Uth: EoR if Flush pays 2:1? and a Progressive question

Quote: aceside

1). How do you get the HE of 0.9%? Flush occurs 3.03% of the hands, so the player EV gain is 3.03%x0.5=1.515%; therefore, the new HE is 2.185%-1.515%=0.67%.

2). Usually, $1 pays flush $40, full house $50, quads $300.
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Quote: aceside

I’d like to get a third opinion on this 0.9% HE number.

The payout numbers of yours are ridiculous high. Of course.
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Quote: aceside

I see the problem with my calculation. The 3.03% is the total flush hands, while the 2.572% is the player winning flush hands. Thank you.
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Quote: 100xOdds

3) What if Flush pays 3:2 but Royal only pays 100x?
What's the HE?

Quote: aceside

Quote: 100xOdds

3) What if Flush pays 3:2 but Royal only pays 100x?
What's the HE?

A Royal occurs 0.0032% of the time and normally pays 500:1. If you change the payout to 100:1, the change of house edge is roughly 0.0032%x400=1.28%. Here I assume a player Royal always wins, but this is not the case. I’ve encountered royals within the community five cards several times. The probability of a Royal on the five community cards is 0.000154%. This is as rare as a lifetime event.
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Quote: gordonm888

Quote: aceside

I’d like to get a third opinion on this 0.9% HE number.

The payout numbers of yours are ridiculous high. Of course.
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A Flush is the outcome on 2.5715% of hands (excluding straight flush and royal flush which don't have a change in payout.)
With an increased payout of 0.5 on a flush hand, the incremental player EV is 1.2857%.

The standard UTH EV is 2.185% with perfect strategy. Thus, the new value of house edge with this payout change on flushes is 2.185% - 1.2857% = 0.899222%

Source: UTH Math on increased payout on flushes See in particular the post by charliepatrick
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Quote: aceside

This is perfectly correct.
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Quote: aceside

Good question. But I believe we are not supposed to use standard deviation here. It’s all about variance. If you bet $0.50 on ante, all your payouts are halved, so your variance is quartered. I’d like to hear what other people may say about this.
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Quote: 100xOdds

1) Normally flush pays 3:2 for the Blind bet.
If it paid 2:1, the HE is 0.9%.
(Edit: https://wizardofvegas.com/forum/questions-and-answers/gambling/36944-math-of-ultimate-texas-holdem/#post841494 )
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Quote: aceside

Your conclusion sentence is very interesting. However, it can be more convincing if you could list one hand example of the strategic change that is caused by blind payout.

I know another strategic change from UTH to Heads-up Hold’em. If there is a quads within the five community cards, player should always raise 1x in Heads-up Hold’em, but not always in UTH.
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Quote: Mental

Quote: 100xOdds

1) Normally flush pays 3:2 for the Blind bet.
If it paid 2:1, the HE is 0.9%.
(Edit: https://wizardofvegas.com/forum/questions-and-answers/gambling/36944-math-of-ultimate-texas-holdem/#post841494 )
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I have been away and just noticed this thread. I calculated the HE for both values of flush payout. My program calculates the basic-strategy HE for known dealer hole cards 2 thru A as a side effect. I included these because there was some interest in these BS dealer hole card calculations recently.

The overall HEs are calculated two ways which must give the same result. The average of the HE for all 13 dealer HC must be the same as the overall HE. So, I present the sum of the 13 HE divided by 13. But, I also calculate the HE for any two specific dealer HC. I don't show the results here, but the program sums them and divides the sum by (52 choose 2) =1326. This is mostly just a sanity check for my new code.

If there is any interest, I can provide these 1326 HE in matrix form in another post.

Flush pays 1.5 HE: -0.0218497138

0.4972237334
0.4080658742
0.3181753448
0.2294870515
0.1711200214
0.1104881192
0.0463495828
-0.0250372845
-0.1224322786
-0.1979561768
-0.3219571567
-0.5332806646
-0.8642924452

-0.2840462789 / 13 = TOT EV: -0.0218497138
-28.9727204494 / 1326 = TOT EV: -0.0218497138

Flush pays 2.0 HE: -0.0089922169

0.5106884901
0.4214458733
0.3314667514
0.2426859398
0.1842243175
0.1234935392
0.0592510662
-0.0122455450
-0.1097568588
-0.1854089761
-0.3095466118
-0.5210158687
-0.8521809363

-0.1168988191 / 13 = TOT EV: -0.0089922169
-11.9236795466 / 1326 = TOT EV: -0.0089922169

I note that charliepatrick's result of -0.899222% is correct to six significant digits in spite of the fact that he did not do the calculation from scratch. He just adjusted the payoffs using the same frequencies from Mike's spreadsheet.This leads to the unsurprising conclusion that the half-unit change in pay table barely affects betting strategy, so the hand frequencies barely change.
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Quote: gordonm888

Quote: Mental

I note that charliepatrick's result of -0.899222% is correct to six significant digits in spite of the fact that he did not do the calculation from scratch. He just adjusted the payoffs using the same frequencies from Mike's spreadsheet.This leads to the unsurprising conclusion that the half-unit change in pay table barely affects betting strategy, so the hand frequencies barely change.
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**bolding added**

I am surprised that the half-unit increase in payout for a flush does not change the strategy to Raise 4x for (T-9 suited).

