Big Hands and Keeping a Clear Head

One of the biggest challenges for amateur players is maintaining an emotionless approach when picking up a huge hand. I cannot count how many times I've seen someone pick up a premium hand and play with little to no regard for the actions of the other players at the table. Playing live gives some insight into the mental state of these players and their thought process goes something like this:

To piggyback on my post from Tuesday, if and when they end up losing their loss is often followed by a lamentation about one or more of A) their terrible luck, B) the inconceivable play of their opponent(s), and C) how they had "no choice" but to play the hand the way they did. "C" is where large amounts of money are needlessly lost. To demonstrate, let's look at a hand I played a few days ago:

Seat 1: Donzo ($50.85 in chips)
Seat 2: UTGplus1 ($39.82 in chips)
Seat 3: UTGplus2 ($70.73 in chips)
Seat 4: Dealer ($52.99 in chips)
Seat 5: Small Blind ($68.21 in chips)
Seat 6: Big Blind ($50.50 in chips)
Small Blind: posts small blind $0.25
Big Blind: posts big blind $0.50
*** HOLE CARDS ***
Dealt to Donzo [Ac Ah]
Donzo: raises $1.50 to $1.50
UTGplus1: calls $1.50
UTGplus2: calls $1.50
Dealer: folds
Small Blind: calls $1.25
Big Blind: calls $1
*** FLOP *** [3s 7d 4c]
Small Blind: checks
Big Blind: checks
Donzo: bets $4.50
UTGplus1: folds
UTGplus2: calls $4.50
Small Blind: folds
Big Blind: folds

We're dealt aces UTG and make a standard 3x raise. Four people call, which is more than we're hoping to see, but we're still the overwhelming favorite heading into the flop.

The flop comes 374 rainbow (all different suits) and the blinds check to us. Although this is a low flop that initially looks ideal, it's actually not great for our hand because A) there are no larger pairs that could have hit, meaning hands that pair the board are going to have a hard time paying us off, B) there are no potential flush draws, and C) there are only gut shot straight draws, as 56 already has a straight on the flop. That said, we're still usually going to have the best hand and want to bet to get value from hands like A7, a pair with a gutshot (e.g. 45), and inferior overpairs to the board. We bet $4.50 into $7.50 for 60% pot and get one caller.

*** TURN *** [3s 7d 4c] [2s]
Donzo: bets $8.50
UTGplus2: raises $17 to $17

Unless the villain floated us with A5 or 22 (or inexplicably was in the hand with 72), the turn is effectively a blank. We again bet for value and are quickly min-raised.

At this point it's helpful to talk about the Baluga Whale Theorem. If you don't want to click the link, it's the idea that when faced with a turn raise, the strength of single pair hands must be seriously reevaluated. When an opponent just calls the flop and raises the turn, they're saying "my hand was strong enough on the flop that I wasn't scared of anything and was willing to let another card come so that I could win more money from you firing a second barrel." It is of course possible that some villains will float with air on the flop and raise the turn, but that's generally a more sophisticated play and not one that you see often at lower limits. Do some players know the Baluga Whale Theorem and use it to bluff? Absolutely, and it's something I expect to see more as I move up in stakes.

Furthermore, if a player is on a draw and is inclined toward playing draws more aggressively, the likelihood of a raise happening on the flop is much greater because A) there is less money invested on the flop than the turn, making a bluff attempt less costly, and B) a raise on the flop, if called, has a strong chance to induce a check on the turn from the original bettor (particularly if the original bettor is first to act), leading to the option for a free card for the raiser.

So, to return to our hand and our villain, what are the reasonable hands for him/her to have? If we consider the hands that had us beat on the flop, there are 9 possible set combinations and 16 straight combinations. Let's also include the 1 combination of A7 of spades, as that's a hand that might've originally called for value, but wants to turn the hand into a bluff now that we've fired a second barrel. If we assume that the villain is only raising those hands and we plug them into Equilab, we have an atrocious 7.5% equity with one card to come. But what if they're sometimes raising worse overpairs for value? If we add 88-KK into the villain's range, our equity skyrockets to 58.6%. But if we just take a moment to review the preflop action, we will see how unlikely most of those hands are. The villain is in the cutoff and there was a caller prior to the action getting to him/her. While a flat call with a big pocket pair sometimes happens in response to just the original raiser, it's something one rarely sees when a third person is involved in a pot, not to mention the dealer and blinds who have yet to act. Consequently, the likelihood of KK or QQ is slim to none, as is the likelihood of JJ or TT, although I'll leave a slightly possibility for those to the arbitrary tune of 25% of the time. 99 and 88 are believable calling hands both preflop and on the flop, but what would the purpose of the turn raise be? Value raising my extremely specific UTG betting range of A7, or just hoping for a tilted call with AK? It doesn't make much sense, but allowing the possibility that the villain also plays those hands that way 25% of the time, and factoring in our removal of KK and QQ and the reduced likelihood of JJ and TT, we come up with equity of 24%.

