Poker: cbet frequency

Below is a pretty rough look at how frequently you should c-bet.

In particular, I will look at an arbitrary scenario where you were the preflop opener and someone called behind you. It is easier to analyze than in position c-betting, since you have less information and must play more straightforwardly.

To keep it simple, lets say we opened UTG in full ring with the range detailed in the last post (77+, ATs/AQo+, KQs). The flop comes A92 rainbow. Then 40 of the 92 hands (ATs/AQo+) hit the A, 12 hands (AA and 99) flop a set, and the rest 40 combos are worse than top pair. Again following last post, bet sizing for OOP play on this sort of flop should be ~1/2x-3/4x pot.

Bet Sizings of 1/2 pot

Let's say we choose to standardize it at 1/2x. Then let's say we bet the 40 top pair hands, and try to check raise the sets. How many hands can we bluff? To make opponent indifferent to bluff-catching vs folding (ignoring subsequent streets for now), we bluff 1/3 of what we can value bet- 13 hands. This means that we would choose 77 and 88 to bluff with, and check only TT-KK and KQs.

So in this scenario, we are c-betting 56% of the time, and of the 44% we check, we are checking a set (13%) almost 1/3 of the time. Let's say someone bets half pot after we check. Suppose we checkraise all of our sets and we lay them 3:1 odds (ie. will match + raise half pot, or make the total pot size 3x initial flop pot). Then again, we should bluff 1/3 of the time that we check raise with a legit hand = 4 hands- we can bluff and play KQs like we play our sets for this round of betting. That is another 4% of our opening range.

So now we are left with the TT-KK (26%) that we checked. Maybe KK/QQ warrants one street of check calls depending on the bet sizes.

Let's assume that the opponent doesn't have anything. Then we (UTG opener) are taking down the pot ~75% of the time given this board. That sounds about right.

Bet Sizings of 3/4 pot

So let's say we choose to standardize our bets at 3/4 pot. Maybe we can be more conservative and check call with hands like AJs/ATs. Then we are betting 32 hands outright, and should bluff 32*3/7 = 14 hands. So 77, 88, and half the KQs.

You check raise the sets again, but size it to lay 7:3 odds. So then you can bluff 12*3/7 = 5 hands. Lets say 5 out of 6 TT.

So you are left with 23% of hands (JJ-KK, 2 KQs, 1 TT) that you might checkfold here, and another 9% of hands that you are probably check calling (ATs AJs). Of course, whether your check call or check fold these 32% of hands actually just depends on your opponent's bet size.

Bet Sizes and Clairvoyance

Also note that if we were more aggressive and bet AJs/ATs, then we could fit all the KQs into the bluffing range as well, and we are left with 20% (JJ-KK, 1TT) in our checking-call/fold range. This makes sense because the bigger the bet size the more the opponent will have to fold.

However, this requires clairvoyance- we are making the implicit assumption that AJs/ATs are good- when in fact they might be losing hands. By clairvoyance I mean knowing for sure that you are ahead while your opponent is not sure.

Clairvoyance tends to promote heavier betting to maximize EV by replacing parts of our checking range with our bluffing range. However, note that when we do bet, our EV is the same regardless of bet sizing.

For example, if you were deep stacked with the nuts and no hand can improve to beat it, you could bet 100x pot and bluff at close to the same frequency (100/101x nut frequency). Whenever your opponent is faced with this bet, he should theoretically call to bluff catch 1/101 of the time. Of all the hands that you jam, this allows you to take down ~99% of pots right away and showdown with just over 50% chance of having the nuts in a 202xinitial size pot the remaining 1% of time. Notice that even in the scenario where opponent bluff catches, you are expected to get 101.5x initial pot. In fact you expect to get 1.5x initial size when they call. So when you do jam, you get 99% * 1 + 1% * (101/202 * 202 - 100) = 1x initial size.

Compare this to betting only 1x pot. Then you can only bluff 1/2 of the time, and opponent bluff catches 1/2 of the time. So only 50% of pots are taken down right away and you are 2/3 chance of winning if they bluff catch. So the final payoff = 1/2 * 1 + 1/2 * (2/3 * 3 - 1) = 1x initial size.

