Overestimating Your Implied Odds Against Short Stacks in No-Limit Hold´Em

One of the biggest mistakes that I find players making in the online micro and small stakes game is that they often think that they have way more implied odds than what they really have. They make preflop calls with hands like small pocket pairs and figure if they play purely to hit a set and win then they are still making a +EV call. If you are a 100BB deep, as you often are in many games today, then this is fine and dandy. However, there are often times when the stacks are a little bit shorter. One of the key factors in deciding your strategy is stack sizes. As the stacks get shorter you cannot make as many calls for implied odds.

Making incorrect calls based on implied odds often stems from misunderstanding what implied odds really are, underestimating the cost of flopping a set and losing, and overestimating how much you really will win from an opponent. Let us take a look at a typical short-stacked example to see what I am getting at.

Say you are playing $.50/1 No-Limit Hold Em with $50(50BB) effective stacks and a very tight raiser opens to $4 from early position. You assume that his range is TT+/AK. You are sitting on the button with 22. You know that your hand has poor equity against his range, but you assume that you are getting implied odds to call. Also, since he has a narrow range he will have a hand that he is willing to stack off with a reasonable amount of the time. At this point in time it would be a good idea to try to estimate your implied odds. However, a lot of good players misbelieve that their implied odds are the total amount of money they could win when they hit compared to the cost of the call. This is incorrect. To get specific with our example, there is $4 in the pot now (pretend that the blinds are raked away to keep the math fairly simple) and there is $46 left to win. That means you could win as much as $50 and it only cost you $4 to call. That means 50/4 or 12.5-to-1 implied odds. This is not implied odds. This is stack odds. Implied odds are the average amount you will win when you hit your hand as compared to the cost of the call. The average amount you will win when you hit your hand is not your opponent´s entire stack. Sometimes your opponent wont put all of their stack in(which will often be the case if they have AK and didn´t flop a pair). Sometimes your opponent will put their stack in and you will be behind. Sometimes your opponent will put their stack in and happen to catch a better hand on the turn or the river. You will not win your opponent´s stack 100% of the time that you flop a set. Your implied odds aren´t ever as good as your stack odds because you are never 100% to win your opponent´s stack when you hit your hand.

Implied odds = (Average amount of win when you hit your hand)/(price of the call)

Let´s do a raw EV calculation just to get an idea of what your implied odds look like in the above example. Assume that you will not continue unless you flop a set and that your villain will stack off 100% of the time he flops an over-pair/top-pair or better. Below is how often he flops an over-pair or top pair with each hand in his range when you flop a set:

AA – 100%
KK – 83%
QQ – 66%
JJ – 48%
TT – 31%
AK – 26%

AA

This hand is always stacking off. However, about 8.5% of the time that you flop a set, it does also and about 9.2% of the time that it doesn´t flop a set it will catch up. To simplify, you will win 82.3% of the time and lose 17.7% of the time. Thus, your EV looks like what follows:

(% of time we flop a set and win)(net amount won) + (% of time we flop a set and lose)(net amount lost) = EV

(.823)($50) + (.177)(-$50) =
$41.15 – $8.85 = $32.3

KK

KK gets a little more complicated because it only gets all-in with you if he has an over-pair or a set. This basically works out that he gets it in 83% of the time he has an over-pair and 8.5% he has a set. When he has an over-pair you have about 90.8% equity. I am ignoring the spot where you both flop a set and there is an A on the board. This assumes that he flops an over-pair 83% of the time, a set 8.5%, and an overcard comes the rest of the time.

(% of time we flop both flop sets)(net amount lost) + (% of time our opponent flops an over-pair) [(%of time our set holds up)(net amount won) + (% of time our opponent catches a set on the turn or river)] + (% of time villain folds when an overcard hits the flop)(net amount won) = EV

(.085)(-$50) + (.83)[(.908)($50) + (.092)($-50)] + (.085)($8) =
(-$4.25) + (.83)[$45.4 – $4.6] + $0.68 =
-$4.25 + $33.86 + $0.68 = $30.29

QQ

QQ-TT follow the same logic as KK only they flop an over-pair less often.

(.085)(-$50) + (.66)[(.908)($50) + (.092)(-$50)] + (.255)($8) =
-$4.25 + (.66)[$45.4 – $4.6] + $2.04 =
-$4.25 + $26.93 + 2.04 = $24.72

JJ

(.085)(-$50) + (.48)[(.908)($50) + (.092)(-$50)] + (.435)($8) =
-$4.25 + (.48)[$45.40 – $4.60] + $3.48 =
-$4.25 + $19.58 + $3.48 = $18.81

TT

(.085)(-$50) + (.31)[(.908)($50) + (.092)(-$50)] + (.605)($8) =
-$4.25 + (.31)[$45.40 – $4.60] + $4.85 =
-$4.25 + $12.65 + $4.85 = $13.25

AK

AK is a little different than the others because we essentially have 100% equity when we get it all in (Sure, there are backdoor draws and on certain flops he has a straight. Let’s keep the math simple. You won’t hit a set when he flops a straight so his straight will be of the backdoor variety that will make you throw up when it hits!). We are going to assume that if he doesn´t hit his hand you just win a pot-sized c-bet.

