Fundamental Theorem of Poker

1. Introduction

In this article

• Avoid mistakes, while provoking your opponents to make them
  
• How the Fundamental Theorem of Poker is relevant for your game

The renowned American poker player, theoretician and book author David
Sklansky postulated what was, in his opinion, the most important
theorem for the game of poker, his Fundamental Theorem of Poker (FToP):

This article comments on the implications and the conclusions of
Sklansky's FToP and shows how it has an effect on the game itself and
the choice of one's own playing style. It elaborates on different
standard situations like the semi-bluff or the free-card raise and
explains where the relevance and the usable application of the FToP lie
in these situations.

2. The relevance of the FToP for your own game

When I read
Sklansky's theorem for the first time, two thoughts crossed my mind:
one thing was its significance for the actual game was doubtful to me,
as poker is after all a game of incomplete information and thus the
theorem cannot be directly used for concrete game situations. I saw it
as a theoretical construct without practical relevance. The second
thing I thought about was that this theorem is so trivial and obvious
that I could have thought of it myself.

It was much later that I realized I didn't get to the root of the
matter with my objections -  it wasn't always about directly using the
theorem in concrete situations, on the contrary, there are times you
are virtually forced to violate the theorem due to probabilities. If
every card was in the open, it wouldn't be a poker game: you would be
able to calculate the EV of each hand precisely with the help of
mathematical methods and thus use the correct playing style. If a
player deviates from this, he decreases his own EV and increases the
one of his opponent.

But you usually lack the necessary information to act precisely in that
manner, so you are depending on probabilities. This is why it is
sometimes impossible not to violate the FToP. A simple example should
demonstrate this: if you hold KK, you should, of course, raise
pre-flop. If an opponent holds AA though, you've played the hand in a
way that he makes profit from. Had you known that the opponent is holding AA, you would have folded, thus you would have played the hand
differently from the way you would have played it if you knew your
opponent's cards.

Nonetheless, the chosen playing style was, of course, correct.
According to probability deliberations, it is certainly profitable to
raise with KK. So this means: following the theorem and correctly
playing a hand can, in fact, be in contradiction to each other, and the
term mistake
is thus put into perspective in this sense. As we will still see in the
following parts of the article, it is at least under certain situations
promising, if not even necessary, to deliberately violate the FToP to
obtain other strategic advantages. Furthermore, Sklansky did not draw
up his theorem in order to decide proper playing styles; on the
contrary, this theorem is the essence of countless game theories. It is
virtually the summary, the common basis and the theoretical superstructure
of terms like bluff, semi-bluff, hand reading, value bet on the river
etc. The goal of this article is to point out such connections, i.e.
illustrate that the FToP is the overarching element, the common ribbon
of all these terms. We will see that it is like a guiding principle for
every poker decision one has to make.

Poker is a zero-sum game - someone's winnings are another one's losses.
This is a basic prerequisite and has to be the starting situation for
further considerations. If everybody plays perfectly, nobody can win in
the long run (except for the casino, which always wins). So this means
that you can only win at poker when others make mistakes, namely so big
and so numerous that there is still something left after substracting
the rake. From this point of view, Sklansky's theorem also says
something like: Avoid making mistakes and provoke your opponent into making mistakes.

This is a truism as well, because the money can't come from anywhere
else. Next, we will take this as background and we will now analyze
some strategic concepts a bit closer which try to put Sklansky's
concept of "avoiding mistakes and provoking your opponent into making mistakes" to practice.
Some information will be taken into consideration many times from
different perspectives; this is unavoidable, as every concept is tied
to the FToP.

3. Hand reading

Avoid making mistakes means trying to
evaluate the opponent's hand and then reacting accordingly. First of
all, you have opposing imperatives in relation to the board cards at
your own disposal. In poker literature, there are many examples in
which an opposing hand is determined with exactitude (e.g. Sklansky,
Hold'em Poker, p. 89 et seq.). Of course, this is usually not possible
in real life, but it isn't necessary either. It helps a lot to be able
to limit the range of the possible opposing hand, a very complex
endeavor which can be endlessly perfected. If your knowledge of the
opposing hand range influences your own playing style, then you can at
least try to approach the FToP; if you can do this, then you can be a
great player. An example will help to illustrate the concept:

An opponent hasn't shown any particular strength, but bets the turn and
river. If you hold a marginal hand yourself, you should tend towards calling the river bet
if there some draws possible on the turn. The opponent could have been
on a draw which didn't hit and is now bluff-betting the river.

