Go for quads IIb: 2P with kickers

[Minn. Fatz]

Deciding whether to break two pair to go for quads with a kicker involves a combination of the calculations in Parts I and IIa.

Break 2P to draw three to one pair with a possible quad+kicker payoff only if the sum of the following is less than the weighted sum of the payoffs for quads with and without a kicker given in Part IIa; recall this ranges from 1/4 to 1/3 of the payoff for 4K with kicker plus 3/4 to 2/3 respectively of the payoff for 4K without a kicker depending on the number of kicker "penalty cards:"
27 times the payoff for a FH
the approximate value for -(C/45) listed in Part I; recall this depends on the payoffs for 3K, 2P and the pair that will be held
Note that it is never better to hold one pair with a kicker from an initial hand such as 2288A. When we perform the calculations to compare drawing three to 22 to drawing two to 22A, we find that in 16215 trials the former gives us:

     6 fewer 4K+k, 36 more 4K, 34 more FH, 592 more 3K, 176 fewer 2P and 480 fewer unimproved pairs

A brief inspection is enough to show that, for 22A to have a higher expected value than 22, quads with a kicker must pay off at least 6 times more than quads without. If anyone finds a game like that, be sure to post its location on this forum...

Also note that the calculations in Part IIa and here assume that kickers include three available ranks of cards, or 12 cards in all.

Part III will cover a notable penalty card situation in DB...don't touch that dial...

[BobDancer]

[I very much appreciate your efforts in creating and posting these three posts. I'm too busy right now, but I have saved them and will review them when I get the chance.Don't be too surprised if I post a rebuttal, correction, or ask a question a couple of months down the road. We'll see. But a quick glance tells me you've put together some rules I hadn't previously come up with. They may well be useful!Bob

[Vman96]

Like Bob, I keep meaning to look at this. But to lump in all poker games, the rules obviously are a bit messy, so I have been putting it off for now, sorry. :(

[Minn. Fatz]

As always corrections, comments, etc. are more than welcome. I've rounded the numbers here and there for simplicity's sake and haven't done a rigorous check, but I believe this should hold up fairly well.