💰 Optimally Stacking the Deck – Texas Hold ‘Em – Math ? Programming

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A Puzzle is Solved and then some!
And we found quite a bit more than we expected!
For the sake of this post, the betting rules will not matter.
Optimal Stackings In card games, especially gambling games, trust is rarely given freely.
Instead, we incorporate rituals into standard play that assure fairness.
Assuming the two players are not colluding and there is no sleight of hand, this click here absolves the dealer of any guilt.
Even if the dealer had stacked the deck to deal himself favorable cards, the randomness of the cut will necessarily ruin his efforts.
An optimal stacking circumvents the tacit fairness assumed after a deck is cut.
In particular, we search for an ordering of the cards in the deck which is so mischievous that no matter where it is cut, a specific player always wins.
Instead, the existence or nonexistence of an optimal stacking is an interesting combinatorial property of card games, and it appears to be a measure of complexity.
As we sawsimple games like sometimes called Indian Poker and do not have optimal stackings.
Adding complexity to these games, we invented Kicsi Poker and discovered it has 11 total distinct optimal stackings up to cyclic permutations of the deck.
Because we found optimal stackings for Kicsi Poker after adding complexity, we reckoned that adding more complexity would not rule out the possibility of optimal stackings.
In fact, we find much more than this, but we should formalize the notion a bit further and describe our method before constructing such a stacking.
The remainder of this section will be abstract and seemingly unnecessary for the casual reader.
We hold the contention that a proper mathematical analysis should be independent of the real world, and with the right definitions we can logically extend optimal stackings to other situations.
Definition: A cutting permutation of a list is a permutation in with the : and for a fixedwe call the cut at the -th position.
If the reader is not familiar with thethis is simply a formalization of the intuitive idea of a cut.
Applying this results in the listas expected.
Moreover, every cut can be achieved by iterating the process of putting the top card on the bottom of the deck.
Hence, the set of all cutting permutations is the cyclic group.
Since any list imposes an ordering on its contents, we can apply cutting permutations to any list.
Definition: A cut of a list is the list for some cutting permutation.
If the list is denotedwe denote the cut.
see more to card games, we speak of cutting a deck as the process of replacing a list of cards with its cut.
We can now define an optimal stacking.
Definition: Fix a game played with a set of cards and players.
We say that has an optimal stacking for player if there exists an ordering of the cards such that for every cutting permutationplayer can force a win when is dealt.
Moreover, a game is said to simply have an optimal stacking if it has one for any player.
We do not yet know of games which have optimal stackings for some players but not others, and we conjecture that no natural games do that would make the game unfair for some players, although this author has always found asymmetric games interesting.
It may also benefit one to assume in general that all players are infinitely rational, a standard assumption in Game Theory.
Note that we do not want to assume the cheating player knows the order of the deck ahead of time.
Instead, an optimal stacking should allow the player to force a win simply by logical play and the knowledge that the stacking is optimal.
Proposition: The slightly simplified game of Hearts does not have an optimal stacking.
We allow no passing of cards.
Moreover, we restrict our game of Hearts to a single round, so that the winner of the game is the one who takes the smallest number of points.
Hence, if one cannot shoot the moon, one must try to minimize the number of points taken.
The crux of the proof is that all cards in the deck are dealt before play begins.
Suppose first that the goal is simply to collect 0 points.
Indeed, suppose that some stacking wins for player 1.
Then a cut at position 1 shifts all of the hands right by one player, so then player 4 can guarantee collecting 0 points.
If still player 1 can guarantee collecting zero points, then instead a cut at position 2 gives such guarantees to players 3 and 4.
At this point, player 1 can still guarantee taking 0 points, then player 2 can shoot the moon, hence giving all other players 26 points.
If, on the other hand, player 1 can guarantee shooting the moon, the same argument shows that a cut at position 1 gives that guarantee to player 4.
This is a contradiction since two players cannot shoot the moon simultaneously, nor can one shoot the moon while the other takes zero points.
The same proof applies, along with the knowledge that 26 the total number of points is not divisible by 4 the number of playersor more coarsely that the Queen of Spades already represents 13 points, and can only be taken by one player.
One aspect of this proof contradicts the claim that the existence of an optimal stacking is a measure of complexity.
In particular, Hearts is considered a rather complex game certainly more complex than Kicsi Pokerand yet it does not have an optimal stacking.
While we do not have a formal rebuttal, it may instead be the case that the existence of an optimal stacking measures the ability for a game to diverge.
