Boardwalk Philosophy Page

by Mark G. Meyers

Of the Finite Versus the Infinite, or the Mind’s Rise to Infinity

I was a novice when I first created these games as a teenager. I made attempts at forging practical strategic concepts in the face of various random card arrangements, or “deals”. By way of those concepts, I began to orient my mind to the playing field.

With the passage of time, I felt that I had developed some degree of familiarity with the games, and I established a list of what I felt would be the most commonly useful concepts in strategy.

Over the years, I wondered how much my skill could improve. I experienced a number of plateaus in how many good options I could see. At each plateau, I felt like I was seeing all that could be seen, but over time, I would then be surpassed by new breakthroughs, producing even greater spheres of strategic options.

I have now come to see why my development as a player may ultimately be open ended. I have come to see that there may never be a best or most perfect play, such as what may or may not be possible by way of a perfect game-playing computer program.

For any real game situation, there is the chance of success in each specific strategic response, but there is also the variation in the amount of success gained. Both the odds and the amount of success must be considered together, where options range widely. Even the goal of the game itself is just not cut-and-dried, such as winning at solitaire versus outscoring one’s opponent in competition, or by way of what is called “tossing” versus attempting to score a perfect game. (Tossing is where a player chooses greater odds of success by targeting a lesser end point score.)

After 30 years, I have documented a sufficient series of strategic paradoxes to support the position that, in the end, the master of the game may be the player who sincerely sees more than any such thing as a best possible play. I now see so many interesting and provocative options that I find myself amazed by their diversity. In Boardwalk Solitaire, I have come to experience a love and appreciation as a player in the face of its expanding universe.

On Luck Versus Skill in Boardwalk Solitaire

The deal is defined as the arrangement of cards at the beginning of the game.  We will presume any arrangement we consider to be random.  The following paragraphs are to be taken collectively as the most profound truths of Boardwalk Solitaire games.

The number of possible deals is so large, that the odds are very strongly against any one player experiencing the same game twice in a lifetime (or of being able to recollect such an occurrence if it did).  Forming a precise, premeditated strategy is not regarded herein as humanly possible. 

In these games, the element of randomness also gives us the element of luck, and luck cannot be removed.  Because of this, there is no ultimate law by which the player may know precisely what to do. The only way to know what precisely to do is to develop a strategy by which one performs plays that are in perfect accordance with a random deal.  A random (or indiscernible) strategy appears to be needed in order to play perfectly.  Vertigo is used as a Boardwalk term to name the effect produced by the deal's capacity to disorient the player on strategy (all values are transient).

When playing identical deals in competition, a player who knows what to do may be beaten by another player who skillfully draws upon greater possibilities.  But in order to beat the game itself, and not just another player, one must first obtain a position whereby they can accurately predict its outcome.  Where the game itself has been beaten, its outcome has been made known and it should no longer be regarded as a Boardwalk game. The most direct example of knowing the outcome in any attempt to beat the game would be shown by a player's ability to win (the given variation) on every attempt. 

Hysteresis in Boardwalk is a word for saying, "variability in outcome by luck alone". It may be possible to produce an argument for beating the game on this basis; that when the player has exercised absolutely every last bit of skill that is possible, only the hysteresis will remain.  Where hysteresis is the only variation from a totally predictable outcome, then the variation in outcome has been reduced to being purely the product of luck, and one might consider arguing the variation as no longer being a Boardwalk game.  Care needs to be taken when producing the statement that all non-hysteresis has been removed, and to say so surely.  Prior to this, the argument should not be honored.

This game offers the player an opportunity to apply great skill, and even to make variations requiring greater skill (such as with a larger deck of cards). My experience as inventor of the games, over the years, is now that the sense of strategy in Boardwalk teases the player into attempting to beat the game itself. Since by its definition it does not appear possible to achieve this result as a player and still call it a Boardwalk game, the game itself is offered as the equivalent of a modern, Buddhist koan, or as a question which cannot be answered.  The difference here is that there may be, but by definition, for a living variation it is never presently known if such a thing can be found.

If a computer player (or game-playing algorithm) were developed such that it could calculate everything, then that would produce a perfection standard. I think we have to bear in mind on this that such an algorithm would have to be able to quantify each possible arrangement of cards, in its mathematical/strategic value with absolute precision. There will be a diversity of play possibilities in tandem with a diversity of end goals, and each play possibility may have its own array of possible end goals, each with its own measurable value. This seems to present an astronomical number of calculations. A par algorithm is the idea of a computer program that would play the game with such absolute mathematical thoroughness and precision, that it would serve as a mathematical perfection standard. One would only be able to beat such a program, on average, less than half of the time. Where an opponent to par does better than that, then at that point the par algorithm would be found to be less than what it was held up to be (and in need of revision).

