#ev math

Introduction: Determining the equity you need to guide you towards a good poker decision is a common math challenge. It typically involves determining the equity value that maximizes your expected value or, at least, insures no loss. In this posting, I focus on calling a bet or folding, using what I call the Fair Chance Equity rule (FCE).

Situation: We will examine three heads-up situations, assuming the pot is 60 and villain bets 20, making the pot total to 80 before hero has to call or fold.

Case A: The bet is not all-in and the street is not the river.

Case B: The bet is not all-in but the street is the river

Case C: The bet is all-in

The Fair Chance Equity Rule: There are several ways to determine how much equity you need to math-justify calling a bet. I believe what I call the Fair Chance Equity rule is the most intuitive. For our sample case, after villain bet, the pot is 80 and you have to call or fold (we are ignoring raises as that leads to much complication). If you call, the pot increases to 100, all of which the winner collects. To justify the call, since you invested 20% of the total pot (20/100) at the decision time, a “fair return” would mean that 20% is the minimum frequency with which you should win, hence the minimum equity.[1]

Let’s now see how this works for our three cases:

Case A: Not All-in, Early Street. There is no reason the Fair Chance Equity rule cannot be used but it would only be exact if the hand were checked down. The reason, of course, is that betting is likely to take place on future streets, so that the calculation can only provide an initial cut at the proper decision. Also, there may be justification for calling even if you do not have the required equity such as floating and implied odds.

Case B: Not All-in and River Bet. Here a call closes the action so the FCE calculation provides an exact result. Clearly, a raise is a possibility and then the hero essentially becomes the bettor. If hero can estimate his equity, then using the same rule can provide insight on how much to bet, but we’ll leave that to another posting.

Case C: An All-in Bet. Here there can be no future betting so the FCE rule is exact in all respects.

Summary: The FCE rule is both simple and intuitive IMO. It provides an estimate of the equity you need to justify calling a bet. Of course, it’s most useful when you can reasonably estimate your equity and that is true for all EV math calculations. Clearly, any initial math-analysis should be adjusted for factors not explicitly considered such as position, stack sizes, villain characterization, etc.

In keeping with my past attempts to simplify EV analysis, I offer the following:

If you are pretty confident about a single opponent’s opening range, then one strategy option is to call with the same range.  If you do so, your equity is 50%, so the EV equation in pot size units is as follows:

EVcall = 0.5*(1 + Bet) - 0.5*Bet = 0.5.

This equation used the setting of the strategy as the decision point, so hero’s investment is the bet plus the blind amount. An EV of 0.5 (half the pot) is to be expected if both players have 50% equity. Note that the EV is independent of the bet size and, if not a river decision, ignores future betting or assumes the hand is checked down.

If you raise rather than call, say to K*Bet, K>1, and villain fold equity is fe, then

EVraise = fe*1  + (1-fe)*(0.5*(1+ K*Bet)- 0.5*(K*Bet)) = 0.5*(1+fe)

If fe > 0, a raise results in a higher expected value than calling. If villain always folds (fe=1), EV = 1; if villain folds half the time, EV = 0.75, and, as shown by the previous equation, if villain never folds (fe=0), EV = 0.5. Again, these results as shown do not depend on the bet or raise size, but generally fe will be an increasing function of K and may dip for large K if villain suspects a bluff.

Note that since EV is not dependent on the raise size, it is suggested that you probably should raise big to increase the fe value, especially if your hand is a good one.

Of course, if your hand is at the higher end of the bet/call range, then the above results are conservative and they are optimistic if your hand is at the lower end of the range.

Counter Argument: If your equity is 50% when your calling range is equal to villain’s betting range, why not call with a tighter range so your equity is greater than 50%.

Answer: Following this reasoning, why not call only when you have aces for that will maximize your EV. True, but it only maximizes it for the hands you play and you will hardly play at all, so your EV is quite negative since you’ve been paying the blinds and possibly antes all the while. Of course, if the pot is large enough, pot odds theory says you can profitably call with realized equity less than 50%.