We make the same amount either way when we bet because everyone is playing optimally. However, we can bet more frequently in the first case vs the second case. ie. Let's say our hand range gives us the nuts 40% of the time. Then if we can jam 100x pot, we can bluff another 39.5%, and say check fold the rest. ie. we get to take the jamming line and make 1x initial size 80% of the time. However, if we bet 1x pot, we only get to bet and make 1x initial size 60% of the time.

In fact, if our hand range was so strong that say we could value bet 67% of the range, then you will see that any bet sizing > 1x pot actually lets us bluff > 33% of our range. ie. we could bet 100% of the time. Once we get past this limit, it is especially profitable to jam. Let us redo the calculations above for expected profit. For any bet sizes up to 1x pot bet, you still make 1x initial size. But for 100x pot bet, you make 99% * 1 + 1% * (67% * 202 - 100) = 1.34x initial size. It turns out that if our range and effective stack size ever allow for a bet size that lets us bet 100% of the time, then jamming above that size potentially lets us make above average profits. In fact, the optimal solution for our opponent if they knew our range would be to fold 100% of the time (ie. they can never try to bluffcatch).

So why do people not jam all the time? As stated above, in practice, it is unclear whether you are ahead or not. A smaller bet size lowers the cost if you make such a mistake. Also, what if your opponent does not play optimally and makes mistakes in subsequent play? Then you may have a reason to give more room/space for them to make errors, instead of forcing them to take the easy correct path.

In real life, the hand that is behind may also have outs. Then, how much you bet is actually affected by how likely you are to get drawn out on. As discussed previously in, this can be calculated by considering pot odds and implied pot odds. In real life, you might not be sure which draw/how many outs your opponent has.

To be safe, you could try to price out the draw with the most outs. Theoretically, overbetting is not a big mistake since your opponent cannot take advantage of it immediately (vs if you underbet, opponent can call and you make < 1x initial size).  However, in practice you are again helping your opponent make the correct decision, and if you were wrong and you were in fact behind, then it will be very costly. So perhaps your should take the most common draw and size your bet according to that.

So it turns out that your bet sizing should be determined by the # of outs/how coordinated the board is, and that your action (value bet/bluff/check) frequency should be affected by your hand range. On a side note, if you needed to bluff catch, you would base your decision off of your opponent's range, not your own range.

Poker: An Overview

Hand Ranges

I was doing some house keeping the other day and decided to re-look at my hand ranges to open and raise in a 9 handed game:

• UTG (7% for 2 opponents) 77+, ATs/AQo+, KQs
• UTG+1 (10% for 2 opponents) 77+, A9s/AJo+,  KTs+/KQo, QJs
• UTG+2 (14% for 2 opponents) 66+, A7s/ATo+, K9s/KJo+, QTs+, JTs
• MP1 lowjack (20% for 2 opponents) 55+, A3s/A8o+, K7s/KTo+, Q9s+/QJo, JTs
• MP2 highjack (25% for 1 opponent) 44+, Axs/A7o+/A5o, K6s/K9o+, Q9s/QTo+, JTs
• cutoff (35% for 1 opponent) 33+, Ax, K3s/K7o+, Q6s/9o+, J8s+/JTo, T9s
• dealer (50% for 1 opponent) 22+, Ax, Kx, Qxs/Q5o+, J5s/J8o+, T7s+/T9o, 98s
• SB (50)

We can see this gives an attempt to steal of ~40-45% over the cutoff/dealer/SB positions. This sounds slightly high but still reasonable in online poker. Notice that our range is also skewed to play less OOP and more IP, because I just find it to be easier that way. This is in comparison to a "typical" opening range for UTG to SB: 12/14/16/19/24/31/45/60. ie. open UTG with 12% and hands, open UTG+1 with 14% of hands etc.

