A Little Bit Of Poker Math

I've been a break-even/slightly losing NLHE player for the past couple of years.

Saturday, I decided to try PLO, starting at Ipoker 10PLO 6max.  A couple of hours later, I quit up 3 buyins.

Monday, I took at shot at 10PLO Rush on Full Til, and 3000+ hands later, I'm hotter than the Sun, averaging more than 30bb/100 winrate.  I'm a little shocked.  You know how they say that gettting to the long run, to know if you're really a winning player takes LOADS of hands?  Well, after 3 days of poker with a standard deviation of 82bb/100, I'm way more than 95% confident I'm a winning player in those Rush PLO game.  Was going to give the number, but this is more fun: =normdist(0,33.25,14.164,1)

Which is nice.

In Mathematics of Poker, Chen and Ankenman give the solution to the AKQ game.

The structure is: 2 players, X and Y each get one card from a deck containing 3 cards.  Betting is limit-betting, and the pot is P bets.  X checks, and cannot raise.

The solution given is:

Y bets all Aces, and 1/(P+1) of his Queens.  He always checks with a King.

Y calls all Aces, and (P-1)/(P+1) of his Kings.  He always folds a queen.

----------------

So, that's "by the book".  What happens if you add a Jack to the deck?

Well, betting and calling with Aces remains the same.

Not betting and calling frequencies with Kings remains the same.

Not calling with Jack, and bluffing is the same.

But what happens with a Queen?  Classic weak bluff-catcher.  What do you do with it?

I've been thinking about Schweig's question on betting into a nut/air range in pot limit with an SPR of 4.  This is my second comment, reposted:

Schweig and I spoke elsewhere, and that comment from above is clearly wrong.

If we assume that the action is on the river, where villain checks
to hero, hero has the option of checking down, so the EV should be the
difference between checking down and betting. So let's rerun the math:

Let x = p(call)

Let y = p(value raise)

=> p(raise) = 1.5y

Since we're value betting, we win more than 50%

Let w = p(win when called) - 0.5

=> EV(check) = pot x p(win when called)

=> EV(check) = 1 x (0.5 + w) = 0.5 + w

EV(called bet) = gain x p(win when called) - loss x p(lose when called)

=> EV(called bet) = 2 x (0.5 + w) - 1 x (0.5 - w)

=> EV(called bet) = 1 + 2w - 0.5 + w

=> EV(called bet) = 0.5 + 3w

Gain from betting when called = EV(called bet) - EV(check) = 0.5 + 3w - (0.5 + w) = 2w

That's consistent with a thin value bet...it's thin.

When Villain raises, hero loses 1 pot. In addition, since 0.5y are bluffs, he loses an additional 0.5y compared to checking.

=> EV(bet-fold) = -1 * 1.5y -0.5y = -2y

To make a profit from betting, the sum of these numbers must be positive.

=> 2w * x - 2y >= 0

=> x/y >= 1/w

To put that in context, if your thin value bet has a 10% edge (w =
5%), then villain needs to be flat-calling 20 times as often as he can
raise for value with the nuts.