This third segment will be focusing on the other half of the decision making process that a player needs to make while playing Ultimate X, which is the state of the game board. The first two segments were entirely based on decision making solely on the cards the players are dealt with. There is more than meets the eye with the multipliers, and will be evaluated in two subparts.
The exhaustive efforts of this segment did not improve the overall result by much.
Part 1 - Evaluating the Need for Greed Plays
Update: This part is less relevant
When a player hits a 3 of a Kind or better and a streak starts, the player is tempted to go for the best opportunity to get another 3 of a Kind or better outcome in order to transform the stack into all 12X multipliers to set up the big win. This part is to explore if the tendency of being greedy has its benefits.
From my understanding of the game I have determined that a potential bonus streak is the following:
If a hand has a stack of 2 or more multipliers AND if the active multiplier is not 8 or 12
If there is only 1 multiplier remaining and if a 3 of a Kind or better is hit, the multipliers are added on for a whole new streak. If there are only two multipliers left in a streak, and it starts at 8x, it will be insignificant to hit a Three of a Kind since the next multiplier will be 12x. If a player hits a 3 of a kind or better on a hand where it is already 12X, it will not increase the multiplier beyond the 12X.
Long story short, this greedy play should only be done when the entire board has a streak ongoing. There are 10 hands on the board, and each hand that is not on the streak will reduce the chance of achieving the bonus streak by 1.2%. If you play greedy with just one hand on an active streak, you have nearly 12% chance at getting a Bonus outcome, and need 1 in 10 chance to hit the hand, hence the 1.2%
You will likely only fill out the board with the help of a dealt Bonus outcome hand.
The part that gets most Ultimate X players excited is when they are dealt a good hand so that the entire board is full of multipliers.
Three of a Kind - 1 in 46.3
Straight - 1 in 254
Flush - 1 in 508
Full House 1 in 693
4 of a Kind of any variant - 1 in 4164
Straight Flush - 1 in 72192
Royal Flush - 1 in 649740
On average, 1 in about every 35 hands. Thus the opportunity to get greedy will happen every few minutes.
Omitted useless information as of 12-4-2016
Part 2 - Considering Multipliers when playing suboptimal holds
This is more important
The following logic is inspired from the Ultimate X playing approaches at the Wizard of Odds website, so I give credit to Shackleford where it is due.
Though it is a constant reminder that increasing the Bonus Percentage comes at a cost of expected return on some card deals, it is important to be aware that the cost will only get magnified with the presence of multipliers. Despite how high the multipliers get, the Bonus Percentage remains constant.
This is where the concept of the “Average Multiplier†comes into factor. Mastering this will give the player a few additional hundredth percentage in return, but it will break the simplicity of mastering the strategy if player chooses to.
Before taking a look at the cards on the deal, for every round the player should total up the current multipliers on each of the ten hands. If there is no multiplier, then add a value of 1 to the overall sum.
If Hand 2 and 4 have 2x and if Hand 5 has 3x, while others do not have multipliers, it will be a total of 14X. The total is divided into the number of hands played, in this case 10, for the AVERAGE MULTIPLIER of 1.4X for the hand.
Take for example
Discard Entire Hand VS 3 to Flush with No High Cards
Cards Drawn: 2 ♦ 5 ♥ 6 ♣ 9 ♥ 10 ♥
Expected Result: 1.6127694777954
Bonus Percentage: 0.0277651197342267
Optimal Cards Held:
Bonus Hold
Cards Held: 5 ♥ 9 ♥ 10 ♥
Expected Result: 1.40148011100833
Bonus Percentage: 0.0499537465309898
The suboptimal play of 3 to a Flush with no High Cards results in a loss of .211 credits to gain about 2.22% in Bonus Percentage. With the Average Multiplier of 1.4X which is considered low, the player will be losing 0.2954 credits. It might still be worth making the play. If the Average Multiplier is 5.5X, it will become a more serious loss of 1.1605 credits to trade off for the 2.22% Bonus Percentage. The Player must decide to his or her own weather the suboptimal play will be worthwhile or not.
