EV Calculation for game with different-paying royals

A progressive VP game has different amounts for the Royal Flush of each different suit. Some are wildly high and some sit near the reset amount.

Certainly, the player can at least be as well off playing this game as if he was playing a game with a single Royal Flush payout equal to the sum of the Royal Flush payouts in the differently-paying suited game divided by four, by playing the appropriate strategy for the latter game.

Adjusting his strategy by using the optimal plays for each combination with suits taken into consideration for live-to-the-royal holds yields an even higher ER. You pretend each suit is its own game, then score the live-to-the-royal holds of each suit's EV along with all the other holds' EV for the global game strategy.

Does the ER for this game equal (ER(hearts game) + ER(clubs game) + ER(diamonds game) + ER(spades game))/4? ?

I'm pretty sure it can't ever be lower than this, but I suspect that it might be actually slightly higher. I am eager to hear others' thoughts, or if this is already a solved problem.

Doing it that way would be a good estimate. I think it would overstate the true EV slightly.

I can't think of any way in which that method would overstate the true EV. How do you figure?

As far as a way in which it would understate the true EV:

The formula I provided is equivalent to thinking of the game as one in which at the beginning of each deal we have 1/4 probability of entering a "sub-game", in which ALL royal-drawing holds have EVs of one of the four suits, chosen at random.

However, in a 9/6 JoB with per-coin royals of :
Hearts: 10,000
Clubs: 3,000
Spades: 2,000
Diamonds: 1,000

Consider the following hands, which are all of the significant forms of a single pattern and all equally likely:
Ah Kh Qc Jc 5d
Ah Kh Qs Js 5c
Ah Kh Qd Jd 5c

Ac Kc Qd Jd 5h
Ac Kc Qs Js 5h
Ac Kc Qh Jh 5s

Ad Kd Qs Js 5h
Ad Kd Qh Jh 5s
Ad Kd Qc Jc 5h

As Ks Qh Jh 5d
As Ks Qd Jd 5h
As Ks Qc Jc 5h

We would hold:
Ah Kh
Ah Kh
Ah Kh

Ac Kc
Ac Kc
Ac Kc

Qs Js
Qh Jh
Qc Jc

As Ks
As Ks
As Ks

, not holding any of the Diamond Royal Flush draws.

This shows that because we can be dealt multiple royal draws of different suits in the same hand, we are actually more likely than 3/4 to enter any of the "sub-games" of Hearts, Clubs, or Spades, and thus global ER>(ER(hearts game) + ER(clubs game) + ER(diamonds game) + ER(spades game))/4.

And it's not just the rare scenario of double RF2 draws, with large jackpot discrepancies even frequent hand patterns like Ah Kc 4s 3s 2d would have us holding Ah and thus being in the Heart "sub-game" (in games where AK unsuited is no better than single high card and K>A with standard payoffs).

Using the figures you stated, if you decided to employ a strategy that used a royal flush payout of 4000 per coin for all suits, it would yield an EV of 109.4248%.

If you estimated a suit-specific strategy by averaging the EVs using the 4 listed amounts, you would get an estimate of 109.9036%. This would be the average of 100.0670%, 103.0897%, 106.2351%, and 130.2224%.

If you would actually employ a suit-specific strategy that maximized EV on each hold, the EV would be 110.0172%. This strategy would yield about 1.5% fewer royals overall than the 4000-per-coin strategy, but about 14.5% MORE of the very valuable heart royals.

Quote: drrock

Using the figures you stated, if you decided to employ a strategy that used a royal flush payout of 4000 per coin for all suits, it would yield an EV of 109.4248%.

If you estimated a suit-specific strategy by averaging the EVs using the 4 listed amounts, you would get an estimate of 109.9036%. This would be the average of 100.0670%, 103.0897%, 106.2351%, and 130.2224%.

If you would actually employ a suit-specific strategy that maximized EV on each hold, the EV would be 110.0172%. This strategy would yield about 1.5% fewer royals overall than the 4000-per-coin strategy, but about 14.5% MORE of the very valuable heart royals.

How did you get the last figure? Very curious, and thank you for the reply.

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