The Incompleat Gamester



How strategy outwits uncertainty

by David Parlett

A paper presented at the 11th annual colloquium of the Board Game Studies Association, Lisbon, 2008
Copyright © 2008-2017 by David Parlett

In the course of a simultaneous display... I [once] said to one of my opponents, "Tell me, Mr McMahon, how long did it take you to learn to play Chess so badly?" He replied, "Sir, it's been nights of study and self-denial" (Gerald Abrahams, Brains in Bridge, 1962)

Like Gaul, games are anciently and popularly divided into three parts: games of skill such as Chess and Go, games of chance such as Snakes & Ladders and Roulette, and games of mixed chance and skill such as Backgammon and Bridge. Such categorisation is patently inadequate.

It is slightly more adequate to demolish the divisions and regard chance and skill as polar opposites of a single continuum, so that any given game (jeu) - or any given instance of one (partie) - may be regarded as involving x per cent skill and (100-x) per cent chance.

But then skill and chance are themselves inadequate terms. Games involve many different forms of chance, some of which are perceived rather than real. A more appropriate term for this end of the spectrum is uncertainty, or unpredictability as to the outcome of a game. All games by definition involve a degree of uncertainty, for if the outcome of a game were ever entirely certain or predictable there would be no point in playing it.

At the opposite end of the spectrum lies the antidote or counter to uncertainty, which is the degree, if any, to which you can control or at least influence the outcome of a game. The opposite of uncertainty is better characterised as controllability rather than skill, as skill itself is not an atomic property: there is no such thing as a single, universal "skill at games" but rather many different types of skill. People tend to play those games for which their particular talents suit them, or, if their talent is not one of controllability, to which they are most attuned by temperament.

My interest lies in exploring the elements of uncertainty or types of chance that may be encountered in games, and the corresponding elements of skill or types of controllability that may be employed to counter them. This exploration takes me into specialised disciplines, such as mathematics, psychology, and pedagogy, in which I have absolutely no qualifications or expertise. I write purely as a games enthusiast and inventor, and can only hope that my comments might be found to have some bearing on the classification, invention, and appreciation of games.

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Aspects of chance


The primary, most fundamental and oldest embodiment of uncertainty is the occurrence of randomising events such as the cast of lots or dice, for which reason many suppose games to have originated in the practice of divination. Equally fundamental, but historically more recent, is the randomisation of an initial position, which is classically embodied in the dealing of playing-cards from a shuffled pack.

Compulsion (vs choice)

An element of uncertainty occurs in randomising games that give you no choice of play. A classic example is the Indian ancestor of our Snakes & Ladders (Chutes & Ladders). Why such a game should continue to exist is well explored by Salen & Zimmerman in Rules of Play (2004, p.179). I would paraphrase their argument by suggesting that Snakes & Ladders may be regarded as an overlap between, on one hand, the playing of games, and, on the other, the performance of plays, a point which I think would have appealed to Johan Huizinga. Compulsion also overlaps with divination, in that it is an essential property of Fate. The opposite of compulsion is choice, or free will, which provides an essential opportunity for the exercise of skill.

Information : perfect/imperfect, complete/incomplete, certain/uncertain

Careless commentators tend to lump randomisation together with imperfect information, but in fact there is a significant difference between the types of informational imperfection. Taking Backgammon and Bridge as exemplars:

The differences are significant in that they call for distinctive skills in order to exert some degree of control over their outcome.

(Further distinctions may be drawn between perfect and imperfect information, certain and uncertain information, and complete and incomplete information; but let this suffice for the moment.)

Inequality and indeterminacy

Inequality is associated with a random and therefore indeterminate opening array such as the initial distribution of cards. Its effect is reduced by sufficiently increasing the number of deals that constitute a whole game. It is also inherent in asymmetric board games like Hnefatafl and Fox & Geese, where the two players have different forces and different objectives; but here, too, it is easily overcome by playing an even number of games and alternating the positions. You might say that it is a feature of most combinatorial games, at least in so far as it is usually advantageous to move first.