If there were no draw possibilities on the turn, you have to assume
that the opponent tried to make a pure bluff on the turn, as well as on
the river. So in the first case, a bluff is more probable, than a
semi-bluff, which is much easier to dish out. If you think it's possible that
he bluffs the river with a busted draw due to previous experiences with
this opponent (either by watching his playing style directly, analyzing
the data or by reading player notes), you can call more easily than if
the turn hadn't offered any draw possibilities.

If there are 5 BB in the pot, you only have to be ahead once in 5 cases
to have a positive EV (when playing this situation five times you put 5
BB into the pot and win 6 BB once on average, so 0.2 BB per hand on
average; if you are ahead in one of 6 cases, it doesn't matter if you
call or not; if you believe you are less often ahead, then you should
fold as you have a negative EV).

So if you guess that the opponent also bets a busted draw in 20% of the
cases, then statistically it doesn't matter if he has a good hand or
not, as the average profit is equal to 0.2BB .If you lose,
you have violated the FToP; nonetheless, calling is profitable in the
long run, as the size of the profit equals out the losses in case the
opponent is bluffing. To put the opposing hand on a possible draw on
the turn and thus a busted draw on the river is inaccurate and often
wrong, but is at least an attempt at limiting the range, which is
considerably better than ignoring it.

In general, you have more information, apart
from the opponent's reactions and the board, via Poker Tracker or
through watching him closely. If you know that he raises 40% of
his hands pre-flop (e.g. in steal position), then you have an equity
edge with e.g. 66 and thus a clear re-raise - against the majority of
his hands, you are ahead. Of course you can run into AA or similar, but
that is improbable. The known raise range of the opponent thus
literally demands re-raising. Whether you yourself have violated the
FToP or the opponent has, remains to be seen. Due to the reads, though,
it is more probable that it was the opponent who made the mistake.

If, on the other hand, the opponent only raises 10% of his hands
pre-flop, then you should fold as soon as possible. Of course there are
also opposing hands in the range against which you have an equity edge,
but there aren't enough of them, so it isn't profitable and the
probability that you are now violating the FToP is too high. Note that,
first of all, it's not about violating the FToP in the actual case, but
it's rather about evaluating how high is the probability that we are
violating it. Poker is, and will remain, a game with incomplete
information.

The value bet on the river also
must also be considered in this context. If a loose opponent who often
goes to the showdown hasn't shown any strength until then, you should
also bet for value with a marginal hand, e.g. TT on a board of J9643.
The opponent could, of course, have a J or a set, but 9x, 6x, 88, 77,
55, 22, AXx are also possible. As he is loose and often goes to the
showdown, he will call down all of them. According to the analysis of
the opponent, you come to the conclusion that you are probably ahead of
the range of his possible hands and the opponent will also call with a
worse hand, thus you should bet for value.

If you rate the opponent as being tight though, then the danger of him
holding a J is much higher. Furthermore, he probably wouldn't call with
hands like A6 on the river anymore. This means that a bet has a
considerably smaller probability for success, be it because you are
behind, or that he will fold anyway.

The basic rule is: You should only bet for value on the river when you
are ahead in about 55% of those cases in which the opponent will call
(50% isn't enough as you can never rule out the opponent having a
monster). Thus the decision to bet depends on the range of hands on
which you put the opponent.

4. Provoke mistakes from the opponent

Provoke your opponent into making mistakes
assumes that you even give the opponent the opportunity to make
mistakes. Therefore, it is necessary to obscure the strength of your
own hand. There are basically two possibilities for this:

• At first, play a strong hand in a way it appears to be weak (slowplaying)
• Play a weak hand in a way it appears to be strong (bluffing).

It is extremely important to include both possibilities into your own
game, because if you only bet and raise the strong hands, and only
check and call the weak ones, you're too easily figured out - the
opponent has no problem adjusting whatsoever and will hardly ever
violate the FToP with you again. This means that you lose.