Summary. Have we found a simple solution to always be a winning player? Of course not. For one, you will never know your opponent’s range. And even if you can estimate how often he plays, say 20% of the time, his 20% hand range may (probably) not coincide with your 20% hand range. Even if you can accurately estimate opponent’s range, that doesn’t mean you should always call or raise his bet with that same range. If you have one of the lower ranked hands in the range, a fold may be best.  But, as with so many math-based strategies, this type of analysis can offer initial guidance to better playing, which can then be modified to include other factors not considered here.

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Addendum: We’ve shown calling at villain’s betting range has EV = 0. 5.  We can also call at a different range if we want to set EV = E, say. Let E(Rh,Rv) be the equity of calling with hero range of Rh when villain is betting with range Rv. Then

EV = E(Rh, Rv)*(Pot + 2Bet) - Bet = E

E(Rh,Rv) = (E + Bet)/(Pot + 2Bet)

For the ranking system I used, the following table shows the calling range to use for various bet sizes that results in a showdown EV equal to 0, thus assuring no loss. These results will vary somewhat for other ranking systems but should serve for general use.

Example: If villain makes a 2xPot size bet with a betting range of top 10%, to assure +EV (E=0), hero needs equity of 40% and achieves this with a top 26% range. If villain bets less than ¾ Pot with this range, hero can always call. Naturally if hero’s hand is at the bottom of a range, discretion should be used.

We note that in an earlier post (HM Quickie 15) we did a similar analysis but there we required that every hand in hero’s range had the required equity.

Introduction

An EV analysis on an early street is done to aid in making a bet decision. Such analysis is subject to the criticism that future bets are likely to occur and therefore results are not very meaningful. Also, the card equity value typically used is a showdown equity and therefore it ignores future fold possibilities. While these are valid points, I feel that an early street analysis has value as a first cut math approach to decision making. Also, there are ways to partially address the criticism.

With regards to the equity issue, one can adjust the showdown equity to represent realized equity, the equity you achieve by including incorrect hero and villain folds. This adjustment will be unique to each individual and can be gotten from one’s database. A future post on equity realization is in the works.

Another approach for an early street EV analysis is to apply an implied odds model, which accounts for the possibility of improving a drawing hand to a made hand on the next street and evaluates the chances for a big win vs. a big loss (reverse implied odds). Of course, you can always anticipate some future action and that is the focus here. We previously posted a series of articles on Implied Odds.

We cover this subject in two parts. In Part 1 we consider estimating a future bet. In the second part, we apply a constant bet factor model for estimating the future bet amounts.

Future Bets

We mentioned using realized equity and implied odds as methods to improve early street EV analysis. In this part we consider a third possibility to counter the criticisms made against early street EV analysis and that is to include an estimate of future bet amounts. Consider two cases:

A. EV analysis assuming no future bets.   B. EV analysis with future bets.

We consider the following:

Street: Any, Pot Size Before Bet = Pot  

Villain Bet = Bet, Hero Call Amount = Bet

Hero Equity = eq , Future Bet = Fbet

A: EVa = eq(Pot+2Bet) – Bet

B: EVb = eq(Pot +2(Bet +Fbet)) – (Bet+Fbet)

The similarities of the two equations should be apparent. For each equation, hero’s potential win is the total pot after both players bet. That amount is multiplied by his winning chances, eq, yielding his expected total return. From that amount is subtracted his risk amount to provide the net expected profit/loss. Note that if the bet was all-in, then EVa is exact, but is only an approximation otherwise. It  would be exact if both players checked down all future streets. EVb is also an approximation for we don’t know the FBet value. Is there any way we can approximate it? That is what we will now address.

Turn Bet Situation.

We shall assume villain makes a bet on the turn. If we call, the hand goes to the river. Only one future bet is involved (assuming no raise, re-raise action). It may be quite reasonable to assume what that future bet amounts to. For example, if villain is short stacked, he may decide to go all-in, especially if in a tournament where he is close to being blinded out.  Or hero may have history with the villain that indicates how much and how often villain bets on the river when being the aggressor on previous streets. Or, hero might consider several possible bet values and weight the resulting EV’s by estimates of their occurrence possibilities. Naturally you won’t be doing exact analysis while playing but the approach can be useful for developing your strategy while studying off-line as well as providing an experience base.