But enough big-picture speculation.
Here it is: Th5d7cAd3cQsJc4h6c8c9dTc6hQc8d9sJh5c7dAhAc9h2cKh5sJs8s7h2d 7s9c4dKd8hQd6d3sKs5h2hAs4s2sTs6sQhJd3h4cTd3dKc p1 p2 community p1 hand p2 hand winner Th7c 5dAd Qs Jc 4h 8c Tc One Pair High Card 1 5dAd 7c3c Jc 4h 6c 9d 6h One Pair One Pair 1 7c3c AdQs 4h 6c 8c Tc Qc Flush One Pair 1 AdQs 3cJc 6c 8c 9d 6h 8d Two Pair Two Pair 1 3cJc Qs4h 8c 9d Tc Qc 9s Flush Two Pair 1 Qs4h Jc6c 9d Tc 6h 8d Jh Straight Two Pair 1 Jc6c 4h8c Tc 6h Qc 9s 5c Flush High Card 1 4h8c 6c9d 6h Qc 8d Jh 7d One Pair One Pair 1 6c9d 8cTc Qc 8d 9s 5c Ah One Pair One Pair 1 8cTc 9d6h 8d 9s Jh 7d Ac Straight One Pair 1 9d6h TcQc 9s Jh 5c Ah 9h Trips One Pair 1 TcQc 6h8d Jh 5c 7d Ac 2c Flush High Card 1 6h8d Qc9s 5c 7d Ah texas holdem primer Kh Straight One Pair 1 Qc9s 8dJh 7d Ah Ac 2c 5s One Pair One Pair 1 8dJh 9s5c Ah Ac 9h Kh Js Two Pair Two Pair 1 9s5c Jh7d Ac 9h 2c 5s 8s Two Pair High Card 1 Jh7d 5cAh 9h 2c Kh Js 7h Two Pair High Card 1 5cAh 7dAc 2c Kh 5s 8s 2d Two Pair One Pair 1 7dAc Ah9h Kh 5s Js 7h 7s Trips One Pair 1 Ah9h Ac2c 5s Js 8s 2d 9c One Pair One Pair 1 Ac2c 9hKh Js 8s 7h 7s 4d One Pair One Pair 1 9hKh 2c5s 8s 7h 2d 9c Kd Two Pair One Pair texas holdem primer 2c5s KhJs 7h 2d 7s 4d 8h Two Pair One Pair 1 KhJs 5s8s 2d 7s 9c Kd Qd One Pair High Card 1 5s8s Js7h 7s 9c 4d 8h 6d Straight One Pair 1 Js7h 8s2d 9c 4d Kd Qd 3s High Card High Card 1 8s2d 7h7s 4d Kd 8h 6d Ks Texas holdem primer Pair Two Pair 1 7h7s 2d9c Kd 8h Qd 3s 5h One Pair High Card 1 2d9c 7s4d 8h Qd 6d Ks 2h One Pair High Card 1 7s4d 9cKd Qd 6d 3s 5h As Straight High Card 1 9cKd 4d8h 6d 3s Ks 2h 4s One Pair One Pair 1 4d8h KdQd 3s Ks 5h As 2s Straight One Pair 1 KdQd 8h6d Ks 5h 2h 4s Ts One Pair High Card 1 8h6d Qd3s 5h 2h As 2s 6s Two Pair One Pair 1 Qd3s 6dKs 2h As 4s Ts Qh One Pair High Card 1 6dKs 3s5h As 4s 2s 6s Jd Flush Flush 1 3s5h Ks2h 4s 2s Ts Qh 3h One Pair One Pair 1 Ks2h 5hAs 2s Ts 6s Jd 4c One Pair High Card 1 5hAs 2h4s Ts 6s Qh 3h Td One Pair One Pair 1 2h4s As2s 6s Qh Jd 4c 3d One Pair High Card 1 As2s 4sTs Qh Jd 3h Td Kc Straight One Pair 1 4sTs 2s6s Jd 3h 4c 3d Th Two Pair One Pair 1 2s6s TsQh 3h 4c Td Kc 5d Straight One Pair 1 TsQh 6sJd 4c Td 3d Th 7c Trips One Pair 1 6sJd Qh3h Td 3d Kc 5d Ad Flush One Pair 1 Qh3h Jd4c 3d Kc Th 7c 3c Trips One Pair 1 Jd4c 3hTd Kc Th 5d Ad Qs Straight One Pair 1 3hTd 4c3d Th 5d 7c 3c Jc Two Pair One Pair 1 4c3d TdKc 5d 7c Ad Qs 4h One Pair High Card 1 TdKc 3dTh 7c Ad 3c Jc 6c Flush One Pair 1 3dTh Kc5d Ad 3c Qs 4h 8c One Pair High Card 1 Kc5d Th7c 3c Qs Jc 6c 9d High Card High Card 1 The first two columns show the pocket cards dealt to player 1 and player 2, respectively, and the fourth and fifth columns give their final hands, again respectively.
