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.

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