A Rook Game

Problem: Two players take turns moving a rook on an 8×8 chessboard. The rook is only allowed to move south or west (but not both in a single turn), and may move any number of squares in the chosen direction on a turn. The loser is the player who first cannot move the rook. What is the optimal play for any starting position?

rook-board

Solution: Take advantage of the symmetry of the board. If the rook is not on the diagonal, the optimal strategy is to move it to the diagonal. Then when the other player moves it off, your next move is to move it back to the diagonal. If your opponent starts their turn with the rook always on the diagonal, then you will never lose, and by the symmetry of the board you can always move the rook back to the diagonal. This provides an optimal algorithm for either player. In particular, if the rook starts on a square that is not on the diagonal, then player 1 can guarantee a win, and otherwise player 2 can.

Symmetry is one of the most powerful tools in all of mathematics, and this is a simple albeit illustrative example of its usage.

Want to make a great puzzle game? Get inspired by theoretical computer science.

Two years ago, Erik Demaine and three other researchers published a fun paper to the arXiv proving that most incarnations of classic nintendo games are NP-hard. This includes almost every Super Mario Brothers, Donkey Kong, and Pokemon title. Back then I wrote a blog post summarizing the technical aspects of their work, and even gave a talk on it to a room full of curious undergraduate math majors.

But while bad tech-writers tend to interpret NP-hard as “really really hard,” the truth is more complicated. It’s really a statement about computational complexity, which has a precise mathematical formulation. Sparing the reader any technical details, here’s what NP-hard implies for practical purposes:

You should abandon hope of designing an algorithm that can solve any instance of your NP-hard problem, but many NP-hard problems have efficient practical “good-enough” solutions.

The very definition of NP-hard means that NP-hard problems need only be hard in the worst case. For illustration, the fact that Pokemon is NP-hard boils down to whether you can navigate a vastly complicated maze of trainers, some of whom are guaranteed to defeat you. It has little to do with the difficulty of the game Pokemon itself, and everything to do with whether you can stretch some subset of the game’s rules to create a really bad worst-case scenario.

So NP-hardness has very little to do with human playability, and it turns out that in practice there are plenty of good algorithms for winning at Super Mario Brothers. They work really well at beating levels designed for humans to play, but we are highly confident that they would fail to win in the worst-case levels we can cook up. Why don’t we know it for a fact? Well that’s the P \ne NP conjecture.

Since Demaine’s paper (and for a while before it) a lot of popular games have been inspected under the computational complexity lens. Recently, Candy Crush Saga was proven to be NP-hard, but the list doesn’t stop with bad mobile apps. This paper of Viglietta shows that Pac-man, Tron, Doom, Starcraft, and many other famous games all contain NP-hard rule-sets. Games like Tetris are even known to have strong hardness-of-approximation bounds. Many board games have also been studied under this lens, when you generalize them to an n \times n sized board. Chess and checkers are both what’s called EXP-complete. A simplified version of Go fits into a category called PSPACE-complete, but with the general ruleset it’s believed to be EXP-complete [1]. Here’s a list of some more classic games and their complexity status.

So we have this weird contrast: lots of NP-hard (and worse!) games have efficient algorithms that play them very well (checkers is “solved,” for example), but in the worst case we believe there is no efficient algorithm that will play these games perfectly. We could ask, “We can still write algorithms to play these games well, so what’s the point of studying their computational complexity?”

I agree with the implication behind the question: it really is just pointless fun. The mathematics involved is the very kind of nuanced manipulations that hackers enjoy: using the rules of a game to craft bizarre gadgets which, if the player is to surpass them, they must implicitly solve some mathematical problem which is already known to be hard.

But we could also turn the question right back around. Since all of these great games have really hard computational hardness properties, could we use theoretical computer science, and to a broader extent mathematics, to design great games? I claim the answer is yes.

