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.

"Insignificant" Hands

I'm going to talk briefly about the importance of capitalizing on spots to extract value, even when it seems like a waste of time. In the hand below, it's folded around to the small blind who limps and I am holding a pretty terrible hand, 73 suited.

Seat 1: Dealer ($13.95 in chips)
Seat 2: Small Blind ($10.60 in chips)
Seat 3: Big Blind [ME] ($31.63 in chips)
Seat 4: UTG ($13.04 in chips)
Seat 6: UTG+1 ($26.57 in chips)
Small Blind : Ante/Small Blind $0.10
Big Blind  [ME] : Big blind/Bring in $0.25
*** HOLE CARDS ***
Big Blind  [ME] : Card dealt to a spot [3d 7d]
UTG : Folds
UTG+1 : Folds
Dealer : Folds
Small Blind : Calls $0.15
Big Blind  [ME] : Checks
*** FLOP *** [3h 5s 5h]
Small Blind : Checks
Big Blind  [ME] : Bets $0.25
Small Blind : Calls $0.25

I could raise the limper, but am perfectly happy to see a free flop. The flop is pretty good for me, as it's a paired board and I'm holding a 3. The SB checks and I bet $.25 into $.48 after rake. I bet the minimum because I'm looking to get called by as wide a range as possible--anything from overcards to just an A high or heart flush draw, a call I might not get if I'm making a pot size bet. 

*** TURN *** [3h 5s 5h] [9s]
Small Blind : Checks
Big Blind  [ME] : Bets $0.50
Small Blind : Calls $0.50

The turn fills in whatever hands he was calling with that have a 9 (A9, K9, suited hearts with the 9), but otherwise is not a particularly concerning card. It also creates another potential draw with spades. The SB checks again and I make another half-size pot bet to get value from the hands mentioned on the flop as well as any high card hands that gained a flush draw with the second spade. 

*** RIVER *** [3h 5s 5h 9s] [Qd]
Small Blind : Checks
Big Blind  [ME] : Checks

The river brings a pair for some of the flush draws, but is otherwise a blank. I debated a small third bet for value here, but was content to check behind with what I thought was usually going to be the best hand. The SB shows A7 of spades for a high card that turned into a flush draw on the turn.

Get Value Whenever and Wherever You Can

When holding a weak hand in a situation like the one described above, many players will simply mentally check out of the hand and prepare for a more interesting one. This is a mistake. If you're playing 12+ tables at a time then such attention to detail is not always feasible. But if it's only 1-2 tables, don't just say "meh." Instead, look at the hand as an opportunity to push the skill advantage you have over your opponent. Or if you don't have a skill advantage, at least look at the situation as a learning opportunity.

One might say, "So what, you only won an extra 75 cents." To which I respond, "I sure did!" Profit is the same in the end, whether it comes from a 400 big blind pot or an 8 big blind pot. To maximize profit, don't pass up those subtle little spots where there's money to be made.

Aces and Evolving Scenarios

While it's often more interesting to analyze the "weird" hands, it's also helpful to make sure that we're reviewing how we play our monster hands. In this example, I'll talk a little about how I played AA and how the dynamic of this particular hand was a little different.

Seat 1: UTG ($28.41 in chips)
Seat 2: UTG+1 ($3.84 in chips)
Seat 3: UTG+2 ($26.03 in chips)
Seat 4: Dealer ($9.92 in chips)
Seat 5: Small Blind [ME] ($36.91 in chips)
Seat 6: Big Blind ($33.69 in chips)
*** HOLE CARDS ***
Small Blind  [ME] : Card dealt to a spot [As Ac]
UTG : Calls $0.25
UTG+1 : Folds
UTG+2 : Folds
Dealer : Folds
Small Blind  [ME] : Raises $0.90 to $1
Big Blind : Calls $0.75
UTG : Calls $0.75
*** FLOP *** [Ah 3h Tc]
Small Blind  [ME] : Checks
Big Blind : Checks
UTG : Bets $1
Small Blind  [ME] : Calls $1
Big Blind : Calls $1

Preflop is standard, although I perhaps could've raised it to $1.25 instead of $1. I was raising to isolate with the UTG limper, but get called by the blind blind as well.

At these stakes I think the check on the flop is mandatory. At higher stakes against more sophisticated players it might make sense to lead this flop, as some will know that your continuation bet range includes some bluffs and are therefore likely to call you with a T or worse. But at .10/.25 we can't assume that's likely and can only expect to be called by flush draws or an ace or better. As there is only one ace left in the deck, a check here is best. The big blind checks behind and UTG bets $1 into about $3, which is small and probably indicates a weak hand. Again, I don't like raising here because I want to keep inferior hands in the pot and a raise declares my hand to be either a flush draw or something very strong. I call and the big blind does as well.

