#holdem math

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.

A poker player asked how to estimate the probability his pair of tens will face an over-card on the flop in Hold ‘em.

It is clear that with a pair, over-cards on the board are threatening. Here is the method to calculate the percentage of seeing at least one over-card to your pair on the flop. The equations for the turn and river are shown near the end of this posting for a ten-pair rank. Assume the pair is of rank R. Then there are 4*(R-2) cards less than R in the deck and 2 cards equal to R. The probability of at least one flop card greater than R is then

1 – the probability that all flop cards are less than or equal to R.

The probability equation is: Pr(one or more over-cards to a pair of rank R on flop)

= 1 - C(4*(R-2)+2, 3)/C(50,3) The following table shows the results of NO over-card for all pairs on the flop, turn and river:

Example: You have a pair of tens. There are 4*(10-2) + 2 = 34 cards less than or equal to 10. Pr(No over-card on flop) = C(34,3)/C(50,3) = 0.305

Pr(No over-card by turn) = C(34,4)/C(50,4) = 0.201

Pr(No over-card by river) = C(34,5)/C(50,5) = 0.131

We continue here with a third post on a small blind pre-flop opening. We noted in Post 29 that in such a situation the SB can assume the BB villain has a random hand. In that case we showed with tables and graphs the attainable EV for three showdown equities, 40%, 55% and 70% for various bet sizes and fold equities.

In Post 30, we provided tables that show critical bet sizes for villain indifference and for hero positive showdown EV for selected hands, mostly applicable for large stacks. For small stacks, the card equity needed for a positive EV shove was provided in a table and graph given bet size and big blind call %. That card equity value then defines the hands with which you can shove. A following table and graph display the associated shoving range for the same bet sizes and villain calling frequency assuming villain calling frequency defines his calling range. We also provided a link describing the Slansky-Chubukov theory on a small blind open.

Here we continue that discussion for more general cases by operating on the EV equation for a small blind open. We note that since future betting will likely take place if the raise is not a shove and the BB action is a call, that these results are only a first-cut look at the situation.

Expected Value Analysis  

While the small blind may not get to open-raise pre-flop too often in a multi-player game, when he does, it can be a good opportunity to profit. The applicable EV equation in big blind units is as follows for SB raise and BB call:

EV = fe*1.5 + (1-fe)*(eq*(1.5 + R – 0.5) – (1-eq)*R)

= fe*1.5 + (1-fe)*(eq*(1 + 2R) – R)

where

fe = fold equity

eq = hero card equity

R = SB raise including the 0.5 blind completion. BB call = R – 0.5

The EV for various values of fe, eq and R are shown in the following table. We only show equity values less than 50%; EV is always positive for values greater than 50%.

It is seen that EV decreases with increasing raise size but increases with increasing equity and fold equity. For a relatively small raise like 3bb, the majority of cases examined show a profit (shaded cells). This is also true for large fold equity like 75%. So, if you think the big blind has a good chance of folding to an opening raise, then it is +EV to do so. Of course, the bigger the bet, the more likely it will cause a fold, but, as shown, big bets decrease EV. Here is where logic and psychology have to be considered along with the math for good decisions. What we can do with the math is to show the card equity or the fold equity needed to achieve +EV which we do in the next two sections.

Required Fold Equity

Here we manipulate the EV equation by setting it to 0 to provide the minimum fold equity for +EV. The equation is as follows:

Min fe = -(eq*(1+2R)- R))/(1.5-( eq*(1+2R)- R))

The results for various raise and card equity values are shown below:

It is seen that the minimum fold equity increases with raise size and decreases with increasing card equity. Blanks signify a plus EV situation for even a 100% call.

Required Card Equity

Here we manipulate the EV equation by setting it to 0 to provide the minimum card equity for +EV. The equation is as follows:

Min eq =  ((1-fe)*R - fe*1.5 )/((1-fe)( 1+2R))

The results for various raise and fold equity values are shown below (another form of this table appeared in Post 30):

As to be expected increasing fold equity results in a decreasing minimum card equity. For raises of 3 and lower, if the big blind folds about 2/3 of the time, the small blind can play just about any two cards. The same is true for raises up to 5bb with fold equity at 75% or greater.

Summary

“I found out that Resource xxx[1] suggests folding 55% against 3x open in a heads-up match. Isn't this exploitable? I mean, if we are defending less than half the time, then villain should raise with any two cards simply because he is making money immediately. What do you think?”