[1] EXP is the class of problems solvable in exponential time (where the exponent is the size of the problem instance, say n for a game played on an n \times n board), so we’re saying that a perfect Chess or Checkers solver could be used to solve any problem that can be solved in exponential time. PSPACE is strictly smaller (we think; this is open): it’s the class of all problems solvable if you are allowed as much time as you want, but only a polynomial amount of space to write down your computations. 

A Case Study: Greedy Spiders

Greedy spiders is a game designed by the game design company Blyts. In it, you’re tasked with protecting a set of helplessly trapped flies from the jaws of a hungry spider.

A screenshot from Greedy Spiders.

A screenshot from Greedy Spiders. Click to enlarge.

In the game the spider always moves in discrete amounts (between the intersections of the strands of spiderweb) toward the closest fly. The main tool you have at your disposal is the ability to destroy a strand of the web, thus prohibiting the spider from using it. The game proceeds in rounds: you cut one strand, the spider picks a move, you cut another, the spider moves, and so on until the flies are no longer reachable or the spider devours a victim.

Aside from being totally fun, this game is obviously mathematical. For the reader who is familiar with graph theory, there’s a nice formalization of this problem.

The Greedy Spiders Problem: You are given a graph G_0 = (V, E_0) and two sets S_0, F \subset V denoting the locations of the spiders and flies, respectively. There is a fixed algorithm A that the spiders use to move. An instance of the game proceeds in rounds, and at the beginning of each round we call the current graph G_i = (V, E_i) and the current location of the spiders S_i. Each round has two steps:

  1. You pick an edge e \in E_i to delete, forming the new graph G_{i+1} = (V, E_i).
  2. The spiders jointly compute their next move according to A, and each spider moves to an adjacent vertex. Thus S_i becomes S_{i+1}.

Your task is to decide whether there is a sequence of edge deletions which keeps S_t and F disjoint for all t \geq 0. In other words, we want to find a sequence of edge deletions that disconnects the part of the graph containing the spiders from the part of the graph containing the flies.

This is a slightly generalized version of Greedy Spiders proper, but there are some interesting things to note. Perhaps the most obvious question is about the algorithm A. Depending on your tastes you could make it adversarial, devising the smartest possible move at every step of the way. This is just as hard as asking if there is any algorithm that the spiders can use to win. To make it easier, A could be an algorithm represented by a small circuit to which the player has access, or, as it truly is in the Greedy Spiders game, it could be the greedy algorithm (and the player can exploit this).

Though I haven’t heard of the Greedy Spiders problem in the literature by any other name, it seems quite likely that it would arise naturally. One can imagine the spiders as enemies traversing a network (a city, or a virus in a computer network), and your job is to hinder their movement toward high-value targets. Perhaps people in the defense industry could use a reasonable approximation algorithm for this problem. I have little doubt that this game is NP-hard [2], but the purpose of this article is not to prove new complexity results. The point is that this natural theoretical problem is a really fun game to play! And the game designer’s job is to do what game designers love to do: add features and design levels that are fun to play.

Indeed the Greedy Spiders folks did just that: their game features some 70-odd levels, many with multiple spiders and additional tools for the player. Some examples of new tools are: the ability to delete a vertex of the graph and the ability to place a ‘decoy-fly’ which is (to the greedy-algorithm-following spiders) indistinguishable from a real fly. They player is usually given only one or two uses of these tools per level, but one can imagine that the puzzles become a lot richer.

[2]: In the adversarial case it smells like it’s PSPACE-complete, being very close to known PSPACE-hard problems like Cops and Robbers and Generalized Geography

Examples

I can point to a number of interesting problems that I can imagine turning into successful games, and I will in a moment, but before I want to make it clear that I don’t propose game developers study theoretical computer science just to turn our problems into games verbatim. No, I imagine that the wealth of problems in computer science can serve as inspiration, as a spring board into a world of interesting gameplay mechanics and puzzles. The bonus for game designers is that adding features usually makes problems harder and more interesting, and you don’t need to know anything about proofs or the details of the reductions to understand the problems themselves (you just need familiarity with the basic objects of consideration, sets, graphs, etc).