How you use these possibilities depends on the opponent and on your own
style. The decisive factor, though, is that you sometimes use them and
sometimes not. This way, certain bet sequences become ambivalent for
the opponent: he doesn't know where he is situated and has ample
opportunities to violate the FToP.

Examples showing the ambivalence of bet sequences can be found in the
recommendations for the heads-up play on the flop: you check-raise the
aggressor on the flop with a made hand as well as with a draw hand -
the aggressor doesn't know where he stands then (for a full explanation
see semi-bluff). To put it the other way around, as aggressor, you only
call a check-raise on the flop with a strong hand to then raise the
turn. After the call, the check-raiser doesn't know where he stands.

Raising for free-card also belongs to this
context: you raise the flop with a draw (so you usually don't hold
much), hoping that the opponent checks to you on the following turn
round so that you (in case the draw doesn't hit) see the river card for
free.

The raise is thus (at least) ambivalent. From the opponent's point of
view it could be for value with a strong hand or with a weak drawing
hand. The theoretical approach of the raiser is interesting though. At
first he violates the FToP himself by investing a further SB against
the call. If the opponent knows that it is only an attempt at receiving
a free card, he will, of course, re-raise – and indeed this is the best
defense against an assumed free-card raise. But maybe he has good
reasons to only call – and already he has violated the FToP because he
hasn't raised.

If you receive your free card, you have saved 1 SB at the turn (1 BB -
1 SB which was previously invested). If the opponent raises, it becomes
expensive. Consequently, you have to closely check what sort of
opponent you are up against, if you dare such an attempt, but due to
this very simple example, the principle becomes clear: you invest a
small amount to entice the opponent into making a mistake. That this
attempt might fail does not weigh against it basically - the opponent
is put under pressure and is likely to make mistakes. After all, the
raise could also have been for value; if so, then his re-raise is a
mistake and will be punished.

There is hardly ever an opportunity for really
sensible slowplaying, as you should only do it with relatively
(depending on the amount of opponents and types, pot size and board)
very strong hands. You have to keep in mind that you give the opponents
the possibility to play their draws cheaply: this makes sense when they
basically only have draws for the second best hand, i.e. that you still
beat most draws even if they hit. If they play their draws anyway, they
violate the FToP because they wouldn't have done so if they knew your
hand. Protection is, in a way, a converse strategy (see 5.2.).

The second possibility for obscuring your own
game is to bluff in a more specific way, i.e. to bet and raise with the
currently non-strongest hand. Slowplaying and bluff calls are also
bluffs in the broader sense, but we don't mean that over here. You have
to include bluffs in your game though.

Again: if you bet and raise only with suitable hands (with those you'd
even play this way if you knew all of your opponents' cards), then your
opponent can adjust. The thing is that you don't win much, but you'll
eventually lose a lot. It would be as if you actually let your opponent
glance at your cards, i.e. he will no longer violate the FToP, and so,
in the end, he will profit.

If you are caught bluffing you do lose money, but you are
also creating a certain insecurity in the opponent's mind: he now no
longer knows how to interpret the bets as they have become ambiguous to
him, so he will make mistakes and violate the FToP.

Bluffing is sensible in many different ways and contexts and the
previously described free-card raising also belongs to it in a certain
way. Next, we'll mention some other possibilities.

The pure bluff – bluffing in the
proper meaning of the word – is betting with a hand which is probably
not the strongest and rarely has the opportunity to become the
strongest. If you get called, you've lost, thus you have to evaluate
precisely if the opponent folds often enough for a bluff to be
profitable, as the profit is potentially greater than the insecurity of
the opponent. Against weak opponents, cops and control freaks who have
to check everything all the time, bluffing is completely useless. But
this type of opponent violates the FToP constantly anyway, as he always
calls and thus is easy to figure out. How you act against him has been
previously explained in the section Value Bet on the River (see 3.3.).