Immediately, the reader will recognize that there are 52!
Searching them all, even at supercomputer speed, would easily take longer than the life of the universe at a trillion stackings per second, the estimate is on the order of years.
So we cannot possibly look through them all.
We use in the discrete space of all orderings of the deck where the function to optimize is the number of cuts that win for the first player dealt.
Neighbors consist of the set of stackings obtainable by swapping pairs of cards, and we use random restarts to avoid local maxima.
Moreover, we parallelized the algorithm to run on aaccess to which is graciously here by the UIC mathematical computer science department.
The result is lightning-fast results.
We compute the value of a deck by dealing out a game for each of the 52 cuts of the deck, and seeing how many times player 1 wins.
Then we compare with the value of all neighboring decks, where a neighbor is obtained by swapping any two cards in the deck.
If any such stacking has a higher value, we take the stacking with the highest value, and repeat the process.
Once we get to the point where no such neighboring deck has a higher value, we stop and output the final stacking.
Then we start the whole process over again with a new random deck, and stop when we find an optimal stacking or continue searching for more.
We leave that for future research.
We chose to implement this algorithm in C++.
In hindsight this was probably unnecessary, but we do admittedly wallow in pride when quoting statistics about texas holdem primer fast the program runs.
All of is available for free on.
We also include a text file containing the list of about 100,000 optimal stackings we found, and similar tables as above detailing the hands dealt for each cut.
In any case, we separated the code into three parts.
First we implemented a general steepest-ascent framework for any problem.
To give a pedantic yet unrealistic example, our posted code includes an instance of optimizing a polynomial function in one variable.
Second, we implemented a poker-hand evaluator and neighbor generator.
Otherwise, enlarging the lookup-table to cover seven-card hands would incur more than a 2,000 times increase in memory use, with only a speedup factor of 10 to 15 times.
And even so, on the supercomputer we can evaluate about 7 million seven-card hands per second, down from about 87 million five-card hands per second.
We gain the potential to go up to 87 million seven-card hands per second with the added memory footprint, but as it turns out we can find plenty of optimal stackings with the slower version.
Third, we parallelized the program using to run on 12 threads, increasing the runtime by a factor of 12 for sufficiently large problem sizes.
For smaller problem sizes, we see a speedup factor of roughly 10x, giving a reasonable 80% efficiency.
In the next section we detail the important aspects of the code.
Enter the Jungle Unfortunately, a primer on C++ is far beyond the scope of this blog.
An equally strong understanding of the algorithm itself can also be gained from our Python post onthough the program is admittedly much slower.
The general steepest ascent algorithm features a virtual C++ class called Position.
A position has three functions: value codes uk bonus deposit casino july no 2020 cirrus, neighborsand showthe last of which returns a string containing a textual representation of a Position.
Here are snippets of the code, taken from my hillclimb.
For a short example on how to use this framework, the reader should investigate our quarticopt.
The poker part required a bit of preprocessing to understand, but it is essentially a simple idea.
First, we note that even though there are a lot of poker hands, many are equivalent up to scoring.
But none of these hands is any better than another.
In other words, there are only 7,462 distinct five-card poker hands, which is a lot smaller than the some 2.
The poker scoring function hence takes as in put a five-card hand and computes a number between 1 and 7,462 representing the absolute score of the hand.
It does this through a combination of accessing a pre-generated look-up table for flushes and high-card hands, and a technique for the remaining hands.
The interested reader will see for more details.
It is quite novel, and results in a lightning-fast scoring algorithm.
We note that the next time we work on this problem, we will likely extend the code to work for any number of hands and any number of players, and replace the ugly assignment operators above with a few loops.
Moreover, we leave out an additional optimization from this code snippet for brevity.
Running the main application gives an output similar to the large table above for each discovered optimal stacking.
