The Fundamental Theorem of Poker And SemiBluff Raising

No-limit hold’em’s complexity eludes nearly all players with nuanced decisions. Experienced players feel like we know the best play from seeing what works and what doesn’t over time; but, close spots in big pots are worth lots in the long run. Ignoring that value is foolish and wasteful if you want to be a winner.

In order to juxtapose two or more decisions at the poker, it’s important that you understand both the Fundamental Theorem of Poker (FTOP) and indifference principals.

The Fundamental Theorem Of Poker

Acclaimed poker author and strategist David Sklansky coined the tongue-in-cheek name, The Fundamental Theorem of Poker, to poke fun at the Fundamental Theorem of Calculus. The FTOP highlights the importance of forcing your opponents into difficult situations that create mistake-making opportunities that increase the value of your own plays.

Intuitively, calling down whenever your opponent bets on some boards is unlikely to force a good player into making mistakes. They will value bet when they should. They will give up when they should. And, it’s necessarily the case that the amount of strategic options that one can correctly implement into one’s strategy not only forces FTOP mistakes, but also creates value for one’s game play. That’s a lot of fancy talk, so let’s consider an example to clarify how the FTOP works.

Semibluff raising and the FTOP are like peas and carrots. Imagine that we have called a flop bet with a six-high flush draw, and are now considering how to play the turn against an aggressive opponent who has bet small enough that we certainly have a profitable call given our equity and implied odds on a river bet. Our question is should we raise, or should we call?

Before delving into numbers, we need to consider all possible outcomes, along with a few simplifying assumptions like we always win the pot if we hit our draw and our opponent always bluffs the river if we call on the turn.

Possible outcomes include:

• We call the bet, we hit the river, and win the pot and our opponent’s river bet
• We call the bet, we miss the river, and fold unimproved, having lost the turn call
• We raise all-in, our opponent calls, and we hit, winning the total pot
• We raise all-in, our opponent calls, and we miss, losing the remainder of our stack
• We raise all-in, our opponent folds, and we win the pot on the flop plus our opponent’s turn bet

Calculating the value of these decisions is more tedious than difficult. First, let’s focus on the important high-level points to advance our utilization of the FTOP.
You can see that on this particular turn card that there are many hands that have a fairly high chance to win the hand by improving to a pair, or by making a straight by the river, especially if we have a pair of sevens to call the turn bet, rather than a jack or a draw. Most notably are hands like 9 8 and A 5. And as you can see, if our opponent always bluffs the river and we miss the hand, then it doesn’t matter what he has; we’ll still be folding our six-high on the river when he bets. The two ideas of our opponent likely having decent equity and an estimated inability for us to raise the turn may lead our opponent to bluff the turn too widely and abuse us! Because we never threaten our opponent’s bluffing hands, we can see that calling fails at forcing our opponent to make significant mistakes in the hand, particularly on the turn and particularly against our weaker holdings like a pair of sevens and draws. This is especially true if we remove our simplifying assumptions and our opponent quits bluffing and maybe even check/folds some one-pair hands on river clubs when we make our flush.

So raising on the turn accomplishes two big goals for our hand, and for our strategy in this spot. First, we force opponents who over-do it with bluffs to fold far too often, which creates profitable semibluffs for us with hands that have enough equity when called. In this instance, our equity when called is typically around 33 percent. Secondly, our opponent’s decision to bet the turn, knowing that he may be raised, creates potential for FTOP mistakes.

If our opponent holds a hand like 9 8, then he actually has 55 percent equity in the hand. However, nine-high isn’t exactly an attractive bet/call on the turn! By forcing our opponent to make these mistakes, we win money on average.

The question is when should we semibluff raise to maximally take advantage of the FTOP? A solid clue has to do with the indifference point between the decision for us to call or to raise the turn.

Semibluffing And Indifference

Indifference is likely the single most important concept to understand in poker theory. It may sound convoluted, but we can use a variable (hello, high school math!) for our opponent’s bluffing percentage on the turn in order to decide which play, calling or raising, is more valuable for our 6 5.

First, we need to understand the value of both plays in terms of this variable opponent turn bluffing percentage, or more accurately, our opponent’s turn folding percentage. Let’s name the folding percentage for our opponent “x.”

Back to the possible outcomes of the hand, but this time, let’s add some monetary values. Let’s say on the turn that there’s $100 in the pot and $300 in effective stacks. Our opponent bets $80 on the turn. If we call or are called, then we improve to a winning hand 33 percent of the time. The other 67 percent of the time, we fold to the river bet or our opponent wins.

First, let’s consider the value of calling the turn bet.

• Outcome 1: We call the bet — we hit the river and win the pot and our opponent’s river bet (+ $400).
• Outcome 2: We call the bet — we miss the river and fold unimproved, having lost the turn call (- $80).
So we have two outcomes, neither of which depend on our opponent’s hand. In both scenarios, our opponent is moving all-in on the river, regardless of the card. Thirty-three percent of the time, we win a big pot, and 67 percent of the time, we lose our turn call. Not too shabby!
• Outcome 1: (0.33)($400) = $132
• Outcome 2: (0.67)(-$80) = -$53.6
Total value of calling: +
= $78.4

And for raising, there are three possible outcomes, all of which depend on our opponent’s folding percentage on the turn. Even if we are called, we still have a 33 percent chance to win the total pot. And just in case it looks confusing, (1-x) is the percentage our opponent calls, since x is the percentage he folds.

• Outcome 1: We raise all-in — our opponent calls and we hit, winning the total pot (+$400).
• Outcome 2: We raise all-in — our opponent calls and we miss, losing the remainder of our stack (-$300).
• Outcome 3: We raise all-in — our opponent folds and we win the pot on the flop plus our opponent’s turn bet ($180).

• Outcome 1: (1-x)(0.33)($400) = $132 – $132x
• Outcome 2: (1-x)(0.67)(-$300) = -$201 + $201x
• Outcome 3: (x)($180) = $180x

Total value of raising: +
+ = -$69 + $249x

And with the value of both plays, calling and raising, we can discover the best play based on our opponent’s turn folding percentage. We do this by setting the values equal to one another and solving for the indifferent point at x, which is the same as solving for the folding percentage on the turn for our opponent at which we are indifferent between calling his turn bet or raising all-in with our six-high draw.

(Total value of calling) = (Total value of raising) —> $78.4 = -$69 + $249x

Which simplifies to show that x = 0.592

Folding percentage at point of indifference between calling and raising the turn = 0.592
Meaning that so long as our opponent’s turn folding percentage is greater than 59.2 percent, it is more profitable to move all-in than it is to call.

And now you know the power of using the Fundamental Theorem of Poker and indifference principals to identify the most profitable decision at the table! For extra credit, consider what happens if your opponent gives up with his turn bluffs some of the time on the river. Another exercise is to use set folding and calling percentages for your opponent (make sure that sum to 100 percent) and solve for the minimum chance of winning that your hand must have when called in order to move all-in on the turn.

If you have any questions, or just want to say hello, tweet me
@TransformPoker anytime! ♠

Reid Young is a successful cash game player and poker coach. He is the founder of TransformPoker.com.