For a tangential motivation, I imagine that students would be much more willing to do math problems if they were based on ideas coming from really fun games. Indeed, people have even turned the stunningly boring chore of drawing an accurate graph of a function into a game that kids seem to enjoy. I could further imagine a game that teaches programming by first having a student play a game (based on a hard computational problem) and then write simple programs that seek to do well. Continuing with the spiders example they could play as the defender, and then switch to the role of the spider by writing the algorithm the spiders follow.

But enough rambling! Here is a short list of theoretical computer science problems for which I see game potential. None of them have, to my knowledge, been turned into games, but the common features among them all are the huge potential for creative extensions and interesting level design.

Graph Coloring

Graph coloring is one of the oldest NP-complete problems known. Given a graph G and a set of colors \{ 1, 2, \dots, k \}, one seeks to choose colors for the vertices of G so that no edge connects two vertices of the same color.

coloring

Now coloring a given graph would be a lame game, so let’s spice it up. Instead of one player trying to color a graph, have two players. They’re given a k-colorable graph (say, k is 3), and they take turns coloring the vertices. The first player’s goal is to arrive at a correct coloring, while the second player tries to force the first player to violate the coloring condition (that no adjacent vertices are the same color). No player is allowed to break the coloring if they have an option. Now change the colors to jewels or vegetables or something, and you have yourself an award-winning game! (Or maybe: Epic Cartographer Battles of History)

An additional modification: give the two players a graph that can’t be colored with k colors, and the first player to color a monochromatic edge is the loser. Add additional move types (contracting edges or deleting vertices, etc) to taste.

Art Gallery Problem

Given a layout of a museum, the art gallery problem is the problem of choosing the minimal number of cameras so as to cover the whole museum.

artgallery

This is a classic problem in computational geometry, and is well-known to be NP-hard. In some variants (like the one pictured above) the cameras are restricted to being placed at corners. Again, this is the kind of game that would be fun with multiple players. Rather than have perfect 360-degree cameras, you could have an angular slice of vision per camera. Then one player chooses where to place the cameras (getting exponentially more points for using fewer cameras), and the opponent must traverse from one part of the museum to the other avoiding the cameras. Make the thief a chubby pig stealing eggs from birds and you have yourself a franchise.

For more spice, allow the thief some special tactics like breaking through walls and the ability to disable a single camera.

This idea has of course been the basis of many single-player stealth games (where the guards/cameras are fixed by the level designer), but I haven’t seen it done as a multiplayer game. This also brings to mind variants like the recent Nothing to Hide, which counterintuitively pits you as both the camera placer and the hero: you have to place cameras in such a way that you’re always in vision as you move about to solve puzzles. Needless to say, this fruit still has plenty of juice for the squeezing.

Pancake Sorting

Pancake sorting is the problem of sorting a list of integers into ascending order by using only the operation of a “pancake flip.”

panackesortJust like it sounds, a pancake flip involves choosing an index in the list and flipping the prefix of the list (or suffix, depending on your orientation) like a spatula flips a stack of pancakes. Now I think sorting integers is boring (and it’s not NP-hard!), but when you forget about numbers and that one special configuration (ascending sorted order), things get more interesting. Instead, have the pancakes be letters and have the goal be to use pancake flips to arrive at a real English word. That is, you don’t know the goal word ahead of time, so it’s the anagram problem plus finding an efficient pancake flip to get there. Have a player’s score be based on the number of flips before a word is found, and make it timed to add extra pressure, and you have yourself a classic!

The level design then becomes finding good word scrambles with multiple reasonable paths one could follow to get valid words. My mother would probably play this game!

Bin Packing

Young Mikio is making sushi for his family! He’s got a table full of ingredients of various sizes, but there is a limit to how much he can fit into each roll. His family members have different tastes, and so his goal is to make everyone as happy as possible with his culinary skills and the options available to him.