Greedy Play & Multipliers on DDB UltX Bonus Streak
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alpax
- Video Poker Master
- Posts: 1913
- Joined: Sat Jun 14, 2014 4:42 pm
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alpax
- Video Poker Master
- Posts: 1913
- Joined: Sat Jun 14, 2014 4:42 pm
EDITED: Useless information
-
alpax
- Video Poker Master
- Posts: 1913
- Joined: Sat Jun 14, 2014 4:42 pm
I was able to find a breakthrough with my findings with Ultimate X Bonus Streak but have not had the opportunity to update during the week. There is much to discuss and it will wrap up my efforts with Ultimate X when the updates are completed.
It turns out that the Average Multiplier approach is vital to the strategic approach, and the Greedy Plays is not as vital. I will need to update the main thread by omitting information that is useless.
Because playing with the Average Multiplier is so vital, the new approach will be discussed primarily, and then the Greedy Plays thereafter for those that want a few more hundredth of a percentage return.
The clearcut breakthrough of the Average Multiplier is the fact of setting a proper value of what the Bonus Percentage is worth. I first discussed that the amount of credits lost exceeding half a coin might be a good guideline, but that turns out not to be a good guideline.
My improved evaluation of the Bonus Percentage:
For a visual example, the following hand will be used:
A ♦ 4 ♥ 10 ♥ Q ♥ K ♥
The Optimal 8/5 DDB Play Hold: 10 ♥ Q ♥ K ♥
Expected Return: 6.54486586493987
Hold Breakdown (1081 Possibilities): 1 Royal Flush, 1 Straight Flush, 34 Flush, 26 Straight, 9 3 of a Kind, 27 Two Pairs, 237 Jacks or Better, 746 No Wins
The Ideal Suboptimal 8/5 DDB Play for Ultimate X Hold: 4 ♥ 10 ♥ Q ♥ K ♥
Expected Return: 5.42553191489362
Hold Breakdown (47 Possibilities): 9 Flush, 6 Jacks or Better, 32 No Wins
Expected Return Difference Between the Plays: 1.11933395
Part 1 of 3 - Separating the value (or weight) of the Small Bonus and Big Bonus
As a constant reminder, when a player gets Three of a Kind, Straight, or a Flush, a multiplier of 2x, 3x, 4x will be added on for the next 3 rounds. Anything less than a Three of a Kind, there will be no multipliers.
The player will be getting 9x worth of multipliers. The player not getting the multipliers will get just three singular 1x games worth 3x. The difference is 6x.
On the other hand, when a player gets Full House, Any Four of Kind, Straight Flush, or the Royal Flush, a multiplier of 2x, 3x, 4x, 8x, 12x will be added on for the next 5 rounds.
The player will be getting 29x worth of multipliers over the 5 rounds, the player not getting multipliers will be getting 5x worth over the 5 rounds. The difference is 24x.
With the Big Bonus being worth 24x and the Small Bonus being worth 6x, the Big Bonus play is 4 times more valuable than the small bonus play. Now for the total bonus percentage, all Full House or Better outcome possibilities will now be weighed by a factor of 4 into the adjusted total.
Back to the Example, Adjusted Bonus Percentages:
Optimal Hold:
2 out of 1081 Outcomes (1 RF and 1 SF) are Big Bonus, multiply that by 4 to get 8/1081.
69 out of 1081 Outcomes (34 FL, 26 ST, 9 3oaK) are Small Bonus, so 69/1081.
Add them up:
8 + 69 = 77 -> 77/1081 -> 0.07123 (Adjusted Bonus Percentage) or 7.123%
Non-Optimal UltX Hold:
9 out of 47 Outcomes (9 FL) are Small Bonus, so 9/47 ->
No Big Bonus possibilities so 9/47 -> 0.19149 (Adjusted Bonus Percentage) or 19.149%
The difference of the Adjusted Bonus Percentage between the holds is -> 0.19149 - 0.07123 -> 0.120259 or 12.0259%
Part 2 of 3 - What is the difference of Bonus Percentage is worth of expected return
The pay table are based on a 5 credit bet even though you are betting 10 coins, the 5 extra coins are reserved for the Bonus if you are able to get them. Playing 5 credits will merely be a 10 play 8/5 DDB game.