The last factors to take into account are the raise size and pot odds. As the villain min-raised, we are getting excellent pot odds of 17%. However, considering the strength indicated by the villain's line, we are looking at a strong-to-inevitable chance of facing an all-in on the river. That means that we're not actually looking at calling $8.50 to win $42, we're calling $36.35 (our current stack size) to win $69.85 (what the pot will be after the villain's shove for my remaining $27.85 after I call $8.50). That gives us odds of 34%. If our above range assignment is accurate (or close to it), we don't have the right odds to make the call and should therefore fold, which I did.

Donzo: folds
*** SUMMARY ***
Seat 3: UTGplus2 showed [3d 3h] and won ($31.85)

The villain didn't show at the time, but we can see now that he/she had 33 for a flopped set. Does that mean that the villain will NEVER have a worse hand? No, but the point is to determine how likely that is, and without a read that the villain is extremely aggressive or willing to raise lighter for value, we can't assume that they're taking that line with only a single pair. And because we didn't allow ourselves to become emotionally invested in our aces and left our brain in the "On" position, we were able to escape the hand losing only 29 big blinds instead of 100.

Easy Hands and Expected Value

I want to take a look at a seemingly straightforward situation and explore the various factors in play as well as the math behind the potential decision. The following is a hand I played a few days ago:

Seat 1: Small Blind ($6.70 in chips)
Seat 2: Big Blind [ME] ($27.07 in chips)
Seat 3: UTG ($23.44 in chips)
Seat 4: UTG+1 ($21.70 in chips)
Seat 5: UTG+2 ($18.10 in chips)
Seat 6: Dealer ($19.86 in chips)
*** HOLE CARDS ***
Big Blind  [ME] : Card dealt to a spot [Jc Js]

Despite its reputation as being "unplayable," pocket jacks, while not a monster, is a very good starting hand. Depending on the preflop action, we will usually be calling or raising with these cards from the big blind.

UTG : Raises $0.75 to $0.75
UTG+1 : Folds
UTG+2 : Folds
Dealer : Folds
Small Blind : Raises $6.50 to $6.60

 The UTG player makes a standard raise, but the small blind comes over the top and goes all-in (technically he/she has $.10 behind, but that's irrelevant). Our previous intent in the hand must undergo immediate re-examination. There are a few important factors at play here:

1)  The small blind has been playing extremely poorly and this is at least the 3rd or 4th time they have shoved all-in either as the initial raiser or as a 3-bet. Two of those prior hands were displayed and one hand was two random cards (something like J4) and another was a weak Ace, I believe A8 or A9. Consequently, I feel confident that I am ahead of the SB's range here.

2) I have a note on the UTG player that he/she is extremely tight. I'm about 30 hands into my session and this the first raise I've seen. Based on that information, plus the fact that they are raising as the first player to act, we can with reasonable confidence assign that player a narrow range along the lines of AJ+ and TT+.

3) If we call and UTG villain stays in the hand, we will be out of position.

4) If my extremely rough math is correct, there is about a 57% chance that one of the cards on the flop will be a Q, K, or A. So if we call and then the UTG player does as well, what flop are we hoping to see?

At this point, I have three options:

1) Call. Even if our range assignment for the SB is correct, we have to worry about the UTG player and what they are going to do. If they're holding a premium hand like QQ+ or AK, it's almost a lock that they're shoving all-in. As for hands like AQ, AJ, and TT, it's more difficult to say, as that will depend on the player and their impatience level and optimism. Regardless, if the villain does in fact shove with the part of his range I've suggested, we can expect him/her to do so over half the time (34 combinations of QQ+ and AK vs 30 combinations of AJ (with my blockers), AQ, and TT. Against the villain's full range we are a coin flip, with 49.5% equity. Against that shoving range, we are faring much worse, with only 36% equity. Returning to influencing factor #4 above, if we are fortunate enough to only be called, we still aren't entirely sure what we want to hit. Over half the time we'll have to contend with an overcard, in which case it will be very difficult to check/call with a dead player in the hand. In other words, we are essentially hoping that the villain either has two overcards and misses and we can check it down, or we are set mining for a terrible price.