Parallelization A thorough introduction to parallel computation is again sadly beyond the scope of this post.
Luckily, we chose the arguably simplest possible library for parallelization:.
With OpenMP, the serial code looks almost identical to the parallel code, with the exception of a few sagely placed pragmas code annotations.
For instance, our main function parallelizes at the top-most level, so that each thread performs steepest ascent trials independently of the others.
From this perspective, the steepest ascent algorithm is.
In the code posted, we tried two different levels of parallelization.
First, we parallelized at the top-most level, as above.
The second way was at the finest level, so that all threads were cooperating to evaluate the score of a single deck and compute neighbors.
The former method turned out to be somewhat more efficient than the latter for large problem sizes.
Specifically, in 1,000 iterations of a steepest ascent, the former was 17% faster.
The speed increase is due to the significantly less system go here overhead to schedule the threads, and the fact that evaluating a single deck is very fast, so at the end of every loop the thread synchronization time begins to outweigh the deck evaluation time.
However, for smaller problem sizes the efficiency seems to drop off at a slower rate.
We also include the full source code for the fine-granularity parallelization, with the mindset that it may perform better in other application domains, and it serves as an example implementation.
In particular, about one in six randomly chosen decks texas holdem primer be improved to an optimal stacking.
Specifically, in all of our tests we never witnessed a deck which https://clearadultskin.com/2020/pokerstars-isle-of-man-rally-2020.html more than 15 steps to reach a local maximum.
We do not have a proof that this is upper bound is sharp.
Moreover, some decks were optimized with as few as 5 steps.
We note the click that some of these decks are redundant, but the probability of that happening is decidedly small.
We encourage the reader to try one out at a friendly neighborhood poker game.
And finally, here is a list of ten optimal stackings, so that casual readers can avoid downloading and navigating the directory.
We encourage the interested reader say, as a programming exercise to implement a steepest-ascent algorithm for optimal stackings in a different game, and post the results in a comment.
Some ideas for other games include poker variants like Omaha and Lowball, along with completely unrelated games like Blackjack or Rummy.
Might we even go so far as to investigate non-standard card games like a favorite during my youth?
A critical step in the analysis of most non-poker games would be to formalize what it means to force a win, or relax the definition to make it sensible.
For instance, how hard is it to find optimal stackings for any given player when there are players?
More specifically, how fast does the complexity grow when you add more players?
Are there significantly more semi-optimal stackings, in which we allow ties for player 1?
What sorts of interesting patterns occur in optimal or semi-optimal stackings?
Many of these sorts of questions could be answered by minor modifications to the source code provided.
We leave that as an exercise for the reader, until we find enough time ourselves to go through and find optimal stackings which we believe to exist for any reasonable number of players and any position of the winning player.
That is a very good question, and I think the answer is no.
In a sense, because these stackings are so plentiful we should expect that there may be some games in which all players but one have no hope, regardless of the cut.
Trying to avoid optimal stackings that just happen to come up in real games would actually make the deal less random, not more random.
The ability to play these stackings correctly requires knowledge that you are dealt an optimal stacking.
After all, many of the hands win with a low pair or even a high card, and others hit low flushes on the river with a single card on suit.
One thing that stacking theory does say is that cutting the deck is by no means fool-proof in preventing cheating.
Of course it takes far more work to stack an optimal stacking than one would be able to do surreptitiously at a table, but a weaselly player could swap out a deck with one that is optimally stacked in his favor.
This raises the obvious the question: what sort of ritual should we adopt that truly ensures no foul play?
Ten of Hearts, of course.
Every chapter includes an application, from cryptography to economics, physics, neural networks, and more!

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Please before continue reading, make sure to read the disclaimer at the bottom of this article.
There learn more here be some kind of algorithm which implements and refreshes those numbers at each draw.
Today I am going to write about one of the first simulations I put down as texas holdem primer and wrapped my head around.
The code I am about to show estimates the probabilities of drawing a pair, a three of a kind tris and a poker.
Here they are: At this stage, if we run the code, R will generate three tables or matrix with texas holdem primer results of each one of the 100 simulate games.
Something like this: Now we need only to look for pairs, tris and pokers.
We need to define 3 functions as follows: The result should look like this: In 100 games, we have got 47 pairs, 2 three of a kind and no pokers!
However, this might be a mere case!
We need to run this a LOT of times to be sure the odds we obtained are at least near the real ones.
For instance, the function below runs n times 100 games and then collect the results.
