2. Balance and MDF

Briefly alluded to in the prior section is the understanding that betting can allow us to win a higher % of the pot for both our value hands and our bluff hands. And the reason that this can happen is due to balance. But before explaining the concept of balance, we must first discuss equilbrium.

Equilibrium:

Using a quick google search, a Nash equilibrium is defined as the set of strategies where when each player is assumed to know the strategies of all the other players, no one has anything to gain by changing only one's own strategy. What this means for poker is a fewfold. Firstly, it means that if your opponent(s) is using an equilibirum (game theory optimal) strategy, you must also use a game theory optimal strategy to maximize your win rate (or more precisely, minimize how much you lose). Secondly, if your opponent(s) is not using a game theory optimal strategy, using a game theory optimal solution yourself will not win the maximum against their strategy as you can deviate and exploit your opponent for further pot share; however, if they then realize the strategy that you are employing is an "exploitative" one (one designed to exploit their strategy), they can then counterexploit you (until after infinite iterations of you guys counterexploiting each other back and forth, you converge on the equilibrium solution).

Getting back to balance, the main discussion of balance here will just be that at equilibrium, the bettor will have a ratio of bluffs vs values that makes the opponent indifferent with their bluff catchers between calling and folding. And in a circular manner, a bluff catcher is thus defined as a hand that is indifferent between calling a bet and folding. One importance of this is that by definition, a bluff catcher beats none of the bettor's value bets, because if it did, it should always call (because now that it beats more than just the bluffs, it can profitably call). Similarly, a bluff catcher must also not lose to any bluffs, because if it did, it should always fold (this may not be 100% true due to blocker effects in some edge case situations). To be clear, the most direct application of this concept appears on the final betting node (i.e. betting all-in on the river). On earlier streets, shifting equities from the turn and river as well as additional betting steets complicates things but the concept still applies.

Now, what is this correct balance between value and bluffs? The exact math for the bluff to value ratio is that given a pot size of P and bet size of S, if the player betting is value-betting v proportion of the time and bluffing b proportion of the time (where by definition b=1-v), the player facing the bet would have an EV of calling with its bluff catchers of b*(P+S)+v*(-S), so if that EV is greater than 0, it should call all of its bluff catchers and if it is less than 0, it should fold all of its bluff catchers. So at equilibrium, v=(P+S)/(P+2S). Practically speaking, the bluff-value ratio for a pot-sized bet must be 1:2 (value-betting 67% of the time), the bluff-value ratio for an infinite-sized bet must be 1:1 (value-betting 50% of the time), and the bluff-value ratio for a third-pot-sized bet must be 1:4 (value-betting 80% of the time).

Because the whole concept of an equilibrium bluff-value ratio is based on the assumption that the opponent's EV of calling with its bluff catchers is 0 (which is the EV of folding), there are some interesting implications for earlier streets. As mentioned in the prior paragraph, assuming a pot-sized river bet, the equilibrium value-bet frequency is 67%. Is the equilibrium value-bet frequency of a pot-sized turn bet the same given we are also betting pot on the river (assuming no equity shifting cards, so say the board is 2222 on the turn and ignoring the concept of blockers)? The answer, importantly, is NO! Because every bluff-catcher on the river has an EV of 0 for calling, for simplicity's sake, we can just assume every bluff-catcher is folding the river (if the opponent actually calls any positive % of its bluff catchers, all that does is shift EV from the bettor's bluff hands to its value hands, but the EV of the strategy for all the hands that bet as a whole will be the same, which is 100% the size of the pot before betting), and so all the hands that bet the river in essence can be considered turn value-bets! What that means is then that given a pot-sized river bet, the equilibrium turn bluff-value ratio is 5:4 (value-betting 44% of the time). Similarly, given a pot-sized turn bet and pot-sized river bet, assuming no equity shifting cards (and assuming the opponent cannot bet or raise themselves), the equilibirum bluff-value ratio of a pot-sized flop bet is now 19:8 (value-betting only 30% of the time). Note that in this case, if the pot size on the flop was P, we've bet a total of P+3P+9P=13P by the river and have an EV of P with 19/8*V bluff hands (where V is the number of value hands we have). If we were to instead bet 13P over only one street of betting, we'd only be allowed to have an EV of P with 13/14*V bluff hands (which in turn, means we have a much higher EV for our overall strategy spreading out our bets equally)! This is the whole reasoning for geometric sizing that you'll hear often talked about in high-level strategy, which simply means betting the same amount every remaining street to get all-in (2e means what % of the pot to bet over 2 streets to get allin; 3e means what % of the pot to bet over 3 streets to get allin). Whenever a player (usually the uncapped player) is significantly more nutted than the opponent and the board is fairly dry, the optimal sizing will usually include a 3e size to maximizize the number of bluff hands that can be included in the strategy of winning 100% of the flop pot share.

Furthermore, since a strategy of betting pot on every street means we can have a bluff-value ratio of over 7:3 on the flop and 5:4 on the turn, in practice, most understudied players are vastly underbluffing earlier nodes for large sizes (this of course is mitigated by them actually decreasing their bet sizes over subsequent streets, so a pot-sized flop bet is usually followed by a smaller than pot-sized turn bet). Still (and this will be further discussed in the exploits section), large flop bets can usually be overfolded when playing against most understudied players.

So far, we've only discussed the concept of balance from the standpoint of the bettor; what about the player facing the bet? Here we will introduce the term Minimum Defense Frequency (MDF), which is the minimum % of the caller's range that must defend (call/raise) vs a given bet size to prevent the bettor from profitably betting all of their bluffs. The math behind MDF is such that given a pot size of P and a bet size of B, if the player facing the bet calls c proportion of their range and folds f proportion of their range (where f=1-c), the bettor would have an EV of betting with its bluffs of f*P+c*(-B), so if that EV is greater than 0, it should bet all of its bluffs and if it is less than 0, it should bet none of them. So at equilibrium, c=P/(P+B). Practically speaking, the MDF for a pot-sized bet must be 50%, the MDF for an infinite-sized bet converges to 0%, and the MDF for a third-pot-sized bet must be 75%. Now, the concept of MDF most directly applies on the river and in spots where the bettor doesn't have a wide range advantage, as it assumes that the EV of checking bluffs is 0. If the EV of checking bluffs is higher than 0 (which is essentially true IP on earlier streets, especially on favorable boards for the bettor), then the player facing the bet can in fact defend lower than MDF.