At first glance, the pure bluff seems to be the most obvious violation
of the FToP. Surprisingly though, bluffing with the correct frequency
is profitable per se according to gaming theory - the ratio of made
hand bet versus bluffs has to be equivalent to the opponent's pot odds.
Let's calculate this with a simple example which can be easily
generalized:

The pot has 8 BB, Player A bets and Player B can only win against a
bluff. Player B now has 9:1 pot odds, as one other BB has entered the
pot. If Player A now bluffs in 1 of 10 cases (so 9:1, the bluffing
frequency corresponds to the opponent's pot odds), it doesn't matter if
Player B calls or folds, he always has an EV of 0:

1st Case, Player B always folds:
This
leads Player A to the following profit: he invests ten times 1 bet
(-10BB) and wins ten times 9 BB (+90BB), overall a profit of 80 BB.

2nd Case, Player B always calls:
This leads Player A to the following profit: he invests ten times 1 bet
and wins nine times 10 BB, overall a profit of 80 BB as well.

If Player A never bluffs (so he only bets when he is ahead and
check-folds his hand when he is behind) and if Player B would know
this, then Player A would only win the pot with 8 BB in 9 of 10 cases,
so that's 72 BB. If Player A bluffed, but with a lower frequency than
the opponent's pot odds though, he would still win more, but not as
much - his profit would be between 72 and 80 BB. However, if Player A
bluffs more often, it reduces his expected value. This can be easily
calculated. The optimal bluffing frequency thus corresponds to the
opponent's pot odds. Player B is powerless: Player A could even tell
him his bluffing strategy and Player B would still not be able to
defend himself. The prerequisite, however, is that Player Player A
really uses his bluffs according to the random principle so that Player
B cannot discern any pattern.

The semi-bluff is a bet or a raise with a
hand which is very probably not the best one for now, but has many outs
to become the best one. Of course, here as well, you make a mistake
in a certain way, as you give the opponent the opportunity to raise,
which he would certainly do if he knew all the cards. You only lead him
to believe that you have a better hand than you really have due to the
bet, which is why the opponent will rather call only (for defense
against semi-bluff, see below).

Not raising but just calling is already a mistake according to the FToP
though. and folding would be an even bigger mistake. You thus invest a
certain, relatively small amount with the goal of enticing the opponent
into making a mistake according to the FToP. As you have several outs
with a semi-bluff, you could hit if the opponent doesn't fold, the
costs according to the equity principle are thus lower than the amount
of the immediate investment. Let's use an example to calculate the
ratio in which the opponent has to fold a better hand for the
semi-bluff to thus become profitable:

If e.g. you have 12 outs on the turn and there are 8 BB in the pot, you
are basically a 3:1 underdog and according to the equity principle, 25%
of the pot thus belongs
to you (you win in 1 of 4 cases on average), so 2 BB. Calling is thus
definitely +EV. With the semi-bluff you start an attack on the
remaining 6 bets in the pot (pot size - equity) with a (in the case of
a further semi-bluff raise) bet. Of course, 25% of your own bet and 25%
of the opposing one in case he calls already belong to you
according to the equity principle. If the opponent folds instead,
you're satisfied anyway and if he calls, the investment costs are thus
only half of the paid bet.

In practice, the actual costs are a bit higher as you also have to
expect an opponent's raise. In case opponent calls, you invest ½ BB to
win 6 BB. The ratio of investment to profit is thus 0.5:6 = 1:12. The
opponent only has to fold in 8% of the cases for the semi-bluff to have
a +EV. (1:12 means that you win on average once and lose twelve times
when playing like this 13 times, so the win probability is 1/13. 1/13
extended to a hundred is ~7.69/100, so about 8%. You can also calculate
it the other way round: in 100 games you invest 100 BB and win +7.69*13
BB = ~100 BB)
A precise calculation is very difficult as you would have to guess
almost exactly how probable it is that the opponent folds, how probable
it is that the opponent raises, and how probable it is that the
opponent still pays in case of hitting, to be able to determine if
either calling or semi-bluffing is more profitable.

In this context, a precise calculation isn't decisive either. The
example calculation shows that the percentage of an opponent's fold
doesn't have to be very big. If you generalize the example calculation,
you see that the more outs you have and the bigger the pot is, the
smaller the percentage an opponent's fold can be.

Keep in mind that you are trying to induce the opponent into making a
big mistake with a small investment. Of course, he cannot defend
himself against this by simply raising every bet as he has to expect
that you don't have a semi-bluff, but really a strong made hand. Here
again, the connection that we have to aim for is shown, which is that
our playing style doesn't become predictable. The opponent's fold on a
semi-bluff is often correct so that – at least against better opponents
who able to fold unprofitable hands – a semi-bluff increases the EV
considerably. Against loose opponents who call everything, a semi-bluff
is, of course, not profitable as the previous calculation showed. The
share of the profit is missing, which results from the occasional
folding of the opponent (comp. previous calculation).