(.26)[(1)($50)] + (.74)($8) =
$13 + $5.92 = $18.92

Calculating the EV when you flop a set versus your opponent´s range

Now that we have our EV versus each part of the range when we hit a set we can combine that all together to get our average EV versus his entire range those times that we do hit a set. All of the pocket pairs have 6 combos and the AK has 16 combos. I will simply multiply the calculated EV of each hand by the number of combos it has and divide by the total number of combos in his range (46 total combos) to get the average.

[(6)($32.30) + (6)($30.29) + (6)($24.72) + (6)($18.81) + (6)($13.25) + (16)($18.92)]/46=
[$193.80 + $181.74 + $148.32 + $112.86 + $79.50+ $302.72] / 46 =
$1018.94/46 = $22.15

Calculating total EV of a preflop call

We will flop a set 13% of the time and that we lose our preflop investment the other 87% to get the total EV of the call.

(.13)($22.15) + (.87)*($-4) =
$2.88 – $3.48 = -$0.60 EV

It is EV to make this call with 22 purely for set value when the stacks are 50BB deep. Let´s calculate how deep the stacks should be for us to be able to make this call. First let´s figure out how much we need to make when we do flop a set for this call to be neutral EV by setting the value of flopping a set to and setting the EV to 0 . We can work backwards from there.

(.13)(X) + (.87)*($-4) = $0
.13X â“ 3.48 = $0
.13X = 3.48
X ≈ $26.77

We need to average winning $26.77 when we flop a set. If we divide by the difference of $26.77 and $22.15 by 46(the total number of combos in our opponent´s range) we can calculate how much more we need to make on average of each combo. From this we can calculate how much we need to make off of each hand in our opponent´s range, and thus what the stack size would need to be to make that amount.

Per-combo value needed to set-mine
($26.77 – $22.15)/46 ≈ $0.1

AA

We previously calculated our EV against AA as $32.3. We need to make $0.6 more off of AA than what we currently make. By setting the stack size equal to and the EV equal to $32.9 we can solve for X.

(.823)(X) + (.177)(-$X) = $32.9
0.646X = $32.9
X ≈ $50.93

KK

The same logic follows for the other hole cards. KK´s EV was previous $30.29 so now we will set the EV equation to $30.89.

(.085)(-X) + (.83)[(.908)(X) + (.092)(-X)] + (.085)($8) = $30.89
-.085X + (.83)[.816X] + $0.68 = $30.89
-.085X + .678X = $30.21
.593X = $30.21
X ≈ $50.94

QQ

(.085)(-X) + (.66)[(.908)(X) + (.092)(-X)] + (.255)($8) = $25.32
-.085X + .66[.816X] + $2.04 = $25.32
-.085X + .539X = $23.28
.454X = $23.28
X ≈ $51.28

JJ

(.085)(-X) + (.48)[.816X] + (.435)($8) = $19.41 -.085X + .392X + $3.48 = $19.41
.307X = $15.93
X ≈ $51.89

TT

(.085)(-X) + (.31)[(.816X)] + (.605)($8) = $13.85 -.085X + .253X + $4.85 = $13.85
.168X = $9
X ≈ $53.57

AK

AK is different because there are 16 combos. We will need to add $1.60 to the EV to account for this.

(.26)[(1)(X)] + (.74)($8) = $20.52
.26X + $5.92 = $20.52
.26X = $14.6
X ≈ $56.15

Average the x´s to find the neutral EV stack size against this opponent.

[(6)($50.93) + (6)($50.94) + (6)($51.28) + (6)($51.89) + (6)($53.57) + (16)($56.15)]/46 [$305.58 + $305.64 + $307.68 + $311.34 + $321.42 + 898.4]/46 =
$2450.06/46 ≈ $53.26

Conclusion

You cannot call with a pocket pair purely for set value against a player who is raising a similar range to the one I provided if you only 50BB stacks. The point of all these calculations is to give you a baseline to make your judgment calls from. Given the fact that you will often be facing wider ranges, you probably can start calling raises purely for set value once stack sizes are in the 60-70BB range . If you make a bad here call over and over again imagine how much this will slash your win-rate! In fact, if you believe that this scenario comes up once every two-hundred hands then this particular leak alone would slash your win-rate by about 0.30 big blinds/100 hands. That is a significant chunk for such an easily fixable mistake! This is a huge leak that I see reasonable players get wrong over and over again.

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