Another name for this problem is bin packing. There are a collection of indivisible objects of various sizes and values, and a set of bins to pack them in. Your goal is to find the packing that doesn’t exceed the maximum in any bin and maximizes the total value of the packed goods.

binpacking

I thought of sushi because I recently played a ridiculously cute game about sushi (thanks to my awesome friend Yen over at Baking And Math), but I can imagine other themes that suggest natural modifications of the problem. The objects being packed could be two-dimensional, there could be bonuses for satisfying certain family members (or penalties for not doing so!), or there could be a super knife that is able to divide one object in half.

I could continue this list for quite a while, but perhaps I should keep my best ideas to myself in case any game companies want to hire me as a consultant. 🙂

Do you know of games that are based on any of these ideas? Do you have ideas for features or variations of the game ideas listed above? Do you have other ideas for how to turn computational problems into games? I’d love to hear about it in the comments.

Until next time!

Classic Nintendo Games are NP-Hard

The heroes and heroines of classic Nintendo games.

Problem: Prove that generalized versions of Mario Brothers, Metroid, Donkey Kong, Pokemon, and Legend of Zelda are NP-hard.

Solutionhttp://arxiv.org/pdf/1203.1895v1.pdf

Discussion: Three researchers (including Erik Demaine, a computer science professor at MIT famous for his work with the mathematics of origami) recently finished a paper giving the complexity of a number of classic Nintendo games (the ones I loved to play). All are proven NP-hard, some are shown to be NP-complete, and some are PSPACE-complete. Recall, a problem is NP-hard if an NP-complete problem reduces to it, and a problem is NP-complete if it’s NP-hard and also in NP. As we have just posted a primer on NP-completeness and reduction proofs, this paper is a fun next step for anyone looking for a more detailed reduction proof. A pre-print is available for free on arXiv, and it’s relatively short and easy to read. I’ll summarize his result here, and leave most of the details to the reader. Each game is “generalized” to the task of determining whether one can get from a “start” location to a “finish” location. So the decision problem becomes: given a finite level of a game, can the player move from the starting location to the finishing location? All of the reduction proofs are from 3-Sat, and they all rely on a common framework which can be applied to any platform game. Pictorially, the framework looks like this:

The framework for reducing 3-Sat to platform games.

The player starts in the “start” gadget, which allows one to set up initial state requirements. For instance, in Super Mario Brothers, the start provides you with a mushroom, and you cannot get to the finish without being able to break blocks by jumping under them, which requires the mushroom power-up. Each “variable” gadget requires the player to make a variable assignment in such a way that the player can never return to that gadget to make a different decision. Then each “clause” gadget can be “unlocked” in some way, and each clause gadget can only be visited by the player once the player has chosen a satisfying variable assignment for that clause. Once the player has visited all variable gadgets, he goes to the “check in” area, and can travel back through all of the clauses to the finish if and only if he unlocked every clause. The crossover of the paths in the picture above requires another gadget to ensure that the player cannot switch paths (the details of this are in the paper).

For example, here is the variable gadget described in the paper for The Legend of Zelda, a Link to the Past:

The variable gadget for A Link to the Past.

Note that here we require Link has the hookshot, which can grapple onto chests, but has limited reach. The configuration of the chests requires him to choose a path down one of the two columns at the bottom, and from there he may never return.

Here’s another example. In the classic Super Mario Brothers game, a possible clause gadget is as follows.

The clause gadget for the original Super Mario Brothers.

Note that if the player can only enter through one of the three columns at the top, then the only thing he can do is kick a red koopa shell down so that it breaks the blocks, unlocking the way for Mario to pass underneath at the end. Note that Mario cannot win if he falls from the top ledge (since must always remain large, he can’t fit through a one-tile-high entryway). Further details include the hole at the bottom, in which any stray koopa shell will necessarily fall, but which Mario can easily jump over. We recommend reading the entire paper, because it goes into all of the necessary details of the construction of the gadgets for all of the games.

