Optimistic Overweighting

I want to talk about an important phenomenon I call "optimistic overweighting." Whether they realize it or not, this is something that almost every poker player does and some do it constantly, to the detriment of their game. By optimistic overweighting, I'm referring to the practice of believing your opponent has more bluffs in his or her range than they actually do. To illustrate, let's look at a hand I played a few days ago:

Seat 2: Small Blind [ME] ($32.09 in chips)
Seat 3: Big Blind ($20.96 in chips)
Seat 4: UTG ($11.64 in chips)
Seat 5: Dealer ($28.33 in chips)
*** HOLE CARDS ***
Small Blind  [ME] : Card dealt to a spot [Qd Ah]
Big Blind : Card dealt to a spot [4s 9h]
UTG : Card dealt to a spot [Jh 5h]
Dealer : Card dealt to a spot [Qc 5c]
UTG : Folds
Dealer : Calls $0.25
Small Blind  [ME] : Raises $0.90 to $1
Big Blind : Folds
Dealer : Calls $0.75

Preflop is pretty standard, we're 4-handed and the dealer limps on the button, so I raise to 4x for value and get called.


*** FLOP *** [9d Tc Ac]
Small Blind  [ME] : Bets $1.50
Dealer : Calls $1.50

Flop is perfectly fine, as I flop top pair and the board has a lot of potential draws in addition to inferior aces. The 9 and T are slightly concerning, as they fill in A9 and AT, which would have been good hands from which to extract value, but those hands are still a small part of the villain's range. I bet $1.50 into $2 and get called. 


*** TURN *** [9d Tc Ac] [2d]
Small Blind  [ME] : Checks
Dealer : Bets $2.62
Small Blind  [ME] : Calls $2.62

I don't like my check here. I sometimes will check the turn after betting the flop because it's a line that says "I took a stab at it and you didn't fold, so now I'm showing weakness." Such a line can often get you a second street of value on the river when a bet on the turn might have persuaded a lower limit player to fold what had been an optimistic hand on the flop. However, in retrospect there are still a ton of draws on the turn, including some suited straight draws and suited aces with diamonds that picked up flush draws with the 2, that I should have bet again for value. In any event, the villain bets about 1/2 pot after I check. This doesn't give me much information, as they could be betting any of their draws, a set, or two pair. It does reduce the likelihood of a weak ace with or without a flush draw, as those hands will usually get checked behind if the villain isn't spewy. I assigned the villain a range of any club draw, a set that isn't 22, any open ended straight draw (OESD) or double belly buster draw, any two pair (although I took out about 2/3 of the combinations of 92 and T2 because those are less likely), and any ace high diamond draw.



*** RIVER *** [9d Tc Ac 2d] [9c]
Small Blind  [ME] : Checks
Dealer : Bets $5.24

The 9 brings in the club flush and creates a bit of a pickle. On one hand the flush has come in, which was the most logical hand to which the villain could be drawing. On the other hand the villain has missed all of their straight draws. To figure out the best course of action, I went back to Equilab to take out all combinations that had just an A (with the exception of AK) as well as A2 and T2 on account of getting counterfeited (since those hands would almost certainly check behind). That leaves us with a range of AA, 99, TT, AT, a flush, or a missed straight draw. There are a total of 44 missed straight draws and 71 made hands that beat us. Taken as a straight percentage, that means we beat 38% of the combinations that could logically be betting here. As the villain bet $5.24 into about $9.50 after rake, we need to win a little over 26% of the time for our call to be profitable. Since the villain has 38% bluffs in his/her range, that should make this an easy call, right?

Weighting the Range

Not so fast. The call is only correct if we are right in thinking that the villain is equally likely to bet all of those missed draws as to value bet all those made hands. Against all but the most aggressive opponents, this is not the case. Sometimes they are going to bet their missed draws and sometimes they're going to check behind. The key is figuring out how often they bet those draws. So if our target percentage to make the call is 26%, that means that the villain needs to be bluffing with 26% of their total range, or 30 combinations: (44+71)*(.26) = 29.9 combinations

We then take that number of combinations and divide it by the total number of missed draws: 29.9/44 = 68%. Therefore, if the villain is bluffing about 2/3 of the time with a missed straight draw, we call. If they are bluffing less often than that, we fold.

So how often is the villain bluffing with a missed draw? Well that's where things get tricky. Playing on Bovada, there is no ability to save notes due to playing with random players, so it's difficult to know how aggressive your opponents are at betting draws. Furthermore, there's no HUD to use for tracking players, so I don't have stats on this opponent's aggression level. The best we can do is extrapolate based on the tendencies shown up to this point at the table as well as any other information. In that vein, there are two pieces of information that I neglected to adequately factor into my decision. First, that I had lost a hand to this villain earlier in the session in which he/she was making a modest bet for value. Second, I have yet to see this villain bet or raise without a strong hand.

If I had taken the time to lay it out in those explicit terms, I might have decided to fold. As it is, I called and the villain showed Q5 of clubs for a rivered flush. That said, I started this post with the intent of showing what a bad decision I made, but after running through the numbers it's not as bad as I thought. Without access to solid stats it's difficult to say definitely how good or bad the call was, but the important takeaway is that it's useful to play with ranges to figure out how likely it is that your opponent is bluffing on the river and what circumstances are necessary to make a profitable call.