*** TURN *** [Ah 3h Tc] [Kh]
Small Blind  [ME] : Bets $3
Big Blind : Calls $3
UTG : Folds

The turn card is mildly concerning, as it brings in the possible flush and a gutshot straight draw with QJ. That said, I'm happier than not to see it, as one or both players are more likely to continue in the hand if they're holding the Q or J of hearts with a pair and/or gutshot, e.g. KJ, KQ, QT or JT w/ the Q or J of hearts, JJ with the J of hearts, or perhaps a tentatively-played AQ or AJ with or without the Q/J of hearts. There's also a small chance of KT. Having flopped top set I now need to attempt to get value and bet $3 into about $6. In retrospect this was too little and I should've made it somewhere closer to $4.50 or $5.

It's important to note that this is a situation where a lot of players might check instead of going for value, due to concern about the possibility that the K of hearts made their opponent's hand. While sometimes it's prudent to check with a big hand when a 3rd flush or straight card hits, the flop action and texture of the board present a situation where there is too much money to be made from weaker hands to pass up betting. And even if one of the villains did hit their flush, we're drawing to 10 outs to hit our quads or boat (one ace, three 3's, three T's, three K's), which we have a 22.73% chance to hit.

*** RIVER *** [Ah 3h Tc Kh] [Kc]
Small Blind  [ME] : Bets $11
Big Blind : Calls $11

The K on the river is the best card in the deck for several reasons: 1) Obviously we hit our boat and can only be beaten by quad kings, which is basically impossible for either villain to have because of how the preflop action went. 2) Hands that were a little speculative on the turn like KQ and KJ are likely to call the river here with trips (or if we're very lucky the villain has KT). 3) If the villain has a flush, we are essentially always getting called here, as we could easily be betting a K ourselves or AQ/AJ.

Though it's conceivable to check/raise here, the villain could easily check behind with two pair or trip kings and might even take a bet/fold line if facing a check/raise from us. Consequently, I go for value and bet $11 into a little less than $12. The villain calls and shows 89 of hearts for a turned flush.


Respecting the Process

While it's tempting to look back at the hand and say, "The villain turned a flush, that means we should've raised the flop" or "we should've check/called the turn," it's important to keep our analysis of the hand dependent on the information available at the time. If our logic is good, it's our logical process that matters, not the results of the hand or what the villain was "actually" holding. Results should only be used to help us investigate whether our logic was in fact good or whether we perhaps didn't come to intelligent conclusions about the villain's range. This will be a theme that I bring up again, as it is perhaps the biggest impediment to amateur players not improving their game.

Overly Optimistic Range Analysis

As I said in my initial post, part of the point of this blog is accountability. With that in mind, I'll make my first "real" post a critique of a hand I played and hopefully explain how I made a pretty poor decision.

A few nights ago I played a quick session (about an hour) of $.10/25 and finished down about a buy-in and a half. I lost a flip and got unlucky in a couple other hands, but I also made some hasty decisions and didn't maximize value in a couple spots. My most egregious error came almost immediately, in my 3rd hand (which is usually an indication that a player didn't sit down completely mentally ready to play):

Bovada Hand #2702260454 TBL#4149508 HOLDEM No Limit - 2013-05-15 20:57:54
Seat 1: Big Blind ($14.83 in chips)
Seat 2: UTG [ME] ($25 in chips)
Seat 3: UTG+1 ($38.84 in chips)
Seat 4: UTG+2 ($29.59 in chips)
Seat 5: Dealer ($32.21 in chips)
Seat 6: Small Blind ($28.74 in chips)
*** HOLE CARDS ***
Big Blind : Card dealt to a spot [xx]
UTG  [ME] : Card dealt to a spot [Qd As]
UTG+1 : Card dealt to a spot [xx]
UTG+2 : Card dealt to a spot [xx]
Dealer : Card dealt to a spot [xx]
Small Blind : Card dealt to a spot [xx]
UTG  [ME] : Raises $0.75 to $0.75
UTG+1 : Folds
UTG+2 : Calls $0.75
Dealer : Folds
Small Blind : Calls $0.65
Big Blind : Calls $0.50

Pre-flop is standard--I have AQ off-suit UTG and am obviously raising. I get three callers, on two of whom I have position post-flop.

*** FLOP *** [3h Ah Qh]
Small Blind : Checks
Big Blind : Checks
UTG  [ME] : Bets $2.50
UTG+2 : Folds
Small Blind : Calls $2.50
Big Blind : Folds

The flop is great for me as I flop top two and the monotone board with two face cards means that my opponents potentially have some draws with which to continue. The blinds check to me and I bet $2.50 into just under $3 after rake, which is about 85% of the pot and a perfectly reasonable bet size for getting value. I get one caller and am perfectly happy with the situation going into the turn.

