Mathematics of Poker: Fundamental Error On Pg. 112?

Posted by AF3

Posted by AF3 posted in Mid Stakes

Posted 10 years, 11 months ago

Mathematics of Poker: Fundamental Error On Pg. 112?

In Mathematics of Poker, pages 111-122 introduces the idea of half-street games. The first example, given on page 11 as The Clairvoyance Game, has the rules as follows:

One half-street

Pot Size of P bets

Limit Betting (The only bet size possible is 1 unit)

Y is clairvoyant

Y's range is drawn randomly from a distribution of {1/2 Nuts, 1/2 Air}

The optimal bluffing frequency (b) for such a game would be calculated as the frequency which makes X indifferent to calling when Y bets, therefore satisfying the equation:

<X, Call> = 0

Solving for b gives:

<X, Call> = 0 ==> (b)(P+1) + (1-b)(-1) = 0 ==> bP + b + (-1 + b) = 0 ==> bP + b + b -1 = 0

==> bP + 2b = 1 ==> b (P+2) = 1 ==> b = 1/(P+2)

This is the same answer the Lefort presents in his video on advanced theoretical concepts. He uses variable bet sizing X and gets the optimal bluffing frequency b to be: b = X/(2x+P)

However, on page 112 of Mathematics of Poker, the authors claim that:

"Y, likewise, must bluff often enough to make X indifferent to calling or folding. When calling, X loses 1 bet by calling a value bet, and gains P+1 bets by calling a bluff: If b is the ratio of bluffs to bets, then we have:

1 = (P+1)b ==> b = 1/(P+1)

It seems that they are making an error in setting up the indifference equation. They have forgotten to weight the amount won from calling a bluff to the frequency that the opponent is bluffing (1-b). The equation they start with above is equivalent to:

(P+1)b - 1 = 0

The correct answer is b = 1 /(P+2) , right?

8 Comments

Loading 8 Comments...

BigFiszh 10 years, 11 months ago

They mean a different thing: b in your equation is the percentage of bluffs related to our entire range.

Example: Say, we have $100 in the pot and $75 left, we can only shove or fold. Optimal ratio between valuebets and bluffs (according to x/(2x+P)) is:

bluff% = 75 / (2*75 + 100) = 0.3 (30%)

That means, 30% of our overall-shoving-range (!) should be bluffs, or in other words, we should bet with a ratio of 70:30 between valuebets and bluffs.

Agree?

The "b" in MOP now means the percentage of bluffs related to our valuebets! You can see this from the following equation:

1 = (P+1)b

This means, everytime we bet the nuts (left part of the equation), the player loses one bet. This has to be equal to the profit he makes when Villain bluffs. The "1" is not weighted, which means, we bet our nuts at 100%. So, the "b" on the right side is the ratio we bluff - compared with the 100%.

Take our example again - now with the MOP-formula:

b = 75 / (100 + 75) = 0.42 (42%)

Now, 30 combos out of 70 is "exactly" the same 42%. So, when we bet our 70 combos at 100%, we should bet 42% of air (42% of 70 = 30 combos).

OK?

AF3 10 years, 11 months ago

It seems like this book jumps between implicit assumptions and then does all kinds of calculations it doesn't bother to explain. I don't mean in "math is hard type" way, I mean it's terribly explained.

Now, 30 combos out of 70 is "exactly" the same 42%. So, when we bet our
70 combos at 100%, we should bet 42% of air (42% of 70 = 30 combos).

This is just saying that we first assume we bet our value (since this option dominates checking), then we figure out how much we must bluff to give Villain a break-even call. This is X/(2X+P). It represents how many hands out of each 100 that we bet are bluffs.

Why wouldn't we just calculate the ratio of bluffs: value bets by taking the number of bluffs and dividing it by the number of value bets? I guess this formula is more powerful because our bet-sizing, which determines how often we can bluff, is already built in to the b on page 112?

AF3 10 years, 11 months ago

I still don't understand what the equation 1 = (P+1)b represents in the sense that the authors are going back and forth because formally derived calculations from the definition and EV and verbal arguments they are somehow translating into numbers without bothering to explain them.

I'm stuck because I don't see how the equation is derived. In other, words, you can't say:

<X,Call> = (P+1)b - 1 = 0

That would be false, since the times X loses 1 bet are weighted at 100% and so we would have all kinds of contradictions if b were not equal to zero (which it certainly is not).

The starting point for this equation is not the definition of EV, but the condition that:

Amt Lost On Call = Amt Won On Call

Right?

I don't see why this matters because it's not how I understood defending frequencies of 1-a but I'll keep reading with this in context.