2) Raise/Shove. This is a possibility to consider if we are confident about our range assignment for both players. We have to assume that any hand against which we are doing better than a coin flip (e.g. AJ and TT) is going to fold to our all-in. Depending on the tightness of the player, they may also fold AQ here or perhaps rarely AK or QQ, but we can't depend on that. So in that case we are folding out 37.5% of their range (24 of 64 combinations). Against the remainder of their range (QQ+ and AK) we have 36% equity.

Normally when calculating whether a raise is appropriate we would look at our equity against their calling range vs the amount we win when they fold and how often they do so. But in this hand, things are complicated by the 3rd player who is all-in. If that player were not a factor, the formula for calculating the expected value (EV) from an all-in looks something like this:

((amount of villain's call)*(frequency with which villain calls)*(our equity when villain calls)) + ((current size of the pot)*(frequency with which villain folds) - ((size of our bet)*(frequency with which villain calls)*(villain's equity against us))

So if the hand in question were heads up (let's pretend the money in the pot was all from an initial raise of $7.35 and remember that the $23.44 comes from the UTG player's starting stack size), our EV would be determined as follows:

(($23.44-$7.35)*(.625)*(.36)) + ($7.35)*(.375) - ($23.44)*(.64)
(($16.09)*(.625)*(.36)) + ($7.35)*(.375) - ($23.44)*(.625)(.64)
$3.62 + $2.76 - $9.38
EV = -$3.00

If our above assumptions are correct, we stand to lose an average of $3 in this pot by shoving all-in against that opponent. Unfortunately, we still have to account for that third player. Although I noted above that the 3rd player was playing very loose, it's unlikely he/she was just shoving any two cards. However, for the sake of illustration, I will go ahead and assign that range to the 3rd player. After tearing my hair out for awhile I think I've figured how to alter the formula above (changes in bold, "villain" still refers to the UTG player):

((amount of villain's call that exceeds 3rd player's shove)*(frequency with which villain calls)*(our equity just against villain when villain calls)) + ((amount of 3rd player's shove + amount of villain's call that doesn't exceed 3rd player's shove)*(frequency with which villain calls)*(our equity with both players in the pot)) + ((current size of the pot)*(frequency with which villain folds)*(our equity against 3rd player's range) - ((size of our bet that exceeds amount of 3rd player's shove)*(frequency with which villain calls)*(villain's equity against us)) - ((amount of our bet that doesn't exceed 3rd player's shove)*(villain and 3rd player's combined equity against us))

If that's too hard to read, the most important aspects to note are A) our fold equity is less valuable because if our shove gets the UTG player to fold we still have to account for the other player in the pot and B) if the UTG player calls, a portion of the pot will be 3-way. Using Equilab, we have 77.5% equity against any two cards and 31.1% equity against both players' ranges.

(($23.44 - $6.70)*(.625)*(.36)) + ($6.70 + 6.70)*(.625)*(.311) + ($6.70)*(.375)*(.775) - ($23.44 - $6.70)*(.625)*(.64) - ($6.70)*(.689)
(($16.74)*(.625)*(.36)) + ($13.40)*(.625)*(.311) + ($6.70)*(.375)*(.775) - ($16.74)*(.625)*(.64) - ($6.70)*(.689)
 $3.77 +  $2.60 + $1.95 - $6.70 - $4.62
 EV = -$3.00

I didn't draw it up this way, but the expected value is exactly the same. However, bear in mind that our EV would be worse if I had assigned a stronger range to the 3rd player than "any two cards."

3) Fold. Pack it in and wait for a better spot. This is what I elected to do and the hand played out as follows:

Big Blind  [ME] : Folds
UTG : Raises $11.70 to $12.45
Small Blind : All-in $0.10
UTG : Return uncalled portion of bet $5.75
*** FLOP *** [Jh Qs 8c]
*** TURN *** [Jh Qs 8c] [Ah]
*** RIVER *** [Jh Qs 8c Ah] [Jd]
Small Blind : Showdown [Ah Ac Qs Qc Jh] (Two pair)
UTG : Showdown [Kd Kc Jh Jd Ah] (Two pair)
Small Blind : Hand result $12.97

The SB had AQ and the UTG player KK. If I had called or raised , I would've won a big pot by hitting quad jacks. But that is irrelevant and should have no bearing on our analysis of the hand and whether our logic was good and the correct play was made.

Now did I work out all of the above math while making my decision? Of course not, as I imagine that's something only a small percentage of people can do. For the rest of us mortals,it is helpful to go through exercises like the above to get a feel for expected value and learn to be able to make rough estimates on the fly.