For the conventional payout table a Td-9d has the following expected values for the pre-flop decision
Raise 4x EV = 0.185169
Check EV = 0.205485

I am expecting that with a half-unit increase in Flush Payout that Raise 4x EV should be at least 0.02 bigger than the Check EV - although the complexity of the calculation makes me not very confident in my analysis.

Mental, could you possibly be able to show the Raise and Check EVs for a starting hand of suited T-9 given the half-unit increase in Flush payout? I'd like to see if the strategy decision for suited T-9 changes.
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Quote: 100xOdds

Mental,

For Flush paying 3:2, Wiz said his simple uth strategy increases the HE from 2.2% to 2.4%.

What would the HE be if Flush pays 2:1 and using Wiz's uth strategy?
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Quote: Mental

Quote: 100xOdds

Mental,

For Flush paying 3:2, Wiz said his simple uth strategy increases the HE from 2.2% to 2.4%.

What would the HE be if Flush pays 2:1 and using Wiz's uth strategy?
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The Wizard's UTH strategy is of no practical interest to me because I only use use optimal strategy.

I assume the strategy errors would be very similar for both flush payouts, but flush outs would obviously have slightly higher weight with a 2-unit payout.

In principal, I could code up the Wiz strategy into an algorithm and get your answer, but then so could you. Counting outs is really messy to code accurately and might run slowly.
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Quote: Torghatten

Before the river, shouldn’t the Ante/Blind and the Raise be considered completely independent? The only thing that matters for the Raise decision is whether your hand has a greater than 50% chance of winning against two random cards.

On the river, you have to consider all the bets together, so you should bet if you expect to win more than 1/5 (1/4) of the hands.
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Quote: Torghatten

Before the river, shouldn’t the Ante/Blind and the Raise be considered completely independent? The only thing that matters for the Raise decision is whether your hand has a greater than 50% chance of winning against two random cards.
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Quote: aceside

I believe this part is problematic. For the Raise 4x decision, the only criterion is EV_4x > EV_check. There are several check decisions in which the winning probabilities are greater than 50%. Right?
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Quote: Torghatten

Before the river, shouldn’t the Ante/Blind and the Raise be considered completely independent? The only thing that matters for the Raise decision is whether your hand has a greater than 50% chance of winning against two random cards.

On the river, you have to consider all the bets together, so you should bet if you expect to win more than 1/5 (1/4) of the hands.
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Quote: aceside

Quote: Torghatten

Before the river, shouldn’t the Ante/Blind and the Raise be considered completely independent? The only thing that matters for the Raise decision is whether your hand has a greater than 50% chance of winning against two random cards.
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I believe this part is problematic. For the Raise 4x decision, the only criterion is EV_4x > EV_check. There are several check decisions in which the winning probabilities are greater than 50%. Right?
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Quote: Mental

Quote: Torghatten

Before the river, shouldn’t the Ante/Blind and the Raise be considered completely independent? The only thing that matters for the Raise decision is whether your hand has a greater than 50% chance of winning against two random cards.

On the river, you have to consider all the bets together, so you should bet if you expect to win more than 1/5 (1/4) of the hands.
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Increasing the blind pay table never affects any river decisions. You will never fold a straight or better with the standard pay table, so you will never fold a made flush whether it pays 1.5 or 2 units.

I think that the entire strategy is the same no even if the blind-bet pay table changes. This means that the frequencies of all final hands are fixed and it appropriate to just change the payoffs and recalculate the HE using fixed frequencies.
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Quote: Mental

I note that charliepatrick's result of -0.899222% is correct to six significant digits in spite of the fact that he did not do the calculation from scratch.

Quote: Mental

Quote: Mental

I note that charliepatrick's result of -0.899222% is correct to six significant digits in spite of the fact that he did not do the calculation from scratch.

I said above that Charlie's result agreed with my full calculation to 6-figures, which was all the digits he posted. I assume we agree perfectly if we would compare more digits.

My first intuition told me that there would be some strategy changes for the blind bet payouts, but I had not really thought it through. Now, I believe there are zero strategy changes preflop and on the river. I did some new calculations that suggest that there might not be any strategy changes post-flop either. This is because you never fold a flush on the river.
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Quote: aceside

The original element of risk is 2.185%/4.152252 = 0.526%, so the revised EoR will be 0.9%/4.152252=0.217%. This number does not indicate nothing though.
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Quote: aceside

Another related problem in UTH is the Blind payout of 10:1 for a player quads. The current payout of 10:1 is too small; I believe it should be 20:1. With this change of payout, the strategy will change too and the EV gain will be calculated differently, I speculate. This is because player may raise 1x or fold some of their quads.
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Quote: aceside

Let me show you an example strategy change. If the five community cards on the board are

2s, 2c, 2d; 2h, 3s,

and the player holds these two cards,

3c, 6s,

Player should fold if the quads payout is 10:1, but raise1x if 20:1. I guess this part but haven’t tested it myself.
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