June 17th, 2011 at 12:24:35 AM
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On his Pai Gow page the wizard gives the probability of winning and of losing given different strategies. Since the 5% commission is applied on net winnings the house edge should go down as more players get in since the losing hands will cancel the commission on the winning ones. Here are some stats from the wizard's optimal strategy.

Banking/Win/Lose/Tie

No 29.33% 29.64% 41.03%

Yes 30.24% 28.65% 41.11%

Using these formulas in Excel I tried to come up with a house edge against various number of opponents, C4 being the probability to win and D4 the probability to lose:

1 opponent =0.95*C4-D4

2 opponents =2*0.95*(C4^2)-2*D4^2

3 opponents =3*0.95*(C4^3+C4*C4*D4)-3*(D4^3+D4*D4*C4)

4 opponents =0.95*(4*C4^4+2*4*C4*C4*C4*D4)-(4*D4^4+2*4*D4*D4*D4*C4)

Here are the house edges I came up with:

1 opponent 2 opponents 3 opponents 4 opponents

1.776500% 1.776500% 1.776500% 1.776500% (not banking)

-0.078000% -0.9582% -0.8465% -0.814765942444% (banking)

0.849250% 0.864919% 1.120759% 1.258247% (average, assuming banking once per rotation)

When banking the player's edge is the highest against 2 opponents and then goes down as more opponents play. Is there something wrong with my math? If not, is there an explanation for this? Do the odds of winning/losing change when there are more players? I understand that when there is an odd number of opponents you will have less situations where you push, meaning you do not get as much discount on the commission but why doesn't the player's edge increase significantly when there are 4 opponents?

Edit: I just realized I left out the probability of a tie in my HE calculation for 2-4 opponents. Not too sure how to include it without having ridiculously long and complicated formulas.

Banking/Win/Lose/Tie

No 29.33% 29.64% 41.03%

Yes 30.24% 28.65% 41.11%

Using these formulas in Excel I tried to come up with a house edge against various number of opponents, C4 being the probability to win and D4 the probability to lose:

1 opponent =0.95*C4-D4

2 opponents =2*0.95*(C4^2)-2*D4^2

3 opponents =3*0.95*(C4^3+C4*C4*D4)-3*(D4^3+D4*D4*C4)

4 opponents =0.95*(4*C4^4+2*4*C4*C4*C4*D4)-(4*D4^4+2*4*D4*D4*D4*C4)

Here are the house edges I came up with:

1 opponent 2 opponents 3 opponents 4 opponents

1.776500% 1.776500% 1.776500% 1.776500% (not banking)

-0.078000% -0.9582% -0.8465% -0.814765942444% (banking)

0.849250% 0.864919% 1.120759% 1.258247% (average, assuming banking once per rotation)

When banking the player's edge is the highest against 2 opponents and then goes down as more opponents play. Is there something wrong with my math? If not, is there an explanation for this? Do the odds of winning/losing change when there are more players? I understand that when there is an odd number of opponents you will have less situations where you push, meaning you do not get as much discount on the commission but why doesn't the player's edge increase significantly when there are 4 opponents?

Edit: I just realized I left out the probability of a tie in my HE calculation for 2-4 opponents. Not too sure how to include it without having ridiculously long and complicated formulas.

June 28th, 2011 at 1:23:44 AM
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I never managed to find an easy way to calculate all that so I filled an excel spreadsheet with random numbers and used the following to emulate hands:

One opponent =IF(A5<0.2865,-1,IF(A5<(0.2865+0.3024),1,0))

Two opponents =IF(B2<0.2865,-1,IF(B2<(0.2865+0.3024),1,0))+IF(C2<0.2865,-1,IF(C2<(0.2865+0.3024),1,0))

Three opponents IF(D2<0.2865,-1,IF(D2<(0.2865+0.3024),1,0))+IF(E2<0.2865,-1,IF(E2<(0.2865+0.3024),1,0))+IF(F2<0.2865,-1,IF(F2<(0.2865+0.3024),1,0))