We’ll consider this question but make our hero be the bettor.

If you are betting as hero and if villain folds 55% of the time heads up, then hero wins at least 55% of the time. Assume a pot of 1 and an open hero bet equal to Bet that villain either calls or folds:

EV = 0.55*(1) + 0.45*(eq*(1+ 2Bet) – Bet),

where eq is hero’s card equity.

The most pessimistic case is eq = 0;  i.e., hero never wins if villain calls. Then

EV = 0.55 – 0.45*Bet  

In this case, EV is positive if Bet < 55/45 = 1.22 Pot when betting with complete air.

What equity is required so hero is assured of winning, given the bet size?

EV = 0.55 + 0.45*(eq*(1+ 2Bet) – Bet) > 0

Eq > (0.45*Bet- 0.55)/(0.45 + 0.90Bet) = (9*Bet-11)/(9+18*Bet)

The following table and graph show some results:

Example. If villain is expected to fold at least 55% of the time, a bet of twice the pot requires card equity of at least 16% for +EV.

For a 3x bet, EV is positive if hero’s equity is at least 25%, which is generally true for any two cards preflop if the hand goes to showdown. So, poster is almost certainly correct in stating that a high folding probability exposes one to being exploited provided that the exploiter can realize his equity.          

Key Points:

In general, in any situation where hero has better than 50% showdown equity for a particular action, his showdown EV will always be positive. If fold equity is high, say over 50%, then in most cases a bet of 3x or less will be a +EV move. For larger bets, which may result in a higher fold equity, the table provides guidance on minimum card equity.

Future posts will generalize the relationships between card equity, fold equity and bet size relative to expected value.

Finally, whenever showdown card equity is included in an EV analysis, it is important to adjust a result or conclusion to account for equity realization;  that is, a showdown equity assumes hero does not fold in a future round, an assumption that may not be true.

Multiway EV analysis is complicated. Typically, there will be a main pot and one or more side pots dependent on the number of players and how stack sizes are distributed. Yet, it can be done if one in his study is willing to do the math. We present here an example of a 3-way hand a poster on a poker forum asked about:

“If my stack is 10, Villain A stack is 10, and Villain B stack is 5, I can work out the EV for when I go all-In and just A calls, or the EV when I go all-in and just the B calls, but I don't know the proper way to do it for when they both call.”

Admittedly this is a fairly simple multi-way problem compared to some others but one that is useful for indicating the approach. We’ll make the following assumptions:

Pot = 1. Hero goes all in with a bet of 10.

Hero hand/stack: AKs /10.  Villain A range/stack = 15% / 10.  Villain B range/stack = 30% / 5.

First, we obtain Hero’s equity for the following cases using a hand evaluator such as Equilab.

H vs. A vs. B= 46%, 29%, 25%.   (A wins, B wins, C wins)

For the relevant two-way contest, we have using the H and A equities,

H vs. A = 46 / (46+29) = 61%,  

Note this result is very close to what you get if you use Equilab in a two-way confrontation (63%) but that would not account for card removal of the third player, B.

Based on the stack sizes, the 3-way or main pot is 1 + 10 + 5 = 16 and the 2-way side pot is 5. The side pot occurs if B wins the hand leaving H vs. A fighting for it.

The following table summarizes the situation:

Note: Sum of probabilities = 100%

The EV equation is as follows:

EV = Sum [Pr(Win)*Win$] – Sum [Pr(Lose)*Lose$]

EV_hero = 0.46 * 16 + 0.15 * 5 – (0.29 + 0.10) * 10 = 4.2

In this series we established a model for directly considering important factors for determining the implied odds needed for calling a current street bet where the pot odds are insufficient. If P$ is the pot after villain bets on the current street, C$ is the amount hero has to call and F$ is the amount hero will bet on the next street if he hits his outs, the implied odds equation is as follows:

IO = (P$ + F$)/C$

To determine F$, the future bet requirement to achieve +EV, we have the following:

EV =(1-H)(-C$)+ H(C(W(P$+F$) - (1-W)(C$+F$))) + H(1-C)P$

F$ = (C$ - HK$(1+CW- C)) / (CH(2W- 1))

where

K$ = P$ + C$, the pot after all bets are made on the current street

H = the probability hero will hit his outs on the next street

W = the probability hero will win the hand if he hits, bets and villain calls

C = villain call probability when hero makes a future bet

For example, for a set mining situation with a low to middle pair, the chance for a hit is about 12%, meaning if you hit and win, you need pot odds of 88 to 12 or 7.3 to 1. Given that you don’t have these odds, you consider the possibility of hitting the set on the flop and winning big. Recognizing that you may not win with a hit, villain may not call if you do hit or the effective stack you might win is not very big, you often see a comment that you should have implied odds or 2 to 3 times the minimum odds, which would be about 15 to 20 to 1 for set mining. The model we propose is an attempt to quantitatively include these factors so that there is mathematical justification for the required implied odds.