Note that It outputs the probabilities as the mean of the probabilities occurred.
Much of the code here is replicated from the functions above.
I guess this could have been done a lot better!
If you have any idea please let me know or leave a comment!
Here are the results: For debugging purposes our function outputs each poker it finds.
Since usually pokers are not that frequent it should be fine.
By simulating in less than 2 minutes 100.
Anyway the other two probabilities seem fine you can check for more texas holdem primer on Hope you enjoyed!
Disclaimer: This article is for educational purpose ONLY.
Odds generated by this code are calculated by a random simulation.
As such the odds will represent an approximation of the true odds.
They might even be completely wrong or misleading.
This code must NOT be used for anything other than educational purpose.
The provider of this code does not guarantee the accuracy of the results and accepts no liability for any loss or damage that may occur as a result of the use of this code.
Understanding and agreeing to the terms of this disclaimer is a condition of use of this code.
By reading the article you confirm you have understood and will comply with this disclaimer.
To leave a comment for the author, please follow the link and comment on their blog:.
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A Puzzle is Solved and then some!
And we found quite a bit more than we expected!
For the sake of this post, the betting rules will not matter.
Optimal Stackings In card games, especially gambling games, trust is rarely given freely.
Instead, we incorporate rituals into standard play that assure fairness.
Assuming the two players are not colluding and there is no sleight of hand, this ritual absolves the dealer of any guilt.
Even if the dealer had stacked the deck to deal himself favorable cards, the randomness of the cut will necessarily ruin his efforts.
An optimal stacking circumvents the tacit fairness assumed after a deck is cut.
In particular, we search for an ordering of the cards in the deck which is so mischievous that no matter where it is cut, a specific player always wins.
Instead, the existence or nonexistence of an optimal stacking is an interesting combinatorial property of card games, and it appears to be a measure of complexity.
As we sawsimple games like sometimes called Indian Poker and do not have optimal stackings.
Adding complexity to these games, we invented Kicsi Poker and discovered it has 11 total distinct optimal stackings up to cyclic permutations of the deck.
Because we found optimal stackings for Kicsi Poker after adding complexity, we reckoned that adding more complexity would not rule out the possibility of optimal stackings.
In fact, we find much more than this, but we should formalize the notion a bit further and describe our method before constructing such a stacking.
The remainder of this section will be abstract and seemingly unnecessary for the casual reader.
We hold the contention that a proper mathematical analysis should be independent of the real world, and with the right definitions we can logically extend optimal stackings to other situations.
Definition: A cutting permutation texas holdem primer a list is a permutation in with the : and for a fixedwe call the cut at the -th position.
If the reader is not familiar with thethis is simply a formalization of the intuitive idea of a cut.
Applying this results in the listas expected.
Moreover, every cut can be achieved by iterating the process of putting the top card on the bottom of the deck.
Hence, the set of all cutting permutations is the cyclic group.
Since any list imposes an ordering on its contents, we can apply cutting permutations to any list.
Definition: A cut of a list is the list for some cutting permutation.
If the list is denotedwe denote the cut.
Specifically to card games, we speak of cutting a deck as the process of replacing a list of cards with its cut.
We can now define an optimal stacking.
Definition: Fix a game played with a set of cards and players.
We say that has an optimal stacking for player if there exists an ordering of the cards such that for every cutting permutationplayer can force a win when is dealt.
Moreover, a game is said to simply have an optimal stacking if it has one for any player.
We do not yet know of games which have optimal stackings for some players but not others, and we conjecture that no natural games do that would make the game unfair for some players, although this author has always found asymmetric games interesting.
It may also benefit one to assume in general that all players are infinitely rational, a standard assumption in Game Theory.
Note that we do not want to assume the cheating player knows the order of the deck ahead of time.
Instead, an optimal stacking should allow the player to force a win simply by logical play and the knowledge that the stacking is optimal.
Proposition: The slightly simplified game of Hearts does not have an optimal stacking.
We allow no passing of cards.
Moreover, we restrict our game of Hearts to a single round, so that the winner of the game is the one who takes the smallest number of points.
Hence, if one cannot shoot the moon, one must try to minimize the number of points taken.
The crux of the proof is that all cards in the deck are dealt before play begins.
Suppose first that the goal is simply to collect 0 points.
Indeed, suppose that some stacking wins for player 1.
Then a cut at position 1 click all of the hands right by one player, so then player 4 can guarantee collecting 0 points.