Future Work

We note that there are some parts of the paper that only got partial results, mostly due to the variation in the game play between the different titles. For instance, the original Super Mario Brothers is known to be NP-complete, but the added ability to pick up koopa shells in later Super Mario Brothers games potentially makes the decision problem more complex, and so it is unknown whether, say, Super Mario World is in NP. We will summarize exactly what is known in the table below. If readers have additions for newer games (for instance, it’s plausible that Super Mario Galaxy could be adapted to fit the same construction as the original Super Mario Bros.), please leave a comment with justification and we can update the table appropriately. I admit my own unfamiliarity with some of the more recent games.

Super Mario Brothers:

Game Title NP-hard in NP PSPACE-complete
Super Mario Bros.
Lost Levels
Super Mario Bros. 2
Super Mario Bros. 3
Super Mario Land
Super Mario World
Land 2: 6 Golden Coins
Super Mario Land 3
Yoshi’s Island
Super Mario 64
Sunshine
New Super Mario Bros.
Galaxy
New Super Mario Bros. Wii
Galaxy 2
Super Mario 3D Land

Legend of Zelda:

Game Title NP-hard in NP PSPACE-complete
The Legend of Zelda
The Adventure of Link
A Link to the Past
Link’s Awakening
Ocarina of Time
Majora’s Mask
Oracle of Seasons
Oracle of Ages
The Wind Waker
Four Swords Adventures
The Minish Cap
Twilight Princess
Phantom Hourglass
Spirit Tracks
Skyward Sword

Donkey Kong:

Game Title NP-hard in NP PSPACE-complete
Donkey Kong
Donkey Kong Country
Donkey Kong Land
Country 2: Diddy’s Kong Quest
Land 2
Country 3: Dixie Kong’s Double trouble!
Land 3
Donkey Kong 64
Donkey Kong Country Returns

Metroid:

Game Title NP-hard in NP PSPACE-complete
Metroid
Metroid II: Return of Samus
Super Metroid
Fusion
Prime
Prime 2: Echoes
Prime Hunters
Prime 3: Corruption
Other M

Pokemon:

Game Title NP-hard in NP PSPACE-complete
All Games
Only trainers

Optimally Stacking the Deck – Kicsi Poker

A Puzzle is Born

Sitting around a poolside table, in the cool air and soft light of a June evening, a few of my old friends and I played a game of Texas Hold ‘Em. While we played we chatted about our times in high school, of our old teachers, friends, and, of course, our times playing poker. We even reminisced lightly about some of the more dishonest people we played with (whom we invited for want of a larger payout), but especially we recalled the times we caught them cheating. We witnessed dealing from the bottom of the deck, two people teaming up with each other to fix hands, and some very blatant deck stacking.

Often having a player different from the dealer cut the deck (usually the player to the dealer’s right, as the cards are dealt starting to the left) absolves the dealer of any foul play. This is tacit at the poker table, and thinking of this on that midsummer evening, a puzzle popped into my head:

For a fixed set of rules in a game using a standard deck of 52 cards, is it possible to optimally stack a deck, such that no matter where the deck is cut, a specific player always wins?

Of course, my mind was initially set on answering this for Texas Hold ‘Em (with a fixed number of players). A public suggestion that such a stacking might be possible halted activity at the table as each person thought about it, attempting to find some quick flaw. Of course, it occurred to everyone at the table that if a stacking existed, it very well could have happened in one of the many hands we collectively played over our (albeit short) lifetimes. Finally, Forrest, one of the men at the table, simply said, “This is blowing my mind.”

At this point I thought it might actually be a puzzle worth working on.

Preliminary Thoughts

This question can be asked of any card game, so we’ll start with as simple a game as we can: Kuhn Poker. There are two players and three cards: a King, a Queen, and a Jack. Each player is dealt one card, and with after a single round of betting, the player with the highest card wins the pot. From a game-theoretic perspective, there are some interesting results on this game; specifically, the first player has multiple optimal betting strategies, while the second player has only one. Despite this, if the second player plays optimally, he should expect to win at a rate of 1/18th of a betting unit per hand. That is, of course, assuming the first player does not stack the deck.

