Over history, people have found amazing solutions to the world's problems. Build the pyramids, put a man on the moon, send information securely over the internet to name a few. In the pure academic sense we've also solved incredibly tough problems like Fermat's Last Theorem and most of the Hilbert problems.

Long time ago as a budding mathematician I learned a few key principles of problem solving:

• Solve a more specific problem
• Solve a more general problem
• Make the problem easier
• Solve a similar problem

These principles can also help in analyzing poker situations. Often improving and understanding the game is met by road blocks because we don't pose the problem in the correct form. As a specific example consider consider the situation of playing 4-bet pots. Suppose you're playing a very aggressive and bad player. He 3-bets 50% of hands, rarely folds to 4-bets. I'm sure many hu players have played such opponents. It can be frustrating and not obvious how they are losing money. Let's pose the problem a little differently.

Since he defends (never flats) 50% of hands, we can't show an immediate profit by raising any 2. Thus, we obviously fold the weakest hands. It's not hard to see that that against such an opponent we should never raise/fold a hand (assuming a 3x open) since we don't shove immediate profit. Even air hands should be either folded or 4-bet. So say we pick some range, say 40% of 'playable' hands. I don't know what's optimal here, it depends on his postflop play, stack sizes, and our relative skill level.

Now pose the problem differently.

Hero (SB) posts an imaginary blind of 3bb (the open)
Villain (BB) posts an imaginary blind of 10bb (3-bet)

Flatting is equivalent to limping. 4-betting is equivalent to open raising. So we 4-bet some value range. With 100bb we have roughly a pot-stack-ratio of 1. We look at our equity against his calling range, his tendencies, and figure out the optimal play. I know you might hear complaints like it's hard to play AK when you miss the flop etc. But we are playing our 'opening' range against his 'flatting' range. If our range is better/stronger/more optimal than his we make money. Reducing the problem of 4-bet pots to a problem we understand better (how to beat opponents who flat too many weak hands oop), the situation isn't as daunting.

To recap, we reduce 4-bet pots against loose and non-folding opponents to a huge blinds game with a PSR of 1. This is an example of solving an analogous problem. The other techniques listed above also apply to poker. To be continued.

It is possible to attack NLHE bet sizing from a GTO perspective. A special case of this gives an interesting result. Consider another 'half street' game where the in position player has the decision to bet on the river. Clairvoyance as Chen calls it, is the situation when your opponent's hand is face up AND it is always a bluff catcher. This occurs often on the river in NLHE against poor players. You know whether your hand is good or not, and thus the only question is what size to bet (and obviously which frequency to bluff). Note that this means we never have a reason to check a hand with showdown value because villain's hand is face up.

Remember, in GTO terms, we must bet with a ratio of valuebets/bluffs such that villain has the same EV for calling versus folding (0 EV). If our range is skewed towards value hands (say 60% value and 40% air), then it is correct to bet our whole stack with our whole range. Yes, we should bet as much as possible with every hand and bluff with the appropriate frequency depending on bet size. For example, betting 2x pot means we should bluff 40% of the time.

Now some intuition to explain the result. The obvious answer seems to be for villain to call 40% of the time and negative our bluffs. This is wrong. Villain should actually fold 100% of the time. Even though we're bluffing, our range is skewed towards strong hands and thus we gain more and more for larger and larger bet sizes. To see this solution more clearly consider the case of villain calling some %, and think about our counter strategy of varying bet sizing for bluffs versus valuebets (ask me if you can't see it).

The point is that when villain's hand is face up, we should valuebet all-in with everything that beats him, and bluff accordingly. This isn't really a new concept. You may have heard the concept referred to as " the showdown tax". For example, on the river with a PSB left, bet every better hand (and some % of bluffs). Everyone is a calling station these days, so they tend to deviate from the 'always fold' case when your range is strong. Exploit that!

More pseudo game theory talk. Hand from a recent hu video of mine. I open 96o on button and get a flop of 766 rainbow. I bet and get called. Turn is 7, also bringing a flush draw. I checked turn: villain either has a 7/6/straight draw/A-high with occasional small PP or random float. So there's no value in betting turn. Checking lets him hit a pair or bluff river. Whereas there are almost no worse hands that can check/call turn.

Pretty standard I think. But what about the rest of our range. 96 is essentially AA-QQ here. 88 is a bit different because of the risk of free cards. So you can argue betting 88 for protection, but checking big (JJ+?). A 7 isn't too different, but maybe different enough to warrant a bet in a vacuum. The rest of our range is unpaired hands with SD value (say QT+), and hands with no SD value. No SD value hands gain from betting since villain will sometimes fold A high, and certainly straight draw type hands. SD value hands can also bet for protection. But SD hands can also check and try to pick off bluff or give up to a river bet.