If still player 1 can guarantee collecting zero points, then instead a cut at position 2 gives such guarantees to players 3 and 4.
At this point, player 1 can still guarantee taking 0 points, then player 2 can shoot the moon, hence giving all other players 26 points.
If, on the other hand, player 1 can guarantee shooting the moon, the same argument shows that a cut at position 1 https://clearadultskin.com/2020/20-free-no-deposit-casino-2020.html that guarantee to player 4.
This is a contradiction since two players cannot shoot the moon simultaneously, nor can one shoot the moon while the other takes zero points.
The same proof applies, along with the knowledge that 26 the total number of points is not divisible by 4 the number of playersor more coarsely that the Queen of Spades already represents 13 points, and can only be taken by one player.
One aspect of this proof contradicts the claim that the existence of an optimal stacking is a measure of complexity.
In particular, Hearts is considered a rather complex game certainly more complex than Kicsi Pokerand yet it does not have an optimal stacking.
While we do not have a formal rebuttal, it may instead be the case that the existence of an optimal stacking measures the ability for a game to diverge.
But enough big-picture speculation.
Here it is: Th5d7cAd3cQsJc4h6c8c9dTc6hQc8d9sJh5c7dAhAc9h2cKh5sJs8s7h2d 7s9c4dKd8hQd6d3sKs5h2hAs4s2sTs6sQhJd3h4cTd3dKc p1 p2 community p1 hand p2 hand winner Th7c 5dAd Qs Jc 4h 8c Tc One Pair High Card 1 5dAd 7c3c Jc 4h 6c 9d 6h One Pair One Pair 1 7c3c AdQs 4h 6c 8c Tc Qc Flush One Pair 1 AdQs 3cJc 6c 8c 9d 6h 8d Two Pair Two Pair 1 3cJc Qs4h 8c 9d Tc Qc 9s Flush Two Pair 1 Qs4h Jc6c 9d Tc 6h 8d Jh Straight Two Pair 1 Jc6c 4h8c Tc 6h Qc 9s 5c Flush High Card 1 4h8c 6c9d 6h Qc 8d Jh 7d One Pair One Pair 1 6c9d 8cTc Qc 8d 9s 5c Ah One Pair One Pair 1 8cTc 9d6h 8d click here Jh 7d Ac Straight One Pair 1 9d6h TcQc 9s Jh 5c Ah 9h Trips One Pair 1 TcQc 6h8d Jh 5c 7d Ac 2c Flush High Card 1 6h8d Qc9s does pokerstars have or nothing 7d Ah 9h Kh Straight One Pair 1 Qc9s 8dJh 7d Ah Ac 2c 5s One Pair One Pair 1 8dJh 9s5c Ah Ac 9h Kh Js Two Pair Two Pair 1 9s5c Jh7d Ac 9h 2c 5s 8s Two Pair High Card 1 Jh7d 5cAh 9h 2c Kh Js 7h Two Pair High Card 1 5cAh 7dAc 2c Kh 5s 8s 2d Two Pair One Pair 1 7dAc Ah9h Kh 5s Js 7h 7s Trips One Pair 1 Ah9h Ac2c 5s Js read more 2d 9c One Learn more here One Pair 1 Ac2c 9hKh Js 8s 7h 7s 4d One Pair One Pair 1 9hKh 2c5s 8s 7h 2d 9c Kd Two Pair One Pair 1 2c5s KhJs 7h 2d 7s 4d 8h Two Pair One Pair 1 KhJs 5s8s 2d 7s 9c Kd Qd One Pair High Card 1 5s8s Js7h 7s 9c 4d 8h 6d Straight One Pair 1 Js7h 8s2d 9c 4d Kd Qd 3s High Card High Card 1 8s2d 7h7s 4d Kd 8h 6d Ks Two Pair Two Pair 1 7h7s 2d9c Kd 8h Qd 3s 5h One Pair High Card 1 2d9c 7s4d 8h Qd 6d Ks 2h One Pair High Card 1 7s4d 9cKd Qd 6d 3s 5h As Straight High Card 1 9cKd 4d8h 6d 3s Ks 2h 4s One Pair One Pair 1 4d8h KdQd 3s Ks 5h As 2s Straight One Pair 1 KdQd 8h6d Ks 5h 2h 4s Ts One Pair High Card 1 8h6d Qd3s 5h 2h As 2s 6s Two Pair One Pair 1 Qd3s 6dKs 2h As 4s Ts Qh One Pair High Card 1 6dKs 3s5h As 4s 2s 6s Jd Flush Flush 1 3s5h Ks2h 4s 2s Ts Qh 3h One Pair One Pair 1 Ks2h 5hAs 2s Ts 6s Jd 4c One Pair High Card 1 5hAs 2h4s Ts 6s Qh 3h Td One Pair One Pair 1 2h4s As2s 6s Qh Jd 4c 3d One Pair High Card 1 As2s 4sTs Qh Jd 3h Td Kc Straight One Pair 1 4sTs 2s6s Jd 3h 4c 3d Th Two Pair One Pair 1 2s6s TsQh 3h 4c Td Kc 5d Straight 2020 no deposit instant bonus poker Pair 1 TsQh 6sJd 4c Td 3d Th 7c Trips One Pair 1 6sJd Qh3h Td 3d Kc 5d Ad Flush One Pair 1 Qh3h Jd4c 3d Kc Th 7c 3c Trips One Pair 1 Jd4c 3hTd Kc Th 5d Ad Qs Straight One Pair 1 3hTd 4c3d Th 5d 7c 3c Jc Two Pair One Pair 1 4c3d TdKc 5d 7c Ad Qs 4h One Pair High Card 1 TdKc 3dTh 7c Ad 3c Jc 6c Flush One Pair 1 3dTh Kc5d Ad 3c Qs 4h 8c One Pair High Card 1 Kc5d Th7c 3c Qs Jc 6c 9d High Card High Card 1 The first two columns show the pocket cards dealt to player 1 and player 2, respectively, and the fourth and fifth columns give their final hands, again respectively.
