Sklansky problem, still early days

The following relates to a problem Sklansky posted where 2 players post an ante and receive a number between 0 and 1.


So far I have only done work on A betting or checking, and B calling or folding (not raising).


A while ago I posted the results of what hands A should bet with to get the best EV.

The results were that A bets (the full pot) for the top 2/9 of hands and the bottom 1/9 of hands. B was calling with hands anywhere between 1/9 and 7/9.


Well I have now done the formula backwards, ie A plays a game according to where B will call, and then we optimise for B. This is the first time that I have done the math this way, mainly because I thought if A has to play first then that is invalid. However I think now that it is valid for a player in front to anticipate what his opponent will do and play according to that for best EV. Don't forget that player B is also anticipating that player A will do this, and then play the best knowing this is going on.


The answer is B calls at 5/9.


A plays according to the formula...

highest bluff = 2*b - 1, or 0 if b

Don't use the greater than and less than signs, as I think that they are interpreted as HTML tags and deleted.

Ahhhhh they got me again.


I will post it again in a minute.


Sklansky, fix this #@$# noticeboard.

The following relates to a problem Sklansky posted where 2 players post an ante and receive a number between 0 and 1.


So far I have only done work on A betting or checking, and B calling or folding (not raising).


A while ago I posted the results of what hands A should bet with to get the best EV.

The results were that A bets (the full pot) for the top 2/9 of hands and the bottom 1/9 of hands. B was calling with hands anywhere between 1/9 and 7/9.


Well I have now done the formula backwards, ie A plays a game according to where B will call, and then we optimise for B. This is the first time that I have done the math this way, mainly because I thought if A has to play first then that is invalid. However I think now that it is valid for a player in front to anticipate what his opponent will do and play according to that for best EV. Don't forget that player B is also anticipating that player A will do this, and then play the best knowing this is going on.


The answer is B calls at 5/9.


A plays according to the formula...

highest bluff = 2*b - 1, or 0 if b lte 0.5

lowest good hand = (b+1)/2


Notice that as if by magic, the values come out to A bluffing 1/9 of hands and betting the high 2/9 of hands; the same results as from doing it the other way round.


However if B calls at a value higher or lower than 5/9, then A can adjust his play according to the formulae above, and get a higher EV.

eg if B calls at 0.5, A will no longer bluff but bet at 0.75 or higher, giving him an EV of 1.1250, up from 1.11111 at optimum.


Above is just the results of all the algebra, if anyone really wants the guts I am happy to put it up.


I am thankful for the 2+2 ers who forced me to check my work that I did a few years ago, because I was wrong, and I have benefited a lot from it. I am now happy that bluffing is intrinsic to playing poker, and I'm solving problems better, both real life and with algebra.


However, I am now going to divert my attention to no limit betting, and leave the Sklansky problem, because I think I will benefit more from it. In time though, I want to do everything, but with work, and this algebra is very slow to do; more results are going to take me a long time.

It's interesting to generalize this problem for different pot-sizes and hand values. Assume that the bettor has a hand B in [b,1] and the caller has a hand C in [c,1].


Bettor bluff bets for b0>B, checks for b1>B>b0 and value bets for B>b1. Caller folds for c0>C and calls for C>c0.


b0=c0-(1-c0)/pot

b1=(c0+1)/2

c0=(2*b*pot^2+(3+2*b)*pot+2)/(2*pot^2+5*pot+2)


With pot=1 and b=c=0, you get the above Sklansky problem with b0=1/9, b1=7/9 and c0=5/9.


Check out the link below for the gory details.


cu


Ignatius

That link was was fantastic, like my fairy godmother just saved me 6 months work. I think you've solved no limit betting for me. /images/smile.gif.


I haven't gone over your text yet Ignatius, but on my first skim, it looks brilliant.


I see you just work around the problem of the less than sign too, leaving the young players to get stuck. I must say I do that at work too with Oracle bugs, mostly the stuff we develop on now is old and unsupported by Oracle, so they won't fix their bugs, and I just know where they all are and work around them now seamlessly. Unless you knew, it wouldn't even look like I did it. The new developers all swear and curse though, it probably isn't funny, but I always chuckle to myself when they get stuck on one of them.


You know everything I do in life I always get back somewhere somehow, and this is just one of those things I suppose.

Glad you found it useful. You might also want to check out the article below by Bruno Wolff - he solves the same problem for No Limit (EV, betsizes and all).


In NL, you would always bluff with the worst 1/7 and value bet with the best 3/7 of hands. The calling threshold, of course, depends on the betsize.


cu


Ignatius