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In limit hold 'em, preflop is the easiest street to play, but it is also very easy to come to the right conclusions for the wrong reasons. In this article, I am going to point out a couple common errors in evaluating multiway preflop decisions and present a more precise method of evaluating those decisions. The calculations will reveal a surprising conclusion about preflop decisions in multiway pots.

The PokerStove Fallacy

I like PokerStove. It has a number of great uses. For example, it is very good at estimating postflop equity against wide hand ranges. It is also good at giving new players a rough sense of hand rankings, hand values, and hand ranges. However, one of the most common misuses of PokerStove is to argue for or against a preflop decision.

PokerStove evaluates hands on a hot/cold basis. This means that it does not take into account any postflop action and does not consider the order in which cards fall. For example, PokerStove does not know the difference between flopping a flush, flopping a flush draw and hitting on the turn, flopping a flush draw and hitting on the river, and chasing down a runner-runner flush. But as poker players, we recognize that there is a huge difference between those situations.

I don’t believe that PokerStove is an effective tool for making multiway preflop decisions. For our purposes, multiway pots have at least three limpers in front of you, and the two blinds left to act behind, so that we expect at least 5 players to see the flop regardless of whether we call or raise. One of the major flaws of PokerStove is that it does not estimate the value of volatile hands very well.

What are volatile hands? Volatile hands are those hands whose preflop and postflop equity values are often very different against the same range of hands. What hands does this include? Perhaps it would be easier to talk about the hands that it excludes, as this list is short. AA-KK are hands that are strong preflop and usually remain strong postflop. For example, AA against 4 random hands has approximately 56% equity. If you pick a random flop and repeat the equity simulation, you'll find that the postflop equity (against the same range) remains close to 56%. Similarly, KK has approximately 50% equity preflop against 4 random hands, and its postflop equity remains near 50% except when there is an ace on the flop (this happens about 17% of the time and the equity falls to 35-40%, depending on the precise flop texture).

In fact, those two hands are the only ones where you can have a highly predictable postflop equity. The offsuit junk hands have medium volatility. They tend to have 10-15% equity preflop. This goes down to 5-10% postflop if they don't pair up and anywhere between 15% and 25% equity postflop for one pair, depending on the size of the paired card. Suited junk has a sharp spike in equity when it flops a flush draw.

All the hands in between AA/KK and the junk are very volatile. For example, AKo has about 32% equity preflop, but it jumps up to 62% when it pairs up, and drops down to around 17% when it doesn't. Q9s is even more volatile. It starts at about 26% equity, which drops down to around 15% when it flops nothing. It goes up to 35-45% equity for flush draws (depending on the texture) and similarly for straight draws. If it pairs up the equity is in the 40-50% range. All of this variation comes as a result of the flop textures, and since PokerStove deals out the flop, turn, and river all together, it fails to see these effects.

The Implied Odds Fallacy

By definition, implied odds are the ratio of the total expected win to the size of the investment. This concept is often used in postflop play when determining whether to call. When the pot odds are not sufficient to make a call you can sometimes justify calling because of the added bets you will win if you make your hand. This implies that the pot is actually bigger than what it seems.

Some players try to use implied odds to justify making raises. “By making the pot bigger,” they say, “we're increasing our implied odds.” In no-limit, this might be a valid statement as a bigger preflop pot may entice a player to become pot-committed, whereas in a smaller pot he still has the option to fold. But in limit hold ‘em, this statement is nonsense. Because implied odds are a ratio of investment to future pot size, a preflop raise will only increase your implied odds if you expect to win at least twice as much as a result of the raise.

For example, suppose that have you have 22 preflop on the button and you expect the pot to be 7-handed postflop. Suppose further that you expect to win 7 BB on average when you flop a set. You implied odds for limping are 14:1. Since you are 8:1 to flop a set, you can play because you have good implied odds. What happens if you raise? If your implied odds are to increase, you might conclude that you will need to win at least 14 BB when you flop a set. This is already a somewhat difficult (not impossible) task, but this number does not correctly factor in the increased losses when you do not flop a set. You do not flop a set about 88% of the time, which means that you must make up for the extra .5 BB loss a vast majority of the time. In essence, your preflop raise is roughly equivalent to facing a preflop raise. It is still a profitable situation and it might even be more profitable because of the extra money in the pot, but your implied odds are cut down because of the larger initial investment and the infrequency with which you actually flop a set.

