The Mathematics of Poker

[Patrick Sileo]

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Is this the appropriate place for extensive content discussion? So far, I like it and expect it to spawn some lively discussion.

I did pick up one error on page 48. The unconditional probabilities p(A_has_the_nuts) and p(A_has_a_bluff) in the equation for <B,call> should be replaced by probabilities conditional on the event "A_bets".

p(A_has_the_nuts|A_bets) = 0.2/(0.2+x)

p(A_has_a_bluff|A_bets) = x/(0.2+x)

Notice that these will sum to 1. This change leaves the critical value x*=0.04 unaffected, but game value will now be seen to be a non-linear function of x. The primary conclusions don't depend on linearity and are unaffected.

A similar error occurs on page 56 in the expression for <A,call>.


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I'm trying to use this formula to solve a problem on river bluffing, and I don't get your comment on non-linearity.

It seems that if you convert to a formula where 0.2 is really 1-x. (In the book example, x=80% as a given value of bluffing, so 1-x = 0.2.)

So if you plug 1-x into everywhere you have 0.20 doesn't the denominator sum to 1 (as you state), and therefore revert to the original formula (which doesn't have a fraction)? And further, then isn't it linear?

Or are you saying that the solution is non-linear if player A doesn't bet 100% of the time? Presumably this would be done with the expectation of sometimes being able to check-raise?

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I was simply pointing out a minor mathematical misstep. The key result of the problem being analyzed depends only on the critical value, which is unchanged by the correction. Hence, the relegation of the matter to the "minor correction" category.