Villain Calling Percent Variations. For simplicity both in concept and exposition, the results we provided in the previous posts had a fairly strong assumption, namely that villain always calls hero’s bet. We now will attempt to show some results where we no longer make this assumption. The following table shows the minimum implied odds for various combinations of Call%, (C), C$, Hit and Win. In every case the implied odds requirement goes down as Hit or Win increase or C$ decreases. This is to be expected for hit or win increasing means hero’s equity is increasing so the future bet requirement decreases. On the other hand, as C$ gets bigger, hero’s risk is greater, thus the reward as represented by the implied odds has to be greater.

For the Call%, the implied odds also decrease for all the table values we considered, Note that the W values, the chance of winning if hero hits his outs, are greater than 50% for all cases. A player with a greater than 50% winning chance normally would prefer a villain call rather than a fold and that is reflected in the table values.  However, we did not include a large range of C$, W and H combinations so this is not always true as there can be instances where greater villain calling does result in a larger future bet and implied odds requirement, usually for low hit values and/or large C$ values.

Example. Assume hero has to call a bet that is 50% of the pot after villain bets on the current street. If the expected hit probability is 20%, such as would be the case for a flush draw, and the win probability is 75%, the following implied odds are required for the following call probabilities:  Call= 10%, IO = 43.5. Call = 50%, IO = 11.5. Call = 100%, IO = 7.5. Because of the high equity given a hit (W=75%), we see that the implied odds and associated future bet requirement decreases as the call probability increases.

Model Application and Summary

Here is my first HM Quickie. A Hold’em Mathology Quickie will be some poker math issue that came up recently, usually through my participation in poker forums, that I thought would be useful to post but that doesn’t require the time, effort or detail for a full posting. It will generally be short and require that the reader have an understanding of basic poker math.

In a poker forum a poster wanted to confirm his EV analysis. He held AThh with a flop of 8h 3h Kc. With a pot of 380, villain bet 370, all-in. Hero could call or fold, the latter having an EV of 0.

EVcall = eq*(Pot + Bet) – (1 – eq)*Bet

Hero has the nut flush draw which has 9 outs. The 2x 4x rule then would provide an estimate of 4*9 = 36% for a hit. The exact hit percentage is

Pr(hit on turn) + P(don’t hit on turn but hit on river) = 9/47 + 38/47 * 9/46 = 35%.

Assuming that a flush hit wins the hand, 35% would be hero’s equity and his EVcall is as follows:

EVcall = 0.35* (380 + 370) – 0.65*(370) = 22

So, a positive EV would suggest a call is profitable compared to a fold.

The poster also asked what is the maximum bet size for a +EV call. If we let X represent the bet amount   and set EVcall to 0, we have

EVcall = 0.35*(380 + X) – 0.65*X = 0

After a little algebra,

X= 380 * 0.65/0.35 = 443.3,

assuming both player’s stack sizes have that amount.

A comment by a very competent poster followed:

“It's impossible to calculate the largest bet size you can call (and at least break even) unless you provide villain's exact range.”

While I understand the thinking, I don’t agree with the “impossible” conclusion. As an example, if you know how many outs you have for the absolute nuts, you then know your card equity and can employ the applicable EV equation to solve for other variables.

Also, you never or at least rarely know villain’s exact range. I stated in the analysis I did that I assumed that given the nut flush hits, it’s a win. That implicitly defines opponent’s range. The best villain can have is KK, 88 or 33 each of which has 7 outs for a full house or quads, thus beating a flush. But the chance for those villain hands is only 3 * 3/C(47,2) = 0.0083. And, as noted by the commenter, if a limited range is considered, the probability of KK, 88, or K8 may rise to about 4%. Also noted, hero could win without the flush with his AT, so perhaps the win/lose effects negate each other. There are other possible situations such as hero hitting a straight or villain having two pair. Again, it is not unreasonable to assume a balancing out if there is no obvious advantage to one side.