From Part 1, when a player gets the bonus, they will receive at least 6x worth of multipliers in value. Thus as a starting point, the Bonus should be worth 6 * 5 credit bet = 30 Credits, but that is not all.
The player is also playing a game that is also negative expectation or something that returns less than 100% of the coin they put in, 8/5 DDB with Optimal Play will be a 96.79% returning game, when players make good non-optimal holds to increase their Bonus Percentage, their hold strategy will reflect a 96.39% returning game. Trial and Error is needed to find out the exact precise in between number, but for simplicity purposes, I chose the value to be an even 96.5% which ended up to being a good value on the simulation results.
So the Bonus is worth 30 * 96.5% = 28.95 Credits if the player is able to hit it.
The difference in bonus percentage (adjusted with the Big Bonus counting for 4 times the amounts) between the non-optimal and optimal play will be multiplied by 28.95 to get the Expected Return worth of the increased Bonus Percentage used in the next part.
Back to the Example, the difference of the Adjusted Bonus Percentages was at 0.120259.
To get the Projected Expected Return Difference of the Bonus Percentage on the suboptimal UltX Play that I projected, it will be:
0.120259 (Bonus Percentage Difference) * 30 (Bonus Feature Benefit Units) * 0.965 (The 96.5% return of the play strategies used) = 3.35965572303 Credits (the amount making the suboptimal play is more valuable)
Part 3 of 3 - Using the Average Multiplier to make the final decision
As mentioned on the main post, add up all the multipliers on the board (add 1 if there is no multiplier) and divide them by 10 in order to get the multiplier. Afterwards, take the difference in expected return from the suboptimal play and the optimal play and multiply by the Average Multiplier value.
Back to the Example, the difference of Expected Return between the two plays is 1.11933395 (right before Part 1).
We take the worth of the Projected Expected Return Difference of the Bonus Percentage on the suboptimal play and divide it with the actual expected difference.
3.35965572303 / 1.11933395 = 3.11
If the Average Multiplier ends up being 3.1 or less, the suboptimal hold should be favored. If it is 3.2 or greater, the expected return the player is set to lose far exceed the additional Adjusted Bonus Percentage value, thus the optimal hold is made.
The only time the Average Multiplier is not considered:
The Obvious Optimal Play situation (does not happen often) is as follows :
The player is dealt a Big Bonus hand (Full House or better) to set a stack of 5 multipliers on all the hands. If the player gets ANY dealt Bonus hand on 2x, 3x, or 4x, to fill out the board with 12x, play the optimal holds until the streak runs out the last hand (might need to set up more multipliers for a new streak). If the player reaches 8x without being dealt ANY Bonus hand, play the optimal hold for that one hand.
There is a rather long list (I mean about 20,000 situations, with unique card suit and rank) of Average Multiplier situations.
It would take too much time to list out so I have shared the report in HTML format so that it can be viewed from a web browser. File is about 27 Megabytes so I used DropBox. No need to sign up for an account, public access, download from page.
Link to DropBox
To interpret the entry of the report:
The first column is the 5 Card Hand that is dealt
The second column is the Optimal Hold played in standard 8/5 DDB
The third column is the Expected Return of the Optimal Hold
The fourth column is the High Bonus Percentage of the Optimal Hold
The fifth column is the Low Bonus Percentage of the Optimal Hold
The sixth column is the Highest Bonus Percentage Hold for 8/5 Ultimate X DDB
The seventh column is the High Bonus Percentage of the Highest Bonus Percentage Hold
The eighth column is the Low Bonus Percentage of the Highest Bonus Percentage Hold
The ninth (9th) column is the Average Multiplier amount, if your total Average Multiplier is under the specified value, play the Highest Bonus Percentage Hold for 8/5 Ultimate X DDB, in this case 2.95. If you get a total 3 or higher, you will play the Optimal Hold.
