Two Plus Two Newer Archives - Nate's Theorem on the "Value Bluff"

I stumbled across the aejones thread on this board tonight and decided that there's a lot of nuance that people are missing with respect to the concept of a "value bluff". I'm going to post this here even though I'm a limit player because this is where I assume it will draw the most interest.

Suppose that you are in position with a marginal hand and contemplating a pot-sized bet of $100 on the river. We will assume that your opponent is not allowed to check-raise.

There are three relevant cases for what can happen when we bet.

Firstly, the opponent can call with a worse hand (CW), which nets us a profit of $100.

Secondly, the opponent can call with a better hand (CB), which costs us $100.

Thirdly, the opponent can fold a better hand (FB), which makes us the size of the pot or $100.

There is a fourth case -- the opponent folds with a worse hand (FW) -- but this does not matter since we will win exactly what's in the pot already but nothing more either way.

Suppose furthermore that we have some uncertainty about just what type of opponent we are up against.

Opponent A is a calling station who will sometimes call with worse hands and sometimes call with better hands but will never fold a better hand. We make money against A when he calls with a worse hand and lose it when he calls with a better one.

Opponent B is weak-tight and will sometimes call with better hands and will sometimes fold better hands but will never call with worse hands. We make our money against B when he folds a better hand and lose it when he calls with a better hand.

We don't know what percentage of the time we're up against opponent A and what percentage of the time we're up against opponent B. BTW, A and B can actually be different opponents or the same opponent at different times; it doesn't matter for the sake of this example.

The key thing to recognize is that there is no circumstance under which betting is profitable against neither opponent A nor opponent B but is profitable if we don't know whether we're up against opponent A or opponent B. Put more banally, two wrongs cannot make a right; we cannot combine a -EV bluff with a -EV value-bet to make a +EV "value bluff".

For example, say that Opponent A's action against your river bet is as follows:

45% of the time, he calls with a worse hand (CW)
55% of the time, he calls with a better hand (CB)

Your bet has an expectation of -$10 against opponent A.

On the other hand, for opponent B:

45% of the time, he folds a better hand (FB)
55% of the time, he calls with a better hand (CB)

Your bet has an expectation of -$10 against Opponent B.

It doesn't matter how often we are up against Opponent A and how often we are up against Opponent B. For any distribution of Opponent A and B, the bet will have an expectation of -$10, because it is the weighted average of two individual cases for which the expectation is -$10. This is fairly trivial to prove mathematically.

However, all is not lost for the concept of the value-bluff! Suppose that we tweak our opponent's frequencies slightly. Now, Opponent A will:

CW 55% of the time
CB 45% of the time

Our bet makes a $10 profit.

And opponent B will:

FB 55% of the time
CB 45% of the time.

Our bet, again, shows a $10 profit.

In this circumstance, the bet will be profitable 100% of the time (and it always has an expectation of exactly $10) regardless of how often we're up against Opponent A and Opponent B. Moreover, we don't know whether that profit comes as a bluff or as a value-bet!. All we know is that we make a profit! So this bet might be thought of as a value-bluff, but what it really is is a bet that's unexplotiable.

However, unexploitability is not a pre-condition of a bet having value both as a bluff and as a value-bet. To get at scenarios where the explotability principle is violated, you need to introduce more than two opponent types. For example:

Opponent W (strong calling station), against whom your bet makes $10 as a value-bet.

Opponent X (semi calling station), against whom your opponent loses $8 as a value-bet.

Opponent Y (strongly weak-tight), against whom your bet makes $10 as a bluff, and,

Opponent Z (semi weak-tight), against whom your bet loses $8 as a bluff.

If we are up against an equal distribution of Opponents W, X, Y, and Z, then our bet will show a profit of $1, even though opponents X and Z exploit us. And we don't know whether that profit comes as a bluff or as a value-bet.

Anyway for you tl;dr people, Nate's Theorem is as follows.

Nate's Theorem.

1. If a bet is neither profitable as a bluff against any individual opponent, nor profitable as a value-bet against any individual opponents, then it cannot be profitable against any range of opponents.

2. In order for a bet to be profitable as a "value bluff", it must be profitable as a bluff against some individual opponents and be profitable as a value bet against other individual opponents.

2a. Provided that condition 2 is met, then there may be bets that are profitable even though we do not know whether that profit comes as a value-bet or as a bluff.

2b. If condition 2 is not met, then a bet may be profitable, but it is only profitable as a value-bet or as a bluff.

I'm not going to try and milk some "poker lesson" out of this because I'm not sure there is any; I'm really just trying to get you all to think about what you might and might mean when you use the term "value bluff". Besides that, the "Aejones Theorem" is not a logical epiphany but a strategic one, which is simply that making thinner value bets makes you tougher to play against.