For this combinatorially tractable game we may analyze each of the six cases quickly. We will describe a stacking of the deck by the first letters of its cards in order, with the left end of the string being first card dealt. For example, “KQJ” represents the unique stacking for which the first dealt card is a King, the second a Queen, and the third a Jack. Then, a cut is specified by positions between cards, indexed from zero as follows: “0K1Q2J.” Here cutting at 0 means the deck is left as is, which is usually an option given to the cutter. Cutting at 1 turns the deck into QJK, and at 2 we get JKQ. Here is our analysis of the game:

KQJ: cut at 2, receiving a King against a Queen
KJQ: cut at 1, receiving a Queen against a Jack
QKJ: cut at 1, receiving a King against a Jack
QJK: cut at 2, receiving a King against a Queen
JKQ: cut at 0, receiving a King against a Jack
JQK: cut at 0, receiving a Queen against a Jack

So no such stacking is possible for Kuhn Poker. While this is disheartening to the cheater, the honest card player may now rest assured that with a random cut, no cheater is secure.

Note that this same argument extends to any game in which each player is dealt only one card, and we do not allow a tie to result from a good stacking (i.e. the second player cuts aiming for a tie if he cannot cut to win). The only such one-card poker game I know that is actually played is Indian Poker (also known as Blind Man’s Bluff), and two-player Indian Poker admits an identical analysis. Suppose we have a stacking of a deck of n cards which is optimal for the first player dealt at any cut. Then card k strictly beats card k+1 for all positions k of cards in the deck. By transitivity of “beats,” we have that the first card in the deck beats the last card (in fact, every card beats the last card), and hence the second player may cut at position n-1. Since this forces the first player to receive the last card (the worst card in the deck), the second player will not lose, contradicting our original assumption that the deck was optimally stacked.

So no such optimal stacking exists for Indian Poker either. Despite the easy answer to this problem, we cannot assume the same result holds for Texas Hold ‘Em. Even if a cut results in the first player being dealt low cards, there are sufficiently many rules so as to allow the first player to win (for example, he could have a low flush or straight). So let us try to invent a poker with sufficiently many rules to allow optimal deck stacking, if it is at all possible.

K Poker

We define a new game called K Poker. The K stands for “kicsi,” (pronounced kee-chi) the Hungarian word for “small,” and also for “Kun,” the family name of its creator (also Hungarian).

K Poker is a two player game. A K Poker deck contains nine consecutively ranked cards of the same suit; we use deuce through ten. Each player is dealt two cards face down from the top of the deck. Dealing is done by the poker standard, to the dealer’s left, one card at a time. After a round of betting, three community cards are placed face up in the center for all players to use. After another round of betting, one further community card is dealt, and after a final round of betting, the players reveal their cards and the best five-card poker hand wins. Dealing then switches, and the next hand is dealt.

Actually, the betting scheme here is irrelevant, since we simply care about which player wins in the end. For our purposes, we might as well just say each player is dealt two cards and there are four community cards which both players may use to make a hand. But we might as well define a complete playable set of rules, so there it is. The first player dealt will hereby be referred to as player one, while the second player dealt will be player two.

Note that there are only two hands in K Poker: high card and straight (or, since all cards are the same suit, a straight flush, but this name is inappropriately extravagant). A player receives a straight about one fifth of the time, while high card comparison is the default.

This simple game, while not realistically enjoyable, has theoretical interest to us: optimal stackings exist for both players.

One optimal stacking is very nice: a sorted, ascending K Poker deck is optimal for player two. On one level this is obvious, because the second player will always receive the last card dealt. Since there are no burn cards, the next four community cards, being in ascending order form player two’s straight. Things just happen to work out in the cases where player two has no straight. As an exercise to the reader, prove (or verify) that with an ascending K Poker deck, player one can never receive a straight, no matter the cut. Then, notice that for both cuts which give player one the ten, player two receives a straight. Otherwise, player two obviously receives the highest card.

