House edge of UTH from 8X raise on pocket aces

A casino is doing a promotion on Ultimate Texas Hold’Em. It allows up to 8X raise on player’s pocket aces. How does this improve the expected value?

Nobody responded, so I investigated this myself. A pocket aces hand happens once every 221 hands, and if it happens, the average EV of this hand is +3.6. If we do 8X raise preflop, the player’s EV gain (from the original UTH of 4X raise) is 4X3.6=14.4 on this hand; therefore, the player’s EV gain percentage (divided by the Ante bet amount of 1 unit) of 4X3.6/221=6.5%. This number sounds too good to be true! Is this calculation correct?

Quote: aceside

Nobody responded, so I investigated this myself. A pocket aces hand happens once every 221 hands, and if it happens, the average EV of this hand is +3.6. If we do 8X raise preflop, the player’s EV gain (from the original UTH of 4X raise) is 4X3.6=14.4 on this hand; therefore, the player’s EV gain percentage (divided by the Ante bet amount of 1 unit) of 4X3.6/221=6.5%. This number sounds too good to be true! Is this calculation correct?
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Is that correct? Isn't the original +3.6 already based upon the 4X raise? If so, wouldn't the gain only be another 3.6?

Quote: aceside

Nobody responded, so I investigated this myself. A pocket aces hand happens once every 221 hands, and if it happens, the average EV of this hand is +3.6. If we do 8X raise preflop, the player’s EV gain (from the original UTH of 4X raise) is 4X3.6=14.4 on this hand; therefore, the player’s EV gain percentage (divided by the Ante bet amount of 1 unit) of 4X3.6/221=6.5%. This number sounds too good to be true! Is this calculation correct?
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I believe from the WoV calculator:

https://wizardofodds.com/games/ultimate-texas-hold-em/calculator/

3.601073 is the expected gain in units, which assumes that you would be making the 4x Raise, so the 8x raise would be roughly double that amount. I think it would be a little bit more than double the amount in EV because you aren't adding anything to the Blind Bet, obviously, so I would assume that Ante/Play has a better per unit expectation (even on wired Aces) than does the Blind Bet, though I could be wrong. I'm pretty sure I'm right, however, as a wired pair would make everything that pays out on the blind bet, except FH and Quads, less likely as opposed to more likely...or less likely than other hands, such as being suited for a flush.

If we just call it double, then you would have an expectation of roughly +7.202146 units, except only 3.601073 units of that is actual value added as the House Edge/EoR of the game already assumes that Raising 4x with Wired Aces is a possibility.

If you would normally expect to lose 0.02185 units, on average, relative to initial ante bet, but now you are expecting to win an additional 3.601073 units roughly 1/221 hands, then:

(3.601073 * 1/221) - (0.02185 * 220/221) = -0.00545668325

Which means that the game would still see the player at a disadvantage, overall. Another thing that would be too complicated for me to consider is the fact that the House Edge should actually go slightly up on the right side because we should no longer be counting dealt aces into that. The 3.601073 is good on the left side because that is value added as the House Edge (relative to ante only) assumes that you can get dealt aces anyway.

***Please recall that this isn't exactly right as I think the EV added will be slightly more than double as the Ante/Play certainly has a greater advantage in this situation than the Blind Bet does, but it should be close enough.

You are right. I investigated this a second time and found the problem. The player’s EV gain (besides the original UTH of 4X raise) is 3.6 on this hand; therefore, the player’s EV gain percentage (divided by the Ante bet amount of 1 unit) of 3.6/221=1.6%. This number sounds good now, because the original UTH house edge is 2.2%, so even with this nice promotion, the house still has an edge of +0.6%.

Quote: aceside

You are right. I investigated this a second time and found the problem. The player’s EV gain (besides the original UTH of 4X raise) is 3.6 on this hand; therefore, the player’s EV gain percentage (divided by the Ante bet amount of 1 unit) of 3.6/221=1.6%. This number sounds good now, because the original UTH house edge is 2.2%, so even with is promotion, the house still has an edge of +0.6%.
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Not a math person, forgive if this is an ignorant statement. That is just the house edge on the blind/ante right? This is not the element of risk based on all money bet?