Using our opponent's "typical" opening range, we can calculate how frequently we will open: (7% + 0.88*10% + 0.88*0.86*14% + 0.88*0.86*0.84*20% + 0.88*0.86*0.84*0.81*25% + 0.88*0.86*0.84*0.81*0.76*35% + 0.88*0.86*0.84*0.81*0.76*0.69*50% + 0.88*0.86*0.84*0.81*0.76*0.69*0.55*50%) / 9 = (7% + 0.88*10% + 0.7568*14% + 0.6357*20% + 0.5149*25% + 0.3913*35% + 0.2700*50% + 0.1485*50%) / 9 = 8.8%

Now also consider the number of times that we 3bet. Roughly, we should 3bet if we are in the top half of the range of the raiser. Instead, let us approximate by saying we are going to 3bet/call raises with the top half of our own open raise range for that position (UTG+1 to dealer). From SB and BB, let's say we will play top 5% of range. Let us approximate our 3bet/call frequency as follows:
(0.12*5% + 0.25*7% + 0.36*10% + 0.49*14% + 0.6*20% + 0.73*25% + 0.85*5% + 0.95*5%) / 9 = 6%

This means we have a VP$IP of very roughly 14%-15%. Tracking my own statistics, I see that I have a VP$IP between 12%-20% each session, and PFR 10%-16%. This sounds roughly inline with what we looked at above (a bit looser) and points to a roughly 50/50 split between 3betting (3%) vs calling (3%) when facing a raise. Whereas previously, most of the calling came from set mining OOP because I was getting say 1:3 odds, I am working on reducing that and instead changing some IP 3bets into flat calling to limit pot sizing, especially against calling stations.

Bet Sizing

Preflop Bet Sizing
Whereas previously I have employed a 3bb bet sizing for late position steals, recently I am testing out a smaller size so the pot doesn't get too bloated. I've frequently had hands before where SPR gets so low that I am committed with mediocre hands.

• 2bb for UTG, UTG+1, UTG+2
• 2.5bb MP and LP
• 3.5bb in SB

For 3bets, I tend to do a bit over half pot if IP and close to full pot if OOP.

Flop Cbet Sizing
Previously, I always ran into trouble betting 1x pot OOP because I would lose pots like overpairs/two pairs against sets or better. That itself is not an issue, but I would be way too committed to fold by the time the opponent showed enough strength/their hand range had clarified to the pt where I am behind/have < 50% equity. Hopefully, the smaller preflop bet sizing should help with the "Bayesian updating" before the pot gets too big. In a 3betted/4betted pot though, I might have to check-call more often on the flop/turn/river, or use 1/2x pot bet sizes.

• OOP- usually 1x pot
• IP- usually 1/2x pot

Adjustments:

• adjust size up by 1.5x-2x of normal bet if flop is super wet
• adjust down to 2/3-1/2x if the flop is super dry

However, I have not thought very deeply on how to incorporate probe bets into flop betting yet. Also, what % of flops should you cbet, and what % should you check raise/call/fold? Should your hand range or bet sizes change as the effective stack size changes? I hope to think about these points in another article.

Poker: running cards multiple times

Consider the following a quick exercise in combinatorics. We are investigating the effects of running cards twice. You can see a real life example here. It is known that the EV doesn't change when you run multiple times (but you lower your variance). Let us check this claim.

Let's take the case of KK vs AA allin after a blank flop. After the flop, there are 45 cards left. If we run it once:

EV for KK = Pr(K on turn and no A on river) + Pr(no A on turn and K on river) + Pr(K's on turn and river) = 2/45 * 41/44 + 41/45 * 2/44 + 2/45 * 1/44 = 8.383838...%

Notice that the first two terms are the same because turn/river is interchangeable. Double checking this on pokerstove and using a flop with 0 chances of runner runner flush/straights, we get 8.384%. Nice. Exact.