It is intractable by brute force human effort to find the other optimal stackings, and it is not obvious whether others exist. Trying some ‘orderly’ stackings (descending, alternating, etc.) do not work. So while we could build up an army of lemmas and work toward some theorem, we find it more cost-efficient for this particular game to conduct a computer search, which may later guide our efforts in proving lemmas and theorems.

We chose a nine-card game specifically with this in mind. In particular, there are merely 9! = 362,880 different permutations of the deck. Since each deck has nine cuts, we may further reduce our search to those stackings which begin with a deuce. This reduces the number of decks to 8! = 40,320. Since we may easily determine who wins on each cut of a stacking (bringing us back up to 9! computations), a brute-force search is reasonable.

We Came, We Sought, We Cheated

This time, our excuse for coding in Mathematica is its lightning fast look-up table for permutations. The Mathematica notebook is posted online (along with the code for our followup post on optimally stacking Texas Hold ‘Em) at this blog’s Github page. Since the code is not exactly scintillating material, we leave the reader to download it at his own discretion.

We found nine distinct (up to cuts) optimal stacking for player one, and merely two for player two. We list them here, where the leftmost digit is on the top of the deck:

Player one:
(2, 5, 10, 7, 3, 8, 4, 9, 6),
(2, 5, 10, 7, 3, 9, 4, 8, 6),
(2, 5, 10, 7, 4, 9, 3, 8, 6),
(2, 6, 8, 3, 9, 4, 7, 10, 5),
(2, 6, 9, 3, 8, 4, 7, 10, 5),
(2, 6, 9, 4, 8, 3, 7, 10, 5),
(2, 6, 10, 7, 3, 8, 4, 9, 5),
(2, 6, 10, 7, 3, 9, 4, 8, 5),
(2, 6, 10, 7, 4, 9, 3, 8, 5)

Player two:
(2, 3, 4, 5, 6, 7, 8, 9, 10),
(2, 5, 6, 7, 8, 9, 10, 4, 3)

Wonderful! These are the only optimal stackings, though we admit if a stacker is willing to accept draws, there are many more admissible stackings.

First, notice that there are many more optimal stackings for player one than player two! In fact, counting cuts, player one has 81 optimal stackings, while player two has a mere 18. Though player one still only has one fifth of a percent chance to get such a stacking at random, that is still better than player two’s mere 0.04%. This does not necessarily say that the game is unfairly balanced for player one, though. There may exist more stackings which admit a larger proportion of wins to player two than player one. And, of course, by the symmetry of switching dealers every hand, the game must be balanced across multiple hands.

Second, notice that when we make a slight modification of the rules, when we burn a single card before dealing the community cards, the count of optimal stackings is reversed: player two has nine while player one has just two. In addition, stacking the deck in descending order is optimal for player one.

Forgetting the sinister tricks we might play on friends, this has exciting implications. Adding complexity gave us more optimal stackings! Further, there is a glimmer of symmetry to the optimal stackings. It remains to be seen whether there is a limit to how much complexity we may have while still maintaining the existence of optimal stackings. We conjecture there is such a limit.

We could try adding more cards or allowing for more poker hands, and seeing if we can maintain optimal stackings. However, the factorial growth rate of the number of deck stackings would limit us almost immediately. Clearly, and sadly enough, a brute force search for optimal stackings in two-player Texas Hold ‘Em would take far longer than the life of the universe. So that is out. In addition, the sparsity of optimal stackings in the case of nine cards implies that a Monte Carlo (random sampling) search is unlikely to produce results. While Monte Carlo is still a viable option, perhaps augmented with some local search methods, we find it more likely that the patterns in K Poker generalize to patterns in Texas Hold ‘Em. We leave both tantalizing problems open for future posts.

Until then!