Let’s do an example by estimating hero’s EV for two cases:

Pot = 100: Villain Stack = 80:  Bet= 50: eq = 0.30

A: Future Bet Not Considered:

EVa = 0.30*(100 + 2*50) – 50 = 60 – 50 = 10

B: Future Bet Is Considered:

After the villain bet of 50 on the turn, his stack reduces to 30. Assume he goes all-in on the river after you call.

EVb = 0.30 *(100 + 2*(50+30)) – (50+30) = 78 – 80 = -2

EVa is the case where we do not consider a river bet. It would be totally applicable if villain and hero both checked the river. It has an EV of 10, so based on this, hero should call villain’s turn bet.

EVb is based on assuming villain goes all-in on the river with his presumably short stack. It has a negative EV for hero of -2.  Here, hero calling on the turn is not advisable. If these were the only two possibilities, the action to take would depend on how likely villain is to check or bet all-in. If, for example, you assume equal likelihood, the average EV is 4, so a turn call should be made. The complicating factors now specifically addressed is that of a villain raise, the villain bet size as related to his hand quality and fold equity. But, these are complicating factors for any other decision process.

Clearly, the future bet estimate is more perilous if it is to be made on an earlier street but the general concept holds.

Summary

You’re in a heads-up situation and out of position on a later street. Suppose you decide to bet for you don’t want to possibly give villain a free look at the next board card if he is on a draw. You know that villain may call or fold and that the larger the bet the more likely he will fold. On the other hand, if villain has the better hand, you will lose more with a larger bet. You also know that if your bet is a bluff, a fold is obviously the desired outcome.

Let’s do some math.

The EV equation in this situation is as follows:

EV = Pr(V folds)* Pot + Pr(V calls)*EVcall

EV = fe(B)*Pot +(1-fe(B))*(eq*(Pot + 2B) – B),

where

fe(B) is the probability villain folds when you bet B, which is also the amount villain must call

Pot is the pot before your bet

eq is the chance you win if villain calls

We seek the bet size, B, that maximizes our EV. But to do that we have to quantify fe(B). As noted earlier, players will normally fold more frequently with a larger bet size. However, if they have a very good hand, that is not true and it might not be true if they suspect you are bluffing or betting with a marginal hand hoping to cause a fold.

We’ll consider four cases where your hand is basically a bluff,  a so-so hand, a 50-50 race hand and a monster hand represented by equities of 10%, 33%, 50% and 90%, respectively. We’ll also assume the following fold equity function where the bet, B, is in pot size units:

fe(B)= B/(1+exp(B/2.7)

The fe function graph with a table of values follows:

We see that fe increases with bet size until about a 3.5Pot bet and then fe starts to decrease as villain suspects a bluff. At a ½ Pot bet, villain folds about 23% of the time and his maximum folding probability is about 75%.  Solving for the break-even bet size (one that sets EV to 0) analytically is a challenge I’d rather not tackle. Instead I solved for EV for various bet sizes for each of the assumed equities and the results are shown below:

The following is noted. A bet of 0 means hero checking; it is assumed the hand is then checked down. For each equity, the bet size that maximizes EV is bolded. It is 2.75Pot for both the 10% and 33% card equities and increases to 3.5Pot for the 50% case. Even with a bluff hand of only 10% equity, EV can be positive, in this case for a number of middle bets.  For the two high equity hands, bet the effective stack. It is seen that EV increases with the card equity as would be expected.  It should also be noted that showdown equities were used so unless the bet was all-in, these results are just a first cut at the EV evaluation.

What about the question posed in the title – do you want a fold or call? Remember, a fold earns you the pot, defaulted to 1 in this analysis. So, a call is more desirable if EVcall > 1. Well, for no equity less than or equal to 50% is a call better than a fold. Even for equities over 50%, a fold may be more desirable. For example, if the card equity was 60%, only for a bet of 2Pot or greater is EVcall greater than 1.  Naturally, these results are based on the fold equity function we used and adjustments to that function should be made to reflect your opponent’s tendencies.

A poster asked if his rough approximation for an EV-based decision was reasonable. He actually used the basic EV equation to see if the EV was greater than zero.  But, you don’t normally have to calculate EV. I think an easier way, or at least an equivalent way, is to see if your equity is greater than the amount you have to contribute to the total pot if you call.