Let's say we run it the second time. A couple possibilities in the first run:

• one A came out (2/45 * 41/44 * 2 = 8.2828%)
• then EV for second run is 2/43 * 40/42 * 2 + 2/43 * 1/42 = 8.9701%
• two A's came out (2/45 * 1/44 = 0.1010%)
• then EV for second run is 2/43 * 2 - 2/43 * 1/42 == 2/43 * 41/42 * 2 + 2/43*1/42 == 9.1915%
• one K and one A came out (2/45 * 2/44 *2 = 0.4040%)
• then EV for second run is 1/43 * 41/42 * 2 = 4.5404%
• one K came out and no A's came out (2/45 * 41/44 *2 = 8.2828%)
• then EV for second run is 1/43 * 40/42 * 2 = 4.4297%
• two K's came out (2/45 * 1/44 = 0.1010%)
• then EV for second run is 0%
• no A/K came out (41/45 * 40/44 * 2 )
• then EV for second run is 2/43 * 39/42 * 2 + 2/43 * 1/42 = 8.7486%

The above EVs were also double checked with pokerstove using dead cards (A, K and blanks) and should be the exact probabilities. Adding these all up, the EV over all cases for the second run is 8.3838%- same as the first run.

Poker: Pot Odds 2

Some follow up thoughts on the first post about odds. To get the most precise numbers for your hand's equity/odds, you should compare pot odds with your hand's odds of hitting its outs.

So using the same scenario from the first post, the third/most precise method is as follows:

Method 3 == Pot Odds vs Odds of Hitting Outs
Pot odds == 1:5
Your hand has 8 outs. On the turn, there is 52 - 5 = 47 cards left. Your odds of hitting  outs is 8 : (47-8) == 8:39 == 1:4.875
Since pot odds > odds of hitting outs, you should call.

So note that this is much more precise, and only deals with the turn card (ie. if you had read the second post about bet sizing, this assumes that you will bet optimally on the turn again and that effective stack size is large enough for optimal bet)

However, the disadvantage of pot odds is that when the probability is small, the odds change very rapidly to any small change in probability. ie. 1:49 vs 1:99 is actually only 1% apart. So whereas for equity calculation, you are "safe" if you have say a 5% buffer between required equity vs hand equity, here there is no similar rule and you might be easily mislead as to how much edge you have.

Let's look at what a 5% equity edge means for different pot odds.

Bet Size	Pot Odds	Required # of Turn Outs Without Buffer	Equity + 5% Buffer	Required Hand Odds With Buffer	Required # of Turn Outs With Buffer
1/4x pot	1:5	7.8	21.7%	1:3.6	10.2
1/2x pot	1:3	11.8	30.0%	1:2.3	14.2
1x pot	1:2	15.7	38.3%	1:1.6	18.1
2x pot	1:1.5	18.8	45%	1:1.2	21.4

Without the 5% buffer, the Required Hand Odds should equal pot odds. Instead, as you can see, 1:5 -> 1:3.6, while 1:1.5 -> 1:1.2. Hence there is no easy way to build in some buffer with pot odds. However, you may have noticed that the required outs actually increases by a constant (2.4 outs) when you build in a 5% buffer. This makes perfect sense- 2.4 outs --> an extra 5% chance of drawing out.

Thus the optimal way to give yourself a buffer is to calculate the odds as is, but then require a couple extra outs to be conservative.

Interestingly, if you look at the # of outs required for 1x pot and 2x pot bets on the flop, they are actually "ahead" (> 50% chance of drawing out by the river). So it seems as though any drawing hand that could call a > pot sized bet could also just raise or push all-in. More on this next.

Poker: Bet Sizing

Following the last introductory post to poker, here is an example on how to determine bet sizing based on the texture of the flop.

Let's say effective stack size is 200 BB, we are in button/cutoff, it was limp/folded to us, and we raised to 4-6 BB and one player called. So the pot is ~10 BB (8.5-13.5 BB depending on dead money/size of raise) and effective stack size is 195 BB.

Let's say the flop comes with two flush cards (two cards of same suit) and you have top pair top kicker and you are committed to calling the all-in even if the another flush card comes because you have a weird hunch/you are tilting etc. Let's look at what you need to bet taking into account the implied pot to push a flush draw opponent out of the pot.

The flush draw opponent has 36% equity -> you want to lay 1.8:1 odds for optimal play -> with an implied pot of 10 BB + 195BB, that is a bet of 205/1.8 = 114 BB. This sounds ridiculously huge and incorrect. There are two reasons for this. 