Immediately, the reader will recognize that there are 52!
Searching them all, even at supercomputer speed, would easily take longer than the life of the universe at a trillion stackings per second, the estimate is on the order of years.
So we cannot possibly look through them all.
We use in the discrete space of all orderings of the deck where the function to optimize is the number of cuts that win for the first player dealt.
Neighbors consist of the set of stackings obtainable by swapping pairs of cards, and we use random restarts to avoid local maxima.
Moreover, we parallelized the algorithm to run on aaccess to which is graciously provided by the UIC mathematical computer science department.
The result texas holdem primer lightning-fast results.
We compute the value of a deck by dealing out a game for each of the 52 cuts of the deck, and seeing how many times player 1 wins.
Then we compare with the value of all neighboring decks, where a neighbor is obtained by swapping any two cards in the deck.
If any such stacking has a higher value, we take the stacking with the highest value, and repeat the process.
Once we get to the point where no such neighboring deck has a higher value, we stop and output the final stacking.
Then we start the whole process over again with a new random deck, and stop when we find an optimal stacking or continue searching for more.
We leave that for future research.
We chose to implement this algorithm in C++.
In hindsight this was probably unnecessary, but we do admittedly wallow in pride when quoting statistics about how fast the program runs.
All of is available for free on.
We also include a text file containing the list of about 100,000 optimal stackings we found, and similar tables as above detailing the hands dealt for each cut.
In any case, we separated the code into three parts.
First we implemented a general steepest-ascent framework for any problem.
To give a pedantic yet unrealistic example, our posted code includes an instance of optimizing a polynomial function in one variable.
Second, we implemented a poker-hand evaluator and neighbor generator.
Otherwise, enlarging the lookup-table to cover seven-card hands would incur more than a 2,000 times increase in memory use, with only a speedup factor of 10 to 15 times.
And even so, on the supercomputer we can evaluate about 7 million seven-card hands per second, down from about 87 million five-card hands per second.
We gain the potential to go up to 87 million seven-card hands per second with the added memory footprint, but as it turns out we can find plenty of optimal stackings with the slower version.
Third, we parallelized the program using to run on 12 threads, increasing the runtime by a factor of 12 for sufficiently large problem sizes.
For smaller problem sizes, we see a speedup factor of roughly 10x, giving a reasonable 80% efficiency.
In the next section texas holdem primer detail the important aspects of the code.
Enter the Jungle Unfortunately, a primer on C++ is far beyond the scope of this blog.
An equally strong understanding of the algorithm itself can also be gained from our Python post onthough the program is admittedly much slower.
The general steepest ascent algorithm features a virtual C++ class called Position.
A position has three functions: valueneighborsand showthe last of which returns a string containing a textual representation of a Position.
Here are snippets of the code, taken from my hillclimb.