Playing these Highest Bonus Percentage Hold everytime without regarding Average Multiplier will result in a return of 97.44% (based on simulation results)
Playing with the Average Multiplier will increase the return to 97.63% if you are willing to master this task.
It turns out that the Average Multiplier approach is vital to the strategic approach, and the Greedy Plays is not as vital. I will need to update the main thread by omitting information that is useless.
Because playing with the Average Multiplier is so vital, the new approach will be discussed primarily, and then the Greedy Plays thereafter for those that want a few more hundredth of a percentage return.
The clearcut breakthrough of the Average Multiplier is the fact of setting a proper value of what the Bonus Percentage is worth. I first discussed that the amount of credits lost exceeding half a coin might be a good guideline, but that turns out not to be a good guideline.
My improved evaluation of the Bonus Percentage:
For a visual example, the following hand will be used:
A ♦ 4 ♥ 10 ♥ Q ♥ K ♥
The Optimal 8/5 DDB Play Hold: 10 ♥ Q ♥ K ♥
Expected Return: 6.54486586493987
Hold Breakdown (1081 Possibilities): 1 Royal Flush, 1 Straight Flush, 34 Flush, 26 Straight, 9 3 of a Kind, 27 Two Pairs, 237 Jacks or Better, 746 No Wins
The Ideal Suboptimal 8/5 DDB Play for Ultimate X Hold: 4 ♥ 10 ♥ Q ♥ K ♥
Expected Return: 5.42553191489362
Hold Breakdown (47 Possibilities): 9 Flush, 6 Jacks or Better, 32 No Wins
Expected Return Difference Between the Plays: 1.11933395
Part 1 of 3 - Separating the value (or weight) of the Small Bonus and Big Bonus
As a constant reminder, when a player gets Three of a Kind, Straight, or a Flush, a multiplier of 2x, 3x, 4x will be added on for the next 3 rounds. Anything less than a Three of a Kind, there will be no multipliers.
The player will be getting 9x worth of multipliers. The player not getting the multipliers will get just three singular 1x games worth 3x. The difference is 6x.
On the other hand, when a player gets Full House, Any Four of Kind, Straight Flush, or the Royal Flush, a multiplier of 2x, 3x, 4x, 8x, 12x will be added on for the next 5 rounds.
The player will be getting 29x worth of multipliers over the 5 rounds, the player not getting multipliers will be getting 5x worth over the 5 rounds. The difference is 24x.
With the Big Bonus being worth 24x and the Small Bonus being worth 6x, the Big Bonus play is 4 times more valuable than the small bonus play. Now for the total bonus percentage, all Full House or Better outcome possibilities will now be weighed by a factor of 4 into the adjusted total.
Back to the Example, Adjusted Bonus Percentages:
Optimal Hold:
2 out of 1081 Outcomes (1 RF and 1 SF) are Big Bonus, multiply that by 4 to get 8/1081.
69 out of 1081 Outcomes (34 FL, 26 ST, 9 3oaK) are Small Bonus, so 69/1081.
Add them up:
8 + 69 = 77 -> 77/1081 -> 0.07123 (Adjusted Bonus Percentage) or 7.123%
Non-Optimal UltX Hold:
9 out of 47 Outcomes (9 FL) are Small Bonus, so 9/47 ->
No Big Bonus possibilities so 9/47 -> 0.19149 (Adjusted Bonus Percentage) or 19.149%
The difference of the Adjusted Bonus Percentage between the holds is -> 0.19149 - 0.07123 -> 0.120259 or 12.0259%
Part 2 of 3 - What is the difference of Bonus Percentage is worth of expected return
The pay table are based on a 5 credit bet even though you are betting 10 coins, the 5 extra coins are reserved for the Bonus if you are able to get them. Playing 5 credits will merely be a 10 play 8/5 DDB game.