Implied Value

Before I define implied value, I want to dissect some preflop decisions a little more closely. Suppose you have AKo and there are three limpers in front of you. What is your action? Of course you raise, but why? I suspect the answers will generally sound like “you have a strong hand.” But why is it such a strong hand? Well, it is often the best hand preflop. But if you follow that logic, you should also raise 22 in this situation because 22 has an even better chance of being the best hand. Yet we don't raise 22 preflop because it isn't a “strong hand.” Some players will then commit the PokerStove fallacy and show you what great equity you have as a reason for raising (AKo has 32% equity against 4 random hands, but 22 only has about 18% equity).

What makes AKo a strong hand, but a better 22 not so strong? The answer is that AKo will flop a very favorable hand about 1/3 of the time, but 22 will only flop a favorable hand about 1/9 of the time.

I define the implied value of a hand to be the average value of a hand when it hits a favorable flop. The term encapsulates the germ of truth behind the implied odds fallacy and better reflects how we should really think about preflop decisions. Determining the implied value of a hand comes down to estimating:

1) The frequency of favorable flops
2) How much one should expect to win on average on a favorable flop.

Frequency of Favorable Flops

Pocket pairs: The numbers represent the total number of flops fitting each characteristic for each pocket pair. The location of the PP represents its rank relative to the other cards on the board. For example, PP x y z means that you have an overpair on a board with three different cards. x x PP y is a paired board where the paired card is higher than pocket pair and the unpaired card is lower than the pocket pair. From this chart, you can see that TT will face exactly one overcard on 7,168 + 768 = 7,836 flops. There are 19,600 possible flops, which means that this will happen 7,836/19,600 = 40% of the time.

Non-pairs: This is significantly more complicated because of the wider range of things that can happen. Instead of trying to do the same type of comprehensive list as I did for the pocket pairs, I will present an abbreviated list that highlights flopping top pair or better.

The next chart is a list of suited and connected flops. For suited hands, the number of flopped flushes and flush draws are completely independent of the cards that are held. But for flopped straights and open-ended/double gutshot straight draws, the ranks once again matter. The capital X indicates the hole cards, the lower-case x indicates the board cards, and the dashes indicate blanks on the board. For example, with 98, the notation - x XX x – indicates that the board is T7 and another card that is not a J, T, 9, 8, 7, or 6; hero's hand is an open-ended straight draw.

How can these numbers be used? We can now estimate the number of favorable flops for various starting hands.

Example 1: Hero holds 99. He will flop a set or quads on 2,304 flops. He will also flop an overpair on an unpaired board another 2,240 flops. There are also another 1,008 + 28 flops where the board will be paired and still everything will be below 9. Hero will get a favorable flop 5,580 out of 19,600, or about 28% of the time.

Example 2: Hero holds AKo. He will pair up on 3,320 + 2,640 flops, and he will also flop broadway on 64 flops. This adds up to 6,024 favorable flops, which means it will happen about 31% of the time. It is true that we are not counting every single possibility here. For example, we have only counted top pair on unpaired boards and we have not counted any gutshot draws. However, those situations are more marginal in their favorability, so I have excluded them from these calculations. If you wanted to make a more accurate estimate, you can bump this up a few percent to account for those cases.

Example 3: Hero holds Q9s. He will flop a flush or flush draw on 2,310 flops, plus a straight or 8-out straight draw on another 704 flops. He will also pair his queen as top pair or better on 2408 flops and a top pair of 9s on 1,008 flops. In all, there are 6,430 favorable flops or about 33% favorable. This is a slight overestimate because we have double counted the straight and flush flops, but the size of this error is not more than a few tenths of a percent.

What Does it Mean?

We will calculate the implied value of the three hands listed above. However, we cannot proceed unless we make some assumptions about the future action. This is an unfortunately nebulous process, as it requires one to estimate of how much extra action will result from the preflop raise, which greatly depends on the tendencies of the other players at the table and the specifics of the flop texture. One must also discount the pot size for drawing hands relative to made hands because of the lost money chasing the draws when they do not come in.

