Some UX Math Questions

I've been playing Ultimate X the last several years. I play 9-6 DDB, 7-5 BP, 8-6 Jacks and 15-9 DW, some 10-play, some 5-play. I've even fiddled with the 99% JW program in the Midwest and the 99% DB program elsewhere.

Here are my questions:

1) I note that Wizard has a simple strategy which costs anywhere bewteen .02 to .08 to implement. I think you can improve greatly on that by using two different strategies, one for 1.9x or less and 2x or greater (the Jacks I would make it <=2x and 2.1+) But how much?
2) I noticed some players were betting 5 coins on deals that the average was 4x or more. What might the reason for doing that?
3) What is the correct way to calculate the covar on the game?

Franklly, I'm much more interested in (3). One place I play at, after over 800K hands I'm down the game and +.7%. The game in question 2p pays 2:1. Royals are on time, maybe even a little over, but avg multiplier sucks. Maybe 1.5x average. I'd like to be able to construct an equation to determine range of probabliltiies. I can probably do something rough using the royals.

(1) and (2) are probably more qualitiative than quantitative in practice but I threw it in for discussion purposes.

IME the covar shouldn't be that high, the 10-play fluctuations are a lot smoother than the 5-play ones. But if I can pin it down to an actual number, that would be great. I would guess something along 20% of the variance.

Quote: ET

Here are my questions:
3) What is the correct way to calculate the covar on the game?

Franklly, I'm much more interested in (3).

Variance: For each stating set of multipliers (e.g., in 5 Line one starting set is [1, 3, 12, 2, 1]) compute the probability of each payout (e.g., the probability of RSF or Flush) and multiply this by the probability of the starting state to get the state-payout probability p. From there, just compute and sum (p*Pay) and (p* Pay * Pay), across all the starting states and payouts to get the usual two terms for the variance calculation: i.e., Variance = sum(p*Pay*Pay) – (sum/Pay/n)^2.

See

https://wizardofodds.com/pdf/ultimatex.pdf

or for an updated version that discusses variance (amongst other minor additions and errata):

http://playperfectllc.com/uploads/3/4/9/0/34902374/ultimatex.pdf

In a typical Deuces Wild game there are 343 starting states in 3-Line, 16,807 in 5-Line and 282,475,249 in 10-Line games. One can collapsed the states to those having the same multipliers but in different orders (i.e., state [1, 3, 12, 2, 1] would be lumped with state [1, 1, 2, 3, 12] in the calculations). Then there are 84 starting states in 3-Line, 462 in 5-Line and 8,008 in 10-Line games.

As discussed in the second paper above, the Variance calculation is substantially harder than the expected value calculation. The Wiz first noted that for calculating the expected value, one could lump states together that had the same sum of multipliers. This doesn’t work for the variance. So one can’t simplify the overall computational effort beyond this point.

All of this assumes one has computed the probabilities of the starting states and probabilities of outcomes under optimal play.