1. Being willing to commit 200 BB on a flush board with top pair is an incorrect play. 
2. The existence of a turn bet means that the true "correct bet" amt is much lower. 

We will look at these two lines of thought below and examine what the "correct bet" should be if we take such factors into account..

How much you would typically/normally commit with top pair top kicker
Generally, with top pair or worse, you want to control the pot to be medium sized. Let's look at what that means for the flop/turn/river betting rounds:

1. if betting sizes were pot sized, I would say two bets would already be really pushing it. (ie. you would control the size by either bet/check/bet or bet/bet/check etc). In this case the pot would be 90 BB by showdown and you and your opponent would have each put in another 40 BB.
2. if betting sizes were 1/2 pot, it is probably feasible to bet on all three rounds. In this case the pot would just be 80 BB (35BB each since flop). But betting like this might be stupid with the flush draw on the board.

From these rough estimates, it seems that we should be willing to commit another 35-40 BB to this 10 BB pot. Let's say that the implied pot size is 10 + 35 BB. Then the optimal bet is 25BB, or 2.5x the pre-money pot. This certainly seems more reasonable but from what we know empirically about poker betting, it still seems to be on the high end. Let's move on to consider the fact that there can still be betting on the turn/river.

Betting on the turn

From the conclusion of last post, remember that by betting optimally against a dog, each time you bet, you are effectively causing them to loose money equivalent to them folding to the bet in the long run. I would thus make the following statement:

By betting optimally on the turn, it is as though we have cut off the opponent's chances to draw on the river. ie. it is as though the opponent loses the pot right here on the turn.

Now, we can see that perhaps our opponent doesn't have 36% equity, since after they see one card on the turn, we can bet again, which is equivalent to cutting them out right there. So they have 9 outs out of 47 possible turn cards. So we could lay 38:9 ~= 4.2:1 odds to price them out. With a 45BB implied pot, this means betting 10.7 BB ~= 1.1x the pre-money pot. This is more inline with what we know as "normal" betting. 

What is interesting is that if you follow this line of reasoning/betting on the flop, you are committed to betting the turn 100% of the time if no flush cards come out- otherwise you are actually giving your opponent 36% equity and good odds to call on the flop.

So it turns out that maybe our flop bet sizing could vary depending on our game plan on the turn.

As a matter of exercise, let's look at what our proper bet on the turn is. Let's say we just bet another 1x pot on the flop- so the turn pot is 30BB (and we are ready to commit another 25-30BB). What is the optimal bet on the turn if no flush cards come? They have 9 outs out of 46 river cards, so we lay 37:9 ~= 4.1:1 odds. With an implied pot of 60BB, this means that betting 15xBB = 1/2 the pre-money pot.

So either you bet [1.1x pot on flop AND 0.5x pot on turn], or maybe you need to overbet on the flop by betting (at most?) 2.5x pot if you intend to (sometimes? check the turn.

Poker: Intro to Pot Odds

Here's some useful background for anyone interested in poker. Frequently, when someone bets into you, and you are chasing a draw, you need to decide if you want to call or not. Below, we look at a couple ways to think about this.

Before we start, one important definition to get out of the way. The term equity/pot equity may be confusingly used to refer to a % of the pot or an actual $ amount. I will use %equity and $equity to differentiate between these. Both %equity and $equity look at what would happen on average if your hand was taken to show down without any further betting.

Take an example on the turn where the pot is $100. then the opponent bets $25 into pot. We need to decide if we call or not. 