It turns out to be significantly better to use the size of the win instead of the overall pot size. The reason is that the pot size is bloated by Hero’s bets and raises which do not count towards profit. In different situations, this can both over and underestimate the true value of hands. We will be working with the assumption of a 5-handed pot under various settings, but you are welcome to adjust the numbers to make them more applicable to your specific situations.

4.5 BB wins: This represents the small, limped pot wins in a weak-tight setting. This number comes from

4 SB preflop + 2 SB flop + 1 BB turn + .5 BB river

7.5 BB wins: This represents the medium-sized pot that you get from a raised pot in a weak-tight table

8 SB preflop + 3 SB flop + 1.25 BB turn + .75 BB river.

6 BB wins: This represents a medium-sized win from a limped pot in at a looser table.

4 SB preflop + 3 SB flop + 1.5 BB turn + 1 BB river

9.5 BB wins: This represents a large pot win in a raised pot against a looser table.

8 SB preflop + 4 SB flop + 2 BB turn + 1.5 BB river.

The second aspect of this calculation is determining how Hero will win the pot. For example, if Hero flops a nut flush draw, he should not expect to win as often as if he flops top pair, top kicker. The table below comes as a rough estimate based on equities over a range of flop textures against two random hands. These are only estimates, so you are welcome to adjust them as you see fit. (In fact, it is probably a good exercise to change these numbers around a little bit anyway, just to see how much flexibility the results have.) These numbers will be used to weight the various situations.

 

Finally, we will suppose that Hero will check-fold any unfavorable flop. This is a dramatic oversimplification, as Hero will often have odds to favorably chase with second pair or a weak draw. This makes raising slightly more profitable than the numbers indicate.

I have devised a simple spreadsheet to perform the calculations. The basic idea behind the calculator is that it proportionally weights hero’s equity on the favorable flops and subtracts off the losses due to flop check-folds on the unfavorable flops.

Here are the results for the three examples above:

Example 1 - 99

This confirms what most players already recognize. Even in a multiway pot, pocket 9s has enough value to be raised preflop.

Example 2 – AKo:

This result displays one of the weaknesses of this methodology. Both experience and intuition suggest that AKo is more profitable than 99. Why isn’t this reflected here? The reason is that the postflop play of these hands is very different. AK will often see the turn and be able to chase his overcard outs, whereas almost all of the value of 99 is gone (in multiway pots) if it doesn’t get a favorable flop. Both of these will increase the value of AKo relative to the numerical results. However, the relative result of raising being better than limping still stands.

Example 3 – Q9s:

This is a surprising result. Hands that flop many draws, such as Q9s turn out to more valuable in raised pots than limped pots in loose games. Essentially, they flop decent draws often enough to warrant the extra preflop investment. Once again this calculation actually underestimates the value of these hands, as Hero will have good odds to chase two pair or trips in the raised pots (perhaps also in the limped pots, depending on the action and table conditions).

Perhaps what is most striking is the relative size of the increase in the looser game. Increasing the value by 0.08 BB turns out to be nearly a 20% increase in the value of the hand (but this could be 25-30% depending on the effect of drawing to two pair, trips, and sometimes backdoor flushes and straights).

Finally, to demonstrate that this method does not inflate every hand, here are the results of raising 22:

 

Conclusions and Caveats:

Based on these calculations, it turns out that many drawing hands were made for gambling in multiway pots. Suited connected and even gapped hands show a non-trivial increase in value when they are raised preflop in these situations.

Also, postflop tendencies have a prominent role in these types of gambling raises. The raises may not be as effective in passive or weak games where you do not get paid off as much when you do hit your hand (the Q9s example demonstrates this).

Keep in mind that the calculations do not take into consideration the effects on postflop action, such as often having the players check to you on the flop, which only further increase the value of these hands. Also, the ability to draw out on your opponents on unfavorable flops (such as with the AK example) will increase the value of hand beyond the computed results.

 

 

Note from Two Plus Two: Even though we have run this article, there is one spot where we disagree. It can be correct to raise with highly speculative hands like likes deuces or eight-seven suited in multiway pots if your raise will entice some opponents to call with hands on the flop that they would fold in smaller pots.

For example, if you flop a flush draw argainst a player with top pair and another player with middle pair, you want this second person to call. If a large pot now entices him to do so, you gain even if this player is correct to call with middle pair since the player with top pair is the one to give up expectation while you actually gain in this area. See our book Hold 'em Poker for Advanced Players for more discussion.