I then removed the commission by using IF(G3>0,G3*0.95,G3)

I played each game 250,000 times before taking the average. My results seem quite far from the Wizard`s player edge calculations so either I made a mistake somewhere or I simply didn't use enough significant numbers in my calculations or my sample size is too small. Here are the results for 20 times 250,000 hands worth of simulation:

One opponent Two opponents Three opponents

0.0002446 0.006448601 0.006612534

0.0014134 0.0057513 0.00784

0.0014134 0.0057513 0.00784

-0.0001232 0.0059262 0.006380866

0.0011248 0.006203201 0.0060448

-7.72E-05 0.005016 0.0073838

0.0013098 0.005434301 0.0061244

0.001468 0.0054809 0.007063334

-0.0004784 0.004409501 0.0066222

-0.0004468 0.0061776 0.0084768

0.0014862 0.004679801 0.006669534

0.0012614 0.0054647 0.0069032

-0.001727 0.004408601 0.006194666

-0.0003216 0.0054224 0.0052278

0.000193 0.0048698 0.007208466

-0.001716 0.0046646 0.004891266

-0.0005262 0.0034484 0.007232

0.0028244 0.0057564 0.008266066

-0.00055 0.0031932 0.0061818

0.0026202 0.0036776 0.007856534

Averages:

0.00046964 0.00510922 0.006851003

Standard deviations:

0.00126739 0.000929596 0.000949033

My model shows the house edge and the variance gets lower as we bank against more players however I am still unsure by how much whether my calculations were accurate. I was very surprised to see 9 trials out of 20 ended with a loss after 250,000 hands despite the positive player edge!

One opponent =IF(A5<0.2865,-1,IF(A5<(0.2865+0.3024),1,0))

Two opponents =IF(B2<0.2865,-1,IF(B2<(0.2865+0.3024),1,0))+IF(C2<0.2865,-1,IF(C2<(0.2865+0.3024),1,0))

Three opponents IF(D2<0.2865,-1,IF(D2<(0.2865+0.3024),1,0))+IF(E2<0.2865,-1,IF(E2<(0.2865+0.3024),1,0))+IF(F2<0.2865,-1,IF(F2<(0.2865+0.3024),1,0))

I then removed the commission by using IF(G3>0,G3*0.95,G3)

I played each game 250,000 times before taking the average. My results seem quite far from the Wizard`s player edge calculations so either I made a mistake somewhere or I simply didn't use enough significant numbers in my calculations or my sample size is too small. Here are the results for 20 times 250,000 hands worth of simulation:

One opponent Two opponents Three opponents

0.0002446 0.006448601 0.006612534

0.0014134 0.0057513 0.00784

0.0014134 0.0057513 0.00784

-0.0001232 0.0059262 0.006380866

0.0011248 0.006203201 0.0060448

-7.72E-05 0.005016 0.0073838

0.0013098 0.005434301 0.0061244

0.001468 0.0054809 0.007063334

-0.0004784 0.004409501 0.0066222

-0.0004468 0.0061776 0.0084768

0.0014862 0.004679801 0.006669534

0.0012614 0.0054647 0.0069032

-0.001727 0.004408601 0.006194666

-0.0003216 0.0054224 0.0052278

0.000193 0.0048698 0.007208466

-0.001716 0.0046646 0.004891266

-0.0005262 0.0034484 0.007232

0.0028244 0.0057564 0.008266066

-0.00055 0.0031932 0.0061818

0.0026202 0.0036776 0.007856534

Averages:

0.00046964 0.00510922 0.006851003

Standard deviations:

0.00126739 0.000929596 0.000949033

My model shows the house edge and the variance gets lower as we bank against more players however I am still unsure by how much whether my calculations were accurate. I was very surprised to see 9 trials out of 20 ended with a loss after 250,000 hands despite the positive player edge!