A further advantage of the semi-bluff, is that, if you hit, you collect
further bets as the opponent cannot recognize that the new card on the
board has helped you, e.g.: You defend QTo against a steal raise, the
flop is K9x rainbow. A check-raise on the flop as semi-bluff is
certainly good here, as the opponent has to fold many hands of his
steal range. If he holds the K and the turn brings a J, he cannot
easily see that he is beaten. He thus has to effectively violate the
FToP. You cannot ask for more.

From the example we can deduce that: the bigger the pot, the better the
pot odds are, which means that your opponents have to fold less often
for a semi-bluff to be profitable. From game theory, it means that the
bigger the pot is, the less often you should bluff, at least against
experts (comp. 3.4.1.). In practice, many opponents don't use their
calling strategy correctly though, thus bluffs and semi-bluffs are more
profitable when the pot is big.

The defense against the semi-bluff can also be seen as an aspect of the
FToP - you should fold or raise. Calling is, in general, not profitable
as you eventually let the opponent draw cheaply. As an example, let's
use the 3-bet as our standard move OoP with a strong hand against one
opponent:

Pre-flop:

Hero holds UTG:
,

Hero raises, BU calls.

Flop:

Hero bets, BU raises.

Here, Hero sees the opponent's flush draw or a worse K or
something that's lower and 3-bets to make him pay the most for his
draw. The opponent has violated the FToP and will be bled: to only call
would be a violation of the FToP, as it requires too little from him.
If Hero didn't see right and the opponent holds two pair or a
set, then Hero has violated the FToP and has to pay. Here again you see
the difference between 'useful' and correct (according to the
FToP) playing style. As the probability of the opponent only holding a
drawing hand is much higher, the 3-bet is sensible. If the opponent
holds the better hand, Hero, who tries to follow the FToP, will be
trapped and literally forced to violate the FToP himself. Poker is
after all a game of incomplete information - it's all about
probabilities.

The continuation bet on the flop also
has to be seen against the background of the FToP. If you didn't hit
and maybe don't even have the best hand, you can try to steal the pot
via a continuation bet in certain situations. In a sense, it is then a
pure bluff (if you barely have any outs), or a semi-bluff respectively
(if you do have a few outs).

If you don't bet, the opponent can't fold either. Your own violation of
the FToP induces the opponent into making an even bigger mistake
himself – folding. The opponent only knows that he is probably facing a
(relatively) strong hand and is put under pressure by the bet. Here he
can easily make a mistake because calling can also be a mistake.

The situation on the turn is more complex. King Yao (p. 208) gives the following example:

Pre-flop:

Hero holds:

Hero open-raises, a loose player in the BB calls.

Flop:

Opponent checks, Hero bets, Opponent calls,

Turn:

If Hero bets again, his opponent will call with any pair. Hero prevents
that though, so that he will see the river for free in case the
opponent doesn't hold a pair. Furthermore, he receives the chance that
his loose opponent calls his gut shot, which increases his own EV. The
problem becomes interesting though when you look at it from the
opponent's point of view: if he holds T9 or T8, he doesn't know how
many outs he really has. If Hero has e.g. hit with KQ, he only has 4
outs and a call would be unprofitable. If Hero hasn't hit (and no
pocket pair), he has 10 outs and a call would be profitable, thus he
doesn't know if he has 4 or 10 outs. If he takes 7 (discounted) outs,
he lacks the pot odds. Hero's continuation bet on the turn with AQo
thus gave him the chance to make an incorrect decision.

5. Pot Odds and implied odds

The message from Sklansky's FToP that the
opponent wins when you yourself lose mainly refers to heads-up
situations - if several players are involved, situations in which you
do lose a bit against some are possible, but you win so much against
others that in the end, you do have a +EV. This is especially the case
with strong draws or speculative starting hands respectively, as the
opponents barely have a chance if you do hit. If they also hit, they
are behind nonetheless and have very few redraw possibilities due to
your hand's strength, so you collect a lot from the strong draws and
only give away a bit again (due to smaller redraws).