*** TURN *** [3h Ah Qh] [Jd]
Small Blind : Checks
UTG  [ME] : Bets $7
Small Blind : All-in(raise) $25.49 to $25.49

The turn brings in a couple hands, mostly just KT with the K of hearts and perhaps very rarely JJ with the J of hearts, but overall I'm not concerned that it has changed the hand. I bet $7 into a little under $8. Then the needle scratches the record. The villain check/raises all-in and in short order I need to analyze his range and figure out whether I'm getting appropriate odds to call. That brings me to the title of the post and what I think is sometimes a leak in my game: putting too many bluffs or semi-bluffs in my opponents' range.

For those who aren't familiar, there's a neat little program called EquiLab that allows you to customize your opponent's hand range in order to figure out your likely equity in any given spot. You can choose to assign your opponent a range like "AJ+ and QQ+" or you can get specific and choose individual cards. With that in mind, I came up with the following:




For anyone who finds that too hard to look at, what the above screenshot shows is a range of AA, QQ, 33, rarely JJ and KK with the K of hearts, AQ and A3, a number of possible flushes, and hands that were combo draws on the flop and are either a straight or a draw with a pair on the turn. I will call the above the villain's "possible range." Against that range on the turn, EquiLab calculates that we have 32.93% equity.

Possible vs. Probable Range

All the hands I just listed are what it's possible that the villain has. However, at the end of the day we are most concerned with what is probable. The check/call, check/shove line is usually an indication of a strong hand when there is no history between the players, so we should weight our analysis of range accordingly.

It's possible that the villain played AA or QQ this way, but the lack of a preflop re-raise, particularly considering the fact that they were out of position and there was already one caller, makes both hands extremely unlikely. Consequently, I'm going to remove them from the "probable range." For similar reasons I'm going to do the same with KK. A3 is also possible, but in my opinion less likely because it's vulnerable to being counterfeited and most villains would want to maximize value by leading or check/raising the flop, so I'll take out four of the possible six combinations. As for KT, unless one of the cards is a heart (more likely the K), the line looks overly optimistic from all but the fishiest players, so I'll remove all combinations that don't have a heart. AJ with the J of hearts is also somewhat unlikely, if only because the J on the turn gives the villain excellent showdown value and he doesn't want to shove me off a hand like AK or turn AJ into a bluff against AQ or a set. So, while I still think it's possible, I'll remove one of the two combinations of AJ. Finally, KJ with at least one heart has a large number of combinations, but I'm skeptical of it being played this way a majority of the time and so I'll include just the combinations with the K of hearts.

Where it gets tricky is how many flushes to put in the villain's range. I happened to choose 9, but it could easily be more depending on how wide his calling range is with suited connectors/gappers from the small blind. But if we assume my randomly assigned distribution, we have 26.59% equity. In other words, the difference between the possible and probably range is almost the difference between getting 2-to-1 and 3-to-1 on our money.

So to go back to our decision based on the odds, I'd bet $7 into about $8, after which he put me all-in. I had just under $15 to call a pot of $30, so I needed to be getting odds of 2-to-1 or 33.3%. If my "possible" range is accurate, I'm about even money to call. But if my "probable" range is correct, it's a fold, and a pretty clear one at that.

To look at it another way, the hands that are betting for value are as follows: AQ, A3, 33, and a flush. Hands that are bluffing: pretty much just KJ or perhaps very rarely KK. In other words, I can beat everything but a flush or 33, but the likelihood is that he has one of those hands. As is probably evident by now, I called the shove, the villain showed 33 for a flopped se and the river was a blank.


So why did I call? Two (bad) reasons:

1) I over-weighted the possibility of a bluff. Having taken the time to analyze possible hands, it now seems obvious to me that there were very few possible bluffs in his range. But at the time, I persuaded myself that while I expected to be beaten a majority of the time, there were enough times he had KK, KJ, or some other random draw to substantiate making the call. Furthermore, I was seduced by the possibility of almost always having 4 outs to make the nuts.

2) The last couple years of live play have led me to make faulty assumptions. One thing I've noticed playing almost exclusively live the past couple of years is that players with strong made hands are afraid to check/call on a draw-ish board. While this may be an idiosyncacy of the players I've known, it became such a clear and obvious pattern that I could hardly remember an instance when a player slow played a hand like two pair or a set when there was significant potential for their opponent to improve. But different players play differently and of course online is more solid, and so combined with Reason #1, I took the in-the-moment position that "there was no way he/she wouldn't have raised the flop with a set or weak flush." Which was of course wrong and a good reminder of the need to constantly challenge my assumptions and re-adjust.



TL;DR - Don't persuade yourself that your opponent's range is something other than what it is.