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.