Sklansky uses the example of raising with A8s against loose opponents
when 5 players have limped before (HPFAP, p. 173). As simulation
programs show, the equity of A8s is considerably below the average
equity, thus, at first, a raise seems absurd. The decisive point though
is that your own equity increases dramatically if you at least hit a
2-flush on the flop. Then about 1/3rd of the pot belongs to you as you
complete the flush by the river in about 1/3rd of the cases (9/47 +
38/47*9/46 = ~0.35). Players with e.g. bottom pair, who now have at
least 12:1 pot odds, reduce the equity of a player with e.g. top pair,
because this hand isn't very strong and can still be caught up to.

Our equity will barely be effected though, because we will very probably win if we hit. Due to our raising before the flop we improve
the pot odds for every weak draw on the flop, so that these players
stay longer in the hand. If you hit, it's exactly what you want: to
bind the loose players to the hand and take away their money in case
you hit. This makes the pre-flop raise tempting, despite the shortage
of equity - you make a small mistake pre-flop to later induce
the opponents into making bigger mistakes. You can do the same with
suited connectors and small pocket pairs in the same fashion as with
Axs.

The strategic concept of increasing the pot
odds for the opponents to bind them to the pot can also be reversed:
you decrease the pot odds to chase them out of the pot this practice is
called protection. It should be used if you believe you hold
the best hand for now, but it's still a vulnerable one, e.g. top pair
with good kicker on a board that allows draws. Let's explain the
principle again with an example which can be easily generalized:

If, for example, on the flop there are 8 SB in the pot and an opponent
bets, a player who believes he holds the best hand should raise. This
is because other opponents who have weak draws, e.g. a gut shot (11:1
needed pot odds), and haven't called yet, will have trouble doing so.

Meanwhile, 8+1+2=11 SB would be in the pot but they would have to pay 2
SB, so that they only have 5.5:1 pot odds now. Thus they should fold
and thus give up their share of the pot which belongs to them according
to the equity principle. If they call, they'll pay too much for their
draws anyway. For this purpose, compare the case in which the player
who believes he is ahead only calls:

Now there are 10 SB in the pot and the other players only have to pay 1
SB, thus their pot odds are 10:1. They can call profitably and keep
their equity share, so a raise leads the player to a win/win situation:
either he takes away the opponents' share of the pot, or he misleads
them into an unprofitable call and thus into a mistake. This would also
be fine for him.

In this context, let's add a notice about the bad
beat: just because opponents make mistakes – e.g. in the previous
example, the calling without ample pot odds – that doesn't mean that
they will lose every hand, and not making any mistakes yourself doesn't
mean that you'll win in the short run.

We all get worked up over situations where we had the best hand until
the river and built up a big pot, only to lose it because the opponent
hits the miracle card. The anger is, of course, understandable, but
it's actually completely out of place: if the opponent didn't have the
needed pot odds, he made a mistake according to Sklansky's theorem (or
he doesn't care about probabilities and is a fish).

You can easily calculate how big the mistake was, i.e. how
improbable his win was, compared to the pot size. Nonetheless, he has
the pot. This means that while the opponent might have won this one
time –and you lose money despite playing correctly – he will repeat
this mistake so often in the future that you will win money against him
in the long run. Thus, you should be happy to have such a player at the
table. If you sit at a table where no bad beats happen, you should
consider leaving it because the opponents make too few mistakes.

6. Outlook and recap

This article should only be a stimulus
into thinking again about the FToP and its consequences: we can
literally go on with this topic forever. Studying Poker Tracker data
and watching your opponents' betting sequences, for example, leads to
the detection of different leaks which you should abuse mercilessly.

Some approaches to this have been previously examined and always keep
in mind that every practice has a common basis, namely the FToP: you
always try to either provoke the opponent into making bigger mistakes
with your own smaller mistakes (e.g. deviating from the objectively
best line, which would mean knowing every card), or you try to punish
your opponent for his mistakes by following the objectively best line.
This cannot work all the time, since poker is – as has been pointed out
repeatedly – a game with incomplete information and thus it's all about
probabilities, but the effort to think in these categories will help
improve one's own playing style.