In summary, the no SD value hands and occasional 7 gain the most from bets (44 and such). The medium strength hands like 88 A9 are strong enough to check down or pick off a bluff. The strong non-7 hands do fine by checking and betting or calling river unimproved. Should be even be betting here? Our range for betting will be really polarized, and I can't see a way to effectively merge our range (e.g. it's not like he c/c with K high vs our A high).

How do we gain more from bluffs without skewing balance? Solution is to check monsters and raise a proper proportion of our checked air hands as well as monsters. But the problem is that we rarely have a monster. Thus we can't bluff raise river too often.

The hard part is choosing correction action for hands like QJ (marginal SD value vs a river bet and some vs a check). How often do we pick off river bets with these? How often do we bet river when checked to? Playing rivers correctly is pretty tough. But with position it shouldn't be that hard. Does this balance even matter in this spot? This is a pretty rare board texture, but the ideas probably extend to similar spots. What do you think?

In Chapter 11 of MOP Chen analyzes some half street games. Half street games are poker games with one street of betting, in which only one player can do the betting. This situation comes up enough in practice to be useful to analyze. When at the river in position and the pot is big relative to stacks, the oop player often checks and his only choice is call or fold (effectively a half street game).

In these situations the in position player's optimal strategy is usually +EV. That is a powerful idea. It means that this player is +EV regardless (with the same EV) regardless of what the oop player does. Position is goot! Exploitive strategies need to be better than optimal to deviate. This is an obvious statement, but it illustrates that even players using exploitive strategies should understand some game theory in order to "know" when to deviate.

The key result of the chapter is the calculation of the optimal ratio of bluffs/valubets and the related optimal ratio of calls/folds for the ip and oop players. In the special case of NLHE with PSBs, this reduces to bluffing 1/3 often and calling 1/2 often.

There are a few ways to come up with the 1/3 and 1/2 frequencies. One way is to randomize your frequency with individual hands (call 1/2, fold 1/2 etc). That is the mixed strategy. Mixed strategies seem to occur when hand ranges are small. With large hand ranges, it's probably simpler to use absolute hand values to determine frequencies. For example, say you have 2 hands - tptk and two pair, both of which are the "same" against villain's range. The optimal strategy may have you call with two pair and fold tptk. Of course you could fold two pair and call tptk but that increases probability of bluffs with better hands. For most examples it appears that Chen models the range of hands with the interval [0,1] of real numbers. This makes calculations simpler.

Implementing optimal betting frequencies in practice requires an awareness of your own hand ranges. Obviously you need to know what your range is before choosing the 1/3 to bluff with. The final thought I want to mention is the inverse relation between payoffs and frequency. In optimal play, choices with high payoffs should be chosen with less frequently. The rowshambo example from chapter 10 is a good example. If you change the payoffs so that rocks has a higher payoff, the optimal solution requires that you choose rocks less frequently than in the equal payoff version. The intuitive reason is that you choose an optimal strategy vs the maximal exploitive strategy of villain. Since villain will also try to exploit the higher payoff, you have to do it less often to avoid exploitation. In real poker, you should bluff less often for $100 into $200 than you do when bluffing $200 into $200.

I will flush out my thoughts here on the subject. First, some definitions. An exploitive strategy is a strategy against a specific fixed strategy. For example, villain always bets $100 on the river as a bluff, so you call with all bluff catchers (the latter is the exploitive strategy, the former is an exploitable strategy). What we're familiar with is finding the maximally exploitive strategy (MES) (i.e. choosing the strategy to maximize EV against a particular strategy).

Optimal strategy is then derived from the idea of MES. We are hero, and suppose villain plays the MES against us. In other words, he adapts his strategy to maximize EV against our strategy. Hero's optimal strategy is the strategy with the highest EV against villain's MES. Equivalently, the optimal strategy makes villain indifferent between their various exploitive strategies (I'll leave it as an exercise. It is fairly simple to verify using the definition of optimal). This second definition (the exercise) is most useful in actually finding solutions to particular game theory problems.

There is an important property of optimal solutions worth stating. In practice, optimal strategies tend to be mixed. Mixed meaning that you choose action A X%, action B Y% instead of always choosing A or B. The key property is that the EV from any choice from a mixed strategy is the SAME. Say we're on the river and are deciding whether to bluff. If villain was exploitable we'd either bluff 100% or bluff 0%. If we don't know villain's tendencies we choose an optimal solution (against his MES). Something like bluff 30%. The key is that our EV is the same whether or not we bluff.

Thus, in the game theoretic approach to poker there is no value in choosing inferior strategies (i.e. -EV type plays for 'image'). Choosing inferior plays for future EV is a purely exploitive approach! Put more bluntly, game theory advocates believe that metagame is irrelevant.

I hope all that is correct. More to come.

