This is a great book. I'm nearly at the end now.
It has taken me a long time to get this far because I've constantly being putting it down and going away and working on ranges. In my case it's definitely made me start working on all the things which I previously used to ignore and hope I could get away with. Lots of "a-ha" moments. You really find yourself applying it at the table. For example how textures affect villain check-back ranges.
This is definitely applicable to tournament play, but you need to engage your brain and work out how to apply it. For example in a tournament with 40BB stacks, usually the 4-bet is all-in, so much of what Carroters says about 4-bets actually applies more to our 3-bets (being the last bet before villain might go all-in or fold and should rarely flat) but also some of the things he writes about 3-bets also applies to our 3-bets (in both cases being the first re-raise after the open). The last chapter, which I haven't read yet deals specifically with varying stack sizes, but what I want to say is that you can use the approach Carroters uses to working out how to play well in cash games to work out for yourself how to play well in tournaments.
As the twoplustwo forum is a pretty tough peer-review, and people like to find faults anwyay, - 480 pages in I've found a maths example for calculating balanced river ranges which could be made a lot simpler.
How then do we create balance with respect to our bluff combos? We follow the balancing principle below.
Recall that RE describes the equity % of Villain's hand vs. Hero's range that will cause Villain to break even on a call in an end of action spot. It is his bare minimum equity for calling a bet from Hero.
In Hand 130 Hero's bet-size is a shove as the effective stack is small enough for shoving to be feasible and there is no reason to suppose any other bet-size will be better with Hero's range. Generally speaking, polarised ranges on the river want to bet big as this will generate a higher RE target and thus allow for more bluffs in Hero's range.
Hero bets 59BB into a pot of 82.5BB. What is Villain's RE?
RE = ATC / (ATC + TP)
RE = 59 / (59 + 141.5)
RE = 29.4%
Following the balancing principle, this means that 29.45% of Hero's river betting range should be a bluff. Since his betting range contains 16 value combos we can solve for his required bluffs (X).
X / (16 + X) = 0.294 (1)
X = 0.294 (16 + X) (2)
X = 0.294X + 4.704 (3)
(X - 0.294X) = 4.704
0.706X = 4.704
X = 4.704 / 0.706
X = 6.662
(1) We know that X will be 29.4% of Hero's total betting range. Hero's total betting range will be X + his
value combos (16 + X).
(2) Simplification: multiply both sides of the equation by (16 + X).
(3) Simplification: multiply both 16 and X by 0.294 to remove brackets.
For the algebraically shy, there are readily available algebra calculators online for such a job. Let's round this fractional answer up to the nearest whole number and conclude that Hero needs 7 bluff combos to create the 7:16 or 0.44:1 ratio of bluffs to value that satisfies the balancing principle.
This is all correct but there is a much quicker way to get this result if you use old school British-style odds instead of percentages. When we put money on a horse at 3 to 1 (US odds +300, decimal odds 4.00) the odds are fair (or we are indifferent to backing it) if it loses 3 times for every 1 time it wins.
Same in this case (a sportsbet being more equivalent to a "call" in poker), where villain is being asked to risk 59 BBs to win 141.5 BBs; for villain to be indifferent to making the call he needs to win 59 times for every 141.5 times he loses.
16 x 59 / 141.5 = 6.671
(the minor difference with one in the book is due to rounding the villain's required equity midway though the calculation).
So 7 bluff combos for our 16 value combos.