Equilab is a popular and free equity calculator. You enter your hand or range along with hands or ranges for one or more opponents. The program will then provide your showdown equity, either through simulation or computation.[1] Not only that, you can also enter flop or flop and turn cards for an equity result.

So, with Equilab (and other similar equity calculators) providing this important factor, your winning chances, what is the concern?  It is that an equity calculator assumes the hand gets to showdown, which, of course, is not always true. On an early street, hands get to showdown if a bet is all in or all players check down. Hands don’t get to showdown if all but one player folds. And folding is not an uncommon action. Also, of course, estimating your opponent’s hand or range is subject to error, something which Equilab doesn’t directly consider but you can apply it over a range of inputs.

This leads us into the notion of equity realization, the chance that you will actually realize or even better the showdown equity. I would suggest that 33 has lower realized equity than 89s although it usually has higher showdown equity than 89s against a specific range. For example, 33 has showdown equity of 46% against a top 25% hand while 89s has a lower equity of 37%. But, if the 3 pair doesn’t hit on the flop, there is a good chance it will be folded while the suited connector often will flop a straight or flush draw along with possibly hitting a mid-rank pair, a total probability higher than the 12% chance of the 3-pair setting.

This is not to suggest that Equilab is not useful. Showdown equity provides a first cut to your chances of profiting. And, by recognizing that folds, either by you or your opponent is possible, it’s value can be used as the baseline for equity adjustment depending on the particular circumstances. It would be nice if an equity calculator can also provide realized equity but that would most like have to rely of analyzing many millions of hands under many varying conditions for there is no theoretical or simulation way of obtaining fold equity as far as I know.

As a final point, estimating your equity should not only take into account future folding possibility, by both you and your opponents, but also their calling range as the hand proceeds. An opponent may limp preflop with a top 80% range, but depending on the board, his calling range will almost surely narrow unless he is a maniac bettor. How much it narrows is a challenge a poker player must address.

Introduction

“What is a good bet?” is a question discussed in many introductory poker discussions. The answer usually is something like this:

A good bet gets better hands to fold and worse hands to call.

A sometimes suggested third answer is to bet for protection, a bet made with a vulnerable hand to get a drawing hand to fold or to call with incorrect odds. Also, information bets and blocking bets have been proposed for certain situations.  In this discussion, we will concentrate on betting to get worse hands to call so that your EV exceeds that with a fold.

A Worse Hand and Realized Equity

The first thing we’ll consider is to define a “worse” hand. An obvious definition is to relate the “worse” aspect to equity, the chance of winning the hand. Say you have 22 and you know villain has AKs. By a fraction of a percent, 22 is favored, so by a small margin, AKs is worse. Would you proceed to play 22 as a potential winner based on that? I doubt it and here’s why - realized equity. If you don’t hit the flop with your very small pair, and there’s only a 12% chance you will hit a set or better, what do you do if an ace or king falls and your tight villain bets aggressively? Fold is probably the right decision assuming other factors don’t dictate otherwise.  In fact, you will likely fold to most any missed flop containing one or more premium cards and villain betting. A fold of course means you will not realize your equity. Having AK,s however, you may hit a high pair and the hand has good possibilities for improving to a flush or straight. Even the Ace could win. That is why AKs is rated much higher than 22 even though the latter has a slightly better showdown equity preflop.

For any analysis which includes a showdown equity estimate, that estimate should be adjusted to the extent possible to reflect realized equity. You will increase equity if incorrect villain folds (folding hand he would have won) exceed your incorrect folds; your equity will decrease if the opposite is true.

The Minimum Bet Size Equation

Let’s now suppose you have a much better hand, say you are a 60/40 favorite after the flop or turn, a very good situation and there is a good chance that the equity (or better) is realized. You have the lead. You can check, bet small or bet big. A check would be a consideration if it induces villain to bet. However, it also opens the possibility for a check-back which could give villain a free chance to hit his good draw. Against very aggressive villains this might be a good action, otherwise not so good. Betting small has a similar consequence but at least builds the pot some with a winning hand. Betting big has the desirable result of a much bigger pot but might result in a fold that gives you a minimum win amount. Is there a minimum bet size that provides a potential bigger pot but gives villain the wrong pot odds to call?  Yes, it is given by the following equation:

Bet size for Villain Incorrectly Calling > Pot*(1-eq)/(2*eq-1),  eq > 50%.

Example

For our example case of hero being a 60/40 favorite, assume the pot is 20. Then for villain to incorrectly call, a bet has to be at least the following:

Bet size > 20*(1-0.60)/(2*0.60 – 1) = 8/0.20 = 40.

Any bet greater than 40 will more than triple the win amount, but it will also insure that villain has incorrectly called when he is the underdog. This equation was derived so that on average hero wins at least as much as he would if villain folded. To see this, here is the EV fold and EV bet equations, the latter based on the assumption that villain folds on the next street if the current street is not the river:

EV if villain folds on current street:  EV= Pot = 20

EV if villain calls a bet of 40 but folds next street: EV = 0.60*(Pot + 2*Bet) - Bet = 0.60*100 - 40 = 20

The equality of the two EV’s is as expected for it shows that any hero bet greater than 40 that is called will do better EV-wise than a villain fold. Of course, making a very large bet increases the chance of villain folding, so here is where a read of villain’s tendencies and factors such as position and stack size are very important to guide you to making that bet which does not give villain good pot odds but keeps him in the pot.

Table and Graph

The following table and graph show minimum bet results in pot size units for various equities over 50%. Multiply the given value by the pot size to convert to actual units

Example: If you estimate your realized equity to be 70% on the river, to have villain incorrectly call you need a bet of at least 0.75 *Pot. Thus, if the pot was 100, your bet should be at least 75 or else you would rather have a fold.

Check: Villain Fold: EV = 100. Villain Call: EV = 0.70*(100+150)-75 = 175-75=100

Summary

Here’s what we can conclude from the above:

1.       Raw, hot or cold and showdown equity are terms to describe your winning chance if the hand went to showdown. An equity estimate based on showdown should be adjusted to what can be described as realized equity to reflect the potential you or your opponent might incorrectly fold on following streets even if ahead.

 2.       Being ahead you generally want your opponent to not fold if that will result in a larger winning pot in EV terms. We provided the minimum bet size to achieve the objective.  Of course, betting too large will likely result in a fold, so the non-math art of poker becomes an important factor.

 3.       If the bet is not a closing one, results should be adjusted for factors not explicitly considered.