Last time we analyzed the situation of 3-betting the flop for value. Recall that the board is 986r, and we get checkraised on the flop holding QQ. Let's now explore some of our turn options after calling a flop raise.

Case 1 - Fold unimproved. This is bad against all but the most predictable opponents.

Case 2 - Shove blanks, fold scary cards. We'll discuss this situation in this article.

Case 3 - Call blanks and decide on river, fold scary turns. I dislike this option on this board texture since even if the turn bricks we hate most of the deck on the river. This would be a line I'd take against easy to read players, since you can safely fold the river or easily pick off a bluff. A big draw will bet/call the turn anyways, so this option loses some value.

Case 4 - Call blanks for value, call or shove scary cards as a bluff. This is pretty interesting approach that I may discuss in a future article.

So let's continue with case 2. What are the scary cards? What are the blanks? Now, some of the scare cards are more scary than others, and the same goes for the blanks. But roughly I would say that we have the breakdown

Blank - 2,3,4,J,Q,K,A
Scary - 5,6,7,8,9,T

Note that an ace is somewhat scary (A7 or a pure bluff with AJ etc). A 6 and 8 are also not too scary, but we'll lump them in with the other scare cards. So our plan is to fold on 6 cards, and stack off on the other 7.

Assuming villain has the same range as in part one, we will have about 53% equity against the calling portion of that range on a typical blank turn. Some turns like the A lowers our equity, but 53% is a good approximation if we use a turn card like the J or 4. Our EV of getting allin on the turn is:

(6/13)(-10) + (7/13)[(.53)(130) - (.47)(110)]] = 4.6bb.

Now villain will bet/fold some hands on the turn. Suppose he bets 25 into 37 on the turn. Estimating his bet/folding frequency is a little tough, but let's use 15% (perhaps an underestimation). Then we get the new EV of:

(6/13)(-10) + (7/13)[(.15)(52) + (.85)((.53)(130) - (.47)(110))] = 7.5bb.

7.5bb is significantly lower than the 23bb we got for 3-betting the flop. So is calling to see a turn cleary incorrect? I don't think so. I may have overestimated our allin equity on the flop. I have also probably underestimated villains bluffing frequency on the turn. You can work out the numbers for slightly differennt assumptions to see what the EV difference is. My intuition tells me that the decision is closer, maybe a 4-5bb difference between the two. If the flop contains a flush draw I would guess that calling to wait for a safe turn will be even better. We will probably still stack off to a flush sometimes (call or shove turn depending on the flush card), but we can get more semibluffing hands to bet turn with poorer equity. Sometimes we get bluffed on the scary turns, but remember that we only lose the 10bb from the flop call.

I haven't written about theory since the isurapoker days, but here goes...Getting checkraised on the flop with an overpair on a coordinated board is a common tough situation. Many players have default plays based on experience and maybe a little bit of analysis. But it is instructive to sometimes dig a little deeper to analyze the core difficulties and options in the hand. A hand I recently played illustrates this.

I open from the cutoff and see a flop against the unknown big blind. Stacks are the 120bb, I have QQ, and the flop is 986 rainbow. The game is short handed 2/4. I bet 6bb into the 8bb pot, and get checkraised to 15bb. As a default, I would give my opponent a range of 66-TT, JT, 97s, 87s, 76s, 98s, 86s, A7s, some pure bluffs. The pure bluffs may be 22 or KQ for example. Today I will just analyze the option of 3-betting small and calling an allin. The more interesting options are calling and doing things on various turn cards, but I'll deal with those in the future.

Assume that everything but the pure bluffs will shove over my flop 3-bet. Against that range QQ has a nice edge with 55% equity. I am risking 110bb to win 130bb, so the break even equity is about 46%. Also, assume that villain bluffs the flop 15% of the time. Recall that we win 29bb when he folds, so the EV equation works out to:

EV 3-bet and call = (.15)(29) + (.85)[(.55)(130) - (.45)(110)] = 23bb.

That is a very nice profit, especially considering that 3-betting the flop is the easiest way to play the hand. Incidentally, we gain only 1bb by having villain checkraise and fold the flop (verify).

Of course we sometimes have marginal hands like A9 (41% equity) in this spot. With the A9 we are short of the required 46% needed to play for stacks (assuming no fold equity). Most players will be checkraising and folding the flop in this spot, so the 46% number needed to be adjusted based on that. The simple formula for this (verify!) is:

Adjusted equity = [(Amount lose) - (EV without showdown)] / (Amount win + Amount lose).

In our example, (Amount lose) = 110, (Amount win) = 130, and (EV without showdown) = 29bb * .15 = 4.3bb. Thus, we actually only need 44% equity to play for stacks in this example. What hands would you want to 3-bet shove with in this spot, and why? What other factors should be consider when deciding between a small or allin 3-bet?