Consequently, my sources tell me that the stat will be renamed “All-In Equity” in the upcoming PokerTracker 4.

It doesn’t work. The stat is perfectly accurate for heads-up players. However, unless you only play heads-up poker, a player’s expected win (the All-In EV stat) and actual amount won lines essentially can’t ever actually be expected to converge, regardless of the software you are using. There are two primary reasons why the stat doesn’t work for anything but heads-up play. One is problems accounting for All-In EV in multi-way pots, and the other is trying to account for “EV” (results-oriented win) on online poker sites where the opponent’s holecards are only revealed at showdown if Hero loses, but not if Hero wins.

We’ll discuss those problems in a minute, but first let’s talk about how the stat is supposed to work.



All-In EV in Theory

The first thing that is important to realize is that the All-In EV stat is not calculated every hand. What the stat does is take situations in which Hero is all in and adjusts the amount won accordingly, based on Hero’s equity at the point Hero went all in.

Let’s say it’s a $1-$2 no-limit hold’em game online with $200 stacks.

Preflop ($3): The cutoff ($200) opens with a raise to $7 holding J♦ J♥. Hero ($200) reraises to $22 on the button holding A♠ K♠. Both blinds fold, and the cutoff calls.

Flop ($45): 3♠ 2♠ 2♥. The cutoff checks. Hero bets $40. The cutoff raises all in for $178 total. Hero calls.

Turn ($403): 8♦

River ($403): A♥. Hero wins with two pair, aces and deuces, king kicker.

In this case, according to the odds calculator in the Alpha version of PokerTracker 4, Hero has 49.596% equity on the flop, which is the player’s all-in equity. The player’s expected value (EV) is the total pot size times the player’s equity in the pot, and his EV-adjusted win is this amount minus the player’s contribution to the pot.



EV = (49.596%)($403) = $199.87

All-In EV = $199.87 – $200 = -$0.13.

Player Amount Won All-In Equity Pot Size EV Contribution to Pot All-In EV

Hero $203 49.596% $403 $199.87 $200 -$0.13

Villain -$200 50.404% $403 $203.13 $200 $3.13

Essentially, instead of getting credit for winning $203, the player’s win is adjusted for expected value, and his actual expectation was a net loss of $0.13. On the other hand, the villain gets credit for an All-In EV of $3.13.

In the short run, the Amount Won and EV stats will diverge by default. You can see why this is so from the example: Where the two players had All-In EVs ranging from -$0.13 to $3.13, one player had to win a whole stack, while the other player had to lose a whole stack. And so, over a sample of one hand — the shortest term — these stats will fail to converge.



However, over the long run — say a million hands or so — the Amount Won and EV lines for this situation will tend to converge. As some would say, the results and expectations should eventually even out.

Or at least they would if we could accurately account for multi-way pots, which we can’t.



Problems Accounting for Multi-Way Pots

The problem that both Hold’em Manager and PokerTracker 3 have with accounting for multi-way pots has to do with unknown holecards. In order to make an accurate “EV” (results-oriented) calculation, the software must know the holecards of all players in the hand at the time Hero went all-in — even the holecards of the hands that were folded.



Of course, unless you are a superuser, this is impossible to know.



Let’s say it’s no-limit hold’em and three players put in $25 each before the flop. The starting stack sizes and hands are as follows:



Hero ($100) — A♠ A♦

Villain A ($400) — K♠ K♦

Villain B ($400) — 9♠ 9♦

Flop ($75): 3♠ 2♦ 2♥. Hero pushes all in for $75, and both Villain A and Villain B call.

Turn ($300): XX (any card). Villain A pushes all in for $225, and Villain B (?).



Here’s the problem: Let’s say Villain B only calls if the turn is the 9♥ or 9♣ — that is, if he can beat A-A — but folds if the turn is any other card. What happens is that 41 out of 43 times, for the purposes of the All-In EV calculation, Hero shows up as having 91.616 percent all-in equity against the K-K hand. However, two out of 43 times, Hero shows up as having 82.946 percent equity against both the K-K and 9-9 hand. This happens even though Hero was all in against both opponents at the time Hero went all in.

This is clearly incorrect.

EV if Turn is Non-9 (41 out of 43 cards)

Player All-In Equity Pot Size EV All-In EV

Hero 91.616% $300 $274.84 $174.84

EV if Turn is 9 (2 out of 43 cards)

Player All-In Equity Pot Size EV All-In EV

Hero 82.946% $300 $248.84 $148.84

Consequently, if poker-tracking software were to count this hand in its All-In EV calculation — as Hold’em Manager appears to do — the software will systematically overestimate Hero’s All-In EV. As a result, users as a group will tend to appear unlucky.



PokerTracker 3, on the other hand, does the opposite and does not account for this hand at all for the purposes of calculating All-In EV, and instead only makes the “EV” adjustment when all holecards of callers are known. Unfortunately, this method is not quite optimal either, as accounting for multi-way pots in which all cards are known without accounting for multi-way pots in which hands are folded should result in a smaller bias in the other direction (i.e., the user should theoretically appear slightly lucky rather than very unlucky), but a bias nonetheless.



Problems with Sites That Don’t Reveal Opponent’s Holecards Unless Hero Loses

In addition to multi-way pots, there’s another trouble spot with EV stats, and that is sites that don’t reveal holecards unless the Hero is beat. Most Americans who had played primarily on Full Tilt or PokerStars may not be aware of this, but non-U.S.-facing sites such as Boss, Everleaf, and Yatahay, as well as the U.S.-facing Bodog site, do exactly that — even in all-in situations — if requested by the user. In addition, OnGame reveals mucked cards in their web replayer, but there is no way for poker-tracking software to retrieve this data in an automated fashion.



That said, trying to calculate EV using only hands in which Hero is beat would produce the same bias presented in the multi-way pot example — All-In “EV” would systematically be overestimated, and Hero would tend to appear to be very unlucky — except likely to a much larger degree.



Consequently, PokerTracker 3 does not account for All-In EV at all on these sites.



A Stat with No Meaning

The bottom line is that unless you are a heads-up only player, “EV” stats are going to be of limited value, and they truly are stats with no real meaning. In order for it to work 100 percent of the time and cover all situations, the software would have to be able to create hand ranges for your opponents in these situations, which is another problem entirely. And so, unless the stat-tracking software only makes “EV” or “luck” adjustments in heads-up only situations, any All-In EV stat of any name in any form is going to be biased in one direction or the other.

Summary: 1) It’s not EV, and 2) Unless you only play heads up, the stat doesn’t work — no matter what software you use.