The Complexity of the Game

There are four existing variations in Boardwalk Solitaire; Hopscotch, Lucky Seven, Sqatsi and Cheshire. Hopscotch is the simplest and hardest to win, and Cheshire is the most complex and easiest to win (when played very carefully). In fact, the purpose of Cheshire is to illustrate a point. Where it can be won every time, it is intended to portray itself as mundane.

Boardwalk Solitaire introduces a genre of gaming, pitting skill against random luck, each at its own height. From a design perspective, this is the intended nature of the game. Ideally, it is like golf. The range of all of human skill is to fill a boat that swims in a sea of random possibilities. Recall that in these games there is only one human player, and so a human opponent is not available to implement the level of skill for the game.

The problem that appears to arise with attempts at incorporating an opponent into a deck of cards is the narrowness of the opponent’s range in potential skill, from its floor to its ceiling. To make a simpler opponent for the human player, the skill floor is lower, but the skill ceiling also tends to be badly reduced. When making a variation that can match greater human skill, the skill floor also tends to rise. As an example, Lucky Seven seems about half as complicated to play as Sqatsi. Note that the rules of the two are not greatly different, but that strategy in Sqatsi is significantly more involving than in Lucky Seven.

In order to step up to Sqatsi, the player must face twice the complexity. Any reward for doing so comes in the form of a game with a higher skill ceiling. While it is worth noting that the skill floor for Sqatsi still serves one who plays with lesser thought, there is still the business of having to cope with the complexifying aspect of Sqatsi over Lucky Seven, which is to have to cope with two ascending walks in tandem.

Enter two categories of ideas for achieving greater complexity while remaining within the constraints of the three existing entities; upper walk, lower walk and hand, as well as within the three existing game types; Hopscotch, Lucky Seven and Sqatsi . 1) A larger deck of cards could be used, and 2) cards colored on both sides by suit could be used to thrust the player into a significantly greater world of “information”, or that which can be known by looking at the board.

In the first idea, 5 each of 20 face values could be dealt as "100-card Boardwalk" (such as with a 9-column upper walk, a 10-column lower walk and a 5-card hand). There would be 14 cards face-up on the deal to carefully deliberate upon. Two 5-card row 1 stacks could be cleared against a 10-column lower walk before having to file upon the upper. 5 upper walk stacks could be led from simultaneously against 10 lower walk columns, giving rise to an exponential development in multiple-lead-stack possibilities (off of a 9-column upper walk). The skill floor for 100-card Boardwalk would (I would imagine) have to be higher than for its 52-card counterpart, but I think its ceiling would rise considerably more. The skill floor goes up because of the number of ranks simultaneously involved.

With the second idea, the player would always be able to see all of the instances of each suit in the deck, whether face up or face down. The element of (deterministically) available information rises drastically. In 52-card variations, each face down card is transformed from 1 in 13 possible values to 1 in 52. With 100-card variations, each face down card goes from 1 in 20 possible values to 1 in 100. Conversely, possible values for face down cards can be narrowed based upon how many of the same suit is already showing, for those who wish to count them. The really fantastic thing about this “visible suits” idea is that it would have a negligible effect upon the skill floor – the additional information would not have to be used, whereas its effect on the skill ceiling would be considerable. In the context of this writing, it could be a design freebie.

The architectural vision of Boardwalk can be difficult to implement against, such that the game will be both simple and skillful in its range. While 52-card Boardwalk has been shown to be capable of pitting significant skill against 52 cards worth of random possibilities, it may also be too limiting when wanting to push the potential skill to man’s height. 4 each of 13 ranks just isn’t enough. These new ideas, such as a larger deck of cards and visible suits make great strides in doing so. Of particular interest to me is the idea of visible suits, since it would not affect the skill floor while greatly increasing its ceiling. I am also very interested in the idea of 100-card variations, as this would facilitate an exponential expansion in complex, upper walk multi-stack lead possibilities, the likes of which novices would not have to apply. Nevertheless, it would appear to increase the skill floor to some degree, by way of a longer lower walk and a larger number of face values to cope with simultaneously.

The goal in these genre of games is quite grand. It is to provide for the entire ship of man’s skill to sail within an infinite sea of possibilities. I know of no other game which attempts to do this, except perhaps for any other game that may dream of doing so. In solitaire, this is the finite man against infinite possibilities. In competition, where the same arrangement is dealt to multiple players, man may play against man, where each swims in the same sea.