From Part 1, when a player gets the bonus, they will receive at least 6x worth of multipliers in value. Thus as a starting point, the Bonus should be worth 6 * 5 credit bet = 30 Credits, but that is not all.
The player is also playing a game that is also negative expectation or something that returns less than 100% of the coin they put in, 8/5 DDB with Optimal Play will be a 96.79% returning game, when players make good non-optimal holds to increase their Bonus Percentage, their hold strategy will reflect a 96.39% returning game. Trial and Error is needed to find out the exact precise in between number, but for simplicity purposes, I chose the value to be an even 96.5% which ended up to being a good value on the simulation results.
So the Bonus is worth 30 * 96.5% = 28.95 Credits if the player is able to hit it.
The difference in bonus percentage (adjusted with the Big Bonus counting for 4 times the amounts) between the non-optimal and optimal play will be multiplied by 28.95 to get the Expected Return worth of the increased Bonus Percentage used in the next part.
Back to the Example, the difference of the Adjusted Bonus Percentages was at 0.120259.
To get the Projected Expected Return Difference of the Bonus Percentage on the suboptimal UltX Play that I projected, it will be:
0.120259 (Bonus Percentage Difference) * 30 (Bonus Feature Benefit Units) * 0.965 (The 96.5% return of the play strategies used) = 3.35965572303 Credits (the amount making the suboptimal play is more valuable)
Part 3 of 3 - Using the Average Multiplier to make the final decision
As mentioned on the main post, add up all the multipliers on the board (add 1 if there is no multiplier) and divide them by 10 in order to get the multiplier. Afterwards, take the difference in expected return from the suboptimal play and the optimal play and multiply by the Average Multiplier value.
Back to the Example, the difference of Expected Return between the two plays is 1.11933395 (right before Part 1).
We take the worth of the Projected Expected Return Difference of the Bonus Percentage on the suboptimal play and divide it with the actual expected difference.
3.35965572303 / 1.11933395 = 3.11
If the Average Multiplier ends up being 3.1 or less, the suboptimal hold should be favored. If it is 3.2 or greater, the expected return the player is set to lose far exceed the additional Adjusted Bonus Percentage value, thus the optimal hold is made.
The only time the Average Multiplier is not considered:
The Obvious Optimal Play situation (does not happen often) is as follows :
The player is dealt a Big Bonus hand (Full House or better) to set a stack of 5 multipliers on all the hands. If the player gets ANY dealt Bonus hand on 2x, 3x, or 4x, to fill out the board with 12x, play the optimal holds until the streak runs out the last hand (might need to set up more multipliers for a new streak). If the player reaches 8x without being dealt ANY Bonus hand, play the optimal hold for that one hand.
There is a rather long list (I mean about 20,000 situations, with unique card suit and rank) of Average Multiplier situations.
It would take too much time to list out so I have shared the report in HTML format so that it can be viewed from a web browser. File is about 27 Megabytes so I used DropBox. No need to sign up for an account, public access, download from page.
Link to DropBox
To interpret the entry of the report:
The first column is the 5 Card Hand that is dealt
The second column is the Optimal Hold played in standard 8/5 DDB
The third column is the Expected Return of the Optimal Hold
The fourth column is the High Bonus Percentage of the Optimal Hold
The fifth column is the Low Bonus Percentage of the Optimal Hold
The sixth column is the Highest Bonus Percentage Hold for 8/5 Ultimate X DDB
The seventh column is the High Bonus Percentage of the Highest Bonus Percentage Hold
The eighth column is the Low Bonus Percentage of the Highest Bonus Percentage Hold
The ninth (9th) column is the Average Multiplier amount, if your total Average Multiplier is under the specified value, play the Highest Bonus Percentage Hold for 8/5 Ultimate X DDB, in this case 2.95. If you get a total 3 or higher, you will play the Optimal Hold.
Playing these Highest Bonus Percentage Hold everytime without regarding Average Multiplier will result in a return of 97.44% (based on simulation results)
Playing with the Average Multiplier will increase the return to 97.63% if you are willing to master this task.
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