Wizard on his page IIRC has the element of risk at 0.54% right? So that would mean on all money bet this would maybe be barely a player advantage if we use the element of risk instead of house edge on ante/blind bet?

For some strange reason, all these published house edge numbers (like 2.2%) are referenced to the ante bet amount only. However, I believe, they should be referenced to the sum of ante and blind amount, because this sum is the original bet amount on this game.

The element of risk is not an accurate measure for this game because different people might use very different strategies to play the game to obtain different EORs.

Mission, taking your 3.60 number without checking it, every 221 hands you will gain 3.60 units, so it adds 3.60/221 = 1.63% to the player side, reducing the house edge to about 0.55%. Nothing more complicated than that.

Bah, you'll just get beat by the dealer.

Using the calculator the Woo UTH calculator , and adding 4 to the winning hands and subtracting 4 from the losing hands:

Prize	Combinations	Probability	Return
509	854,100	0.000040718	0.020725716
508	56,700	0.000002703	0.001373187
59	794,700	0.000037887	0.002235313
58	56,700	0.000002703	0.000156781
19	175,977,840	0.008389596	0.159402315
12	1,766,425,440	0.084212847	1.010554166
10.5	329,888,200	0.015727142	0.165134996
10	111,376,440	0.005309778	0.053097781
9.5	51,143,400	0.002438219	0.023163077
9	11,929,203,960	0.568714766	5.118432891
8	3,449,306,700	0.164442796	1.315542367
0	114,023,120	0.005435956	0.000000000
-10	3,046,616,700	0.145244889	-1.452448888
Totals	20,975,724,000	1.000000000	6.417369702

The change in EV is: 2.816297064
2.816297064/221 = 0.012743426

Quote: miplet

Using the calculator the Woo UTH calculator , and adding 4 to the winning hands and subtracting 4 from the losing hands:

Prize	Combinations	Probability	Return
509	854,100	0.000040718	0.020725716
508	56,700	0.000002703	0.001373187
59	794,700	0.000037887	0.002235313
58	56,700	0.000002703	0.000156781
19	175,977,840	0.008389596	0.159402315
12	1,766,425,440	0.084212847	1.010554166
10.5	329,888,200	0.015727142	0.165134996
10	111,376,440	0.005309778	0.053097781
9.5	51,143,400	0.002438219	0.023163077
9	11,929,203,960	0.568714766	5.118432891
8	3,449,306,700	0.164442796	1.315542367
0	114,023,120	0.005435956	0.000000000
-10	3,046,616,700	0.145244889	-1.452448888
Totals	20,975,724,000	1.000000000	6.417369702

The change in EV is: 2.816297064
2.816297064/221 = 0.012743426
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I feel very comfortable with this calculation by Miplet. The calculated change in EV is lower than Mission's assumed value because Miplet's calculation correctly takes into account that the payouts on the Ante and Blind bets are not changed by the size of the Play bet.

Quote: miplet

The change in EV is: 2.816297064
2.816297064/221 = 0.012743426
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Your result shows a change of house edge of about 1.27% in favor of the player. This number is less than my estimate number of 1.6%. I think your number is good. Thank you!

Quote: TheCapitalShip

Wizard on his page IIRC has the element of risk at 0.54% right? So that would mean on all money bet this would maybe be barely a player advantage if we use the element of risk instead of house edge on ante/blind bet?
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No, because the EV has been determined to be a positive number, for the house. It won't matter what number you divide it by, it can't turn negative

Promotions like the 8X raise on pocket aces in Ultimate Texas Hold'em improve the expected value by providing players with a significant advantage on strong starting hands. With aces, the increased raise allows for more aggressive betting, increasing the potential payout and making the game more favorable for players. Have you tried this promotion, and what were your results?