For a short example on how to use this framework, the reader should investigate our quarticopt.
The poker part required a bit of preprocessing to understand, but it is essentially a simple idea.
First, we note that even though there are a lot of poker hands, many are equivalent up to scoring.
But none of these hands is any better than another.
In other words, there are only 7,462 distinct five-card 2020 casino japan hands, which is a lot smaller than the some 2.
The poker scoring function hence takes as in put a five-card hand and computes a number between 1 and 7,462 representing the absolute score of the hand.
It does this through a combination of accessing a pre-generated look-up table for flushes and high-card hands, and a technique for the remaining hands.
The interested reader will see for more details.
It is quite novel, and results in a lightning-fast scoring algorithm.
We note that the next time we work on this problem, we will likely extend the code to work for any number of hands and any number of players, and replace the ugly assignment operators above with a few loops.
Moreover, we leave out an additional optimization from this code snippet for brevity.
Running the main application gives an output similar to the large table above for each discovered optimal stacking.
Parallelization A thorough introduction to parallel computation is again sadly beyond the scope of this post.
Luckily, we chose the arguably simplest possible library for parallelization:.
With OpenMP, the serial code looks almost identical to the parallel code, with the exception of a few sagely placed pragmas code annotations.
For instance, our main function parallelizes at the top-most level, so that each thread performs steepest ascent trials independently of the others.
From this perspective, the steepest ascent algorithm is.
In the code posted, we tried two different levels of parallelization.
First, we parallelized at the top-most level, as above.
The second way was at the finest level, so that all threads were cooperating to evaluate the score of a single deck and compute neighbors.
The former method turned out to be somewhat more efficient than the latter for large problem sizes.
Specifically, in 1,000 iterations of a steepest ascent, the former was 17% faster.
The speed increase is due to the significantly less system time overhead to schedule the threads, and the fact that evaluating a single deck is very fast, so at the end of every loop the thread synchronization time begins to outweigh the deck evaluation time.
However, for smaller problem sizes the efficiency seems to drop off at a slower rate.
We also include the full source code for the fine-granularity parallelization, with the mindset that it may perform better in other application domains, and it serves as an example implementation.
In particular, about one in six randomly chosen decks can be improved to an optimal stacking.
Specifically, in all of our tests we never witnessed a deck which required more than 15 steps to reach a local maximum.
We do not have a proof that this is upper bound is sharp.
Moreover, some decks were optimized with as few as 5 steps.
We note the possibility that some of these decks are redundant, but the probability of that happening is decidedly small.
We encourage the reader to try one out at a friendly neighborhood poker game.
And finally, here is a list of ten optimal stackings, so that casual readers can avoid downloading and navigating the directory.
We encourage the interested reader say, as a programming exercise to implement a steepest-ascent algorithm for optimal stackings in a different game, and post the results in a comment.
Some ideas for other games include poker variants like Omaha and Lowball, along with completely unrelated games like Blackjack or Rummy.
Might we even go so far as to investigate non-standard card games like a favorite during my youth?
A critical step in the analysis of most non-poker games would be to formalize what it means to force a win, or texas holdem primer the definition to make it sensible.
For instance, how hard is it to find optimal stackings for any given player when there are players?
More specifically, how fast does the complexity grow when you add more players?
Are there significantly more semi-optimal stackings, in which we allow ties for player 1?
What sorts of interesting patterns occur in optimal or semi-optimal stackings?
Many of these sorts of questions could be answered by minor modifications to the source code provided.
We leave that as an exercise for the click, until we find enough time ourselves to go through and find optimal stackings which we believe to exist for any reasonable number of players and any position of the winning player.
That is a very good question, and I think the answer is no.
In a sense, because these stackings are so plentiful we should expect that there may be some games in which all players but one have no hope, regardless of the cut.
Trying to avoid optimal stackings that just happen to come up in real games would actually make the deal less random, not more random.
The ability source play these stackings correctly requires knowledge that you are dealt an optimal stacking.
After all, many of the hands win with a low pair or even a high card, and others hit low flushes on the river with a single card on suit.
One thing that stacking theory does say is that cutting the deck is by no means fool-proof in preventing cheating.
Of course it takes far more work to stack an optimal stacking than one would be able to do surreptitiously at a table, but a weaselly player could swap out a deck with one that is optimally stacked in his favor.
This raises the obvious the question: what sort of slotomania not loading should we adopt that truly ensures no foul play?
Ten of Hearts, of course.
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