Simple Pot Odds
We are doing the analysis below with the assumption that opponent is all-in (or that there is no more betting on the river). Intuitively, we might already think that it is cheap to call (we can call because the bet was small)- let's see what the math says:

Method 1 == pot odds required probability vs %equity

Pot Odds

• the opponent just bet 1/4 of the pot.
• the pot odds that you are getting is 25:125 == 1:5
• ie. pot odds are post money (inclusive of bet)
• from the pot odds, the required equity (probability of winning) in order to call is 1/(5+1) = 1/6 = 17%

%Equity (== Your Hand's Actual Probability to Win)

• looking at your hand vs your opponent's range, what is your probability of winning?
• let's say you put your opponent on a pair or better and you just have a straight draw.
• you only win if u hit your draw (8 outs = 18% chance of hitting)

Now compare pot odds probability (17%) to your hand's probability to win (18%) and since hand probability > the probability required from pot odds, this is callable. (maybe in practice you might demand say a 5% buffer before saying it's callable?)

Method 2 == EV calculation

folding = $0 

calling = 18% * 125 + 82% * (-25) = $2

so it's +EV to call. so call

Notice that here, you already take into acct the pot size (125) and the bet size (25).
Compare this to method 1 (the pot odds vs probability method) - the probability calculation doesn't take into acct the bet/pot ratio and that's why you need to compare pot odds to it in method 1.

I think in practice, method 1 is actually easier to work out over the board.

Implied Pot Odds

Now let's say we were not allin (there is more betting on the river). Let's say there's another $50 behind in effective stack size after the call. it's actually very easy to do implied pot odds
25:(125+50) == 1:7

that's it. required equity is 12.5%.
no chg with hand equity. so 12.5% vs 18% == much bigger reason to call/you have much more juice

Looking at the EV method, this is 18% * 175 + 82% * (-25) = $11

Benefits of Offering the Correct Odds

On a side note, i think it is interesting to look at what the EV # means. One way to think about the expected value of your profit at each situation is your $equity - money put in. At each point of decision when you have to decide between raise/bet/call/check/fold, you are seeking to maximize your incremental profit.

Let's go through the scenario above where you and your opponent each put in $50 before the turn, and have $25 each left. At the beginning of the turn, your $equity is $18, so your accumulated profit since the start of the hand is $18 - $50 = -$32.

1. if you could check it down (opponent hadn't bet) == you would get avg $18 from the pot of $100. Your incremental profit in this scenario is $0. Your accumulated profit since the start of the hand is still -$32.
2. Opponent bets and you fold. You $equity dropped to $0 here from $18, and you also didn't put any more money in. So your incremental profit for choosing this option is -$18. Your accumulated profit is now -$32 - $18 = -$50.
3. Opponent bets and you call $25 allin. You would get avg 18%*150 = $27 from the $150 pot. Your $equity increased from $18 to $27, but you also spent $25 calling. Your incremental profit = +$9 -$25 = -$16. Add this to your pre-calling accum profit of -$32 before to see that your post-calling accum profit is -$48.

Note that if you could, you would still much rather get option #1 than having to choose between #2 and #3. In option #3, your are choosing an action that has -EV (you lose another $16). However, choosing option #2 would have even worse consequences (-$18). All this is because your opponent had bet out at you when you had <50% in %equity. You either put in more chips being the underdog, or you fold- effectively losing your pot equity (the chance to draw out on the winner).

This has very interesting implications for when you are playing/betting optimally with the winning hand. 

By making an optimal bet, you win exactly your opponent's $equity since they should be apathetic to folding.

Quick example to show this again: lets say your opponent still has 20% chance (one in five) to draw out on you in a $100 implied pre-money pot. Right now, your accum profit is 80 - 50 = $30. Optimally, you would lay 1:4 odds post-money, or 1/3 of the pre-money pot == $33. If we show that when you make this $33 bet, you are increasing your accum profit from $30 to $50, then we have shown a working example of the statement made above. It is obvious if opponent folds. If opponent calls, then your expected payoff is 80% * 166 = 133 and your cost is 50 + 33 = 83. Accum profit = 133 - 83 = 50. 

Note that this optimal bet sizing is most important to get right when it comes to closer draws (ie. it matter less when you are 90/10 favorite already). I claim this because with a close draw (say 60/40), the opponent still has 40% equity in the pot, so betting correctly to win that 40% equity is likely to be hugely lucrative, vs winning the 10% equity is less so. This may be a reason for why we have more freedom to slow play with trips etc when we are already 90%+ favorites.