Here it might be relevant to note that one of the ways in which games evolve is by deliberate alteration of the rules in order to reduce potential inequalities of players' experience. If you subscribe to the online newsgroup you will be aware of a recent interest in devising forms of Chess with indeterminate openings, such as Fischerchess (aka Chess960). In these variants the opening array is unknown in advance, but is fully open before play begins so there is no lack of information. What is lacking here is an experience of playing with a particular configuration of pieces. On these grounds one might propose novelty as a carrier of uncertainty.

Opacity vs clarity

An element of uncertainty, or at least uncontrollability, is induced by the opacity of a game. By opacity I mean the opposite of clarity, a property I think Robert Abbott was the first to point out in a 1976 edition of Games & Puzzles magazine. In an article entitled "Under the Strategy Tree" Abbott writes:

Clarity is essentially the ease with which a player can see what is going on in a game... [It] has nothing to do with simplicity, or even with elegance. Edward de Bono's L-Game is elegantly minimal - it uses only four pieces and is played on a board of only 4x4 squares. It is not, however, clear. I find it very hard to picture what the board will look like when I turn my L over, I find it harder still to visualize my opponent's responses, and it's impossible for me to look ahead to my next move. A game can be simple yet lack clarity, and conversely a game can be complicated but still clear. Playing a game soon reveals its degree of clarity. The greater the clarity of a game, the farther you can see into it, and therefore the greater its depth for you.

Perception + complexity = perplexity

Chance and skill may differ according to whether they are perceived from the inside, subjectively by a player, or from the outside, objectively by an observer. As Salen and Zimmerman put it, "the perception of randomness is more important than randomness itself". They quote the example of Chinese Checkers with four or more players:

As the game unfolds..., the centre... becomes crowded with a seemingly random arrangement of pieces... even though every single move... is the result of a player making a strategic choice about where to play next. If you closed your eyes and opened them only when it is your turn to move, it might seem like the board is merely reshuffling itself, particularly in the middle... game, when the centre area is most crowded. This feeling of randomness is only an illusion, however, as there is no formal chance mechanism in the game. [Hypothetically] logical players... wouldn't feel any randomness: they could look at the board and immediately trace every move back to a series of strategic decisions. However, for human players, this feeling of randomness is an important part of what makes the game fun to play... [T]he feeling of randomness creates a sense of open-ended possibility and players are rewarded for taking advantage of chance configurations on the board... Seeing a pattern emerge out the chaos that allows you to jump a piece back and forth all the way across the entire length of the game board is a moment of wonderfully meaningful play.

Another subjective perception of chance comes into play when a Chess master plays a novice. Both have perfect information as to the moves and changing positions, but whatever the weaker player does, the stronger sees through it to what led him to make that move, what he has in mind, what situation it might ultimately lead to, and how to circumvent or take advantage of it. The weaker, however, will often be baffled by the master's unforeseen move with what are to himself unforeseeable implications. As far as the novice is concerned, he might as well be confronting a completely random move determined by the roll of a die, and exhibiting no perceptible past cause or future effect. As Philip Ross puts it ["The Expert Mind", Scientific American, August 2006):

The feats of chess masters have long been ascribed to nearly magical mental powers. This magic shines brightest in the so-called blindfold games in which the players are not allowed to see the board...

Ross's reference to magic reminds me of the converse of perceived but unreal chance, namely that of perceived but unreal skill. An example is provided by Dennis Tedlock in his Introduction to Stuart Culin's Games of the North American Indians:

We might call it a "game of chance," which is what Culin calls similar games in this book, but that expresses the point of view of an observer. Meanwhile, the participants constantly think in terms of strategy, pitting their wishes against chance in momentary acts of magic, which is what we all find ourselves doing when we throw dice. / [A] paradox of the Zuñi game of wooden dice is that, technically, it is not what Culin calls a "game of dexterity", and yet the players do try slightly different ways of handling the sticks, as if they could influence the outcome of a throw. [...] So if there is any dexterity here, it must remain on the side of magic [...]
Tedlock, Dennis, Introduction to Culin, S, Games of the North American Indians - Volume 1, Games of Chance (Nebraska University Press, 1992), p. 23


We might mention, for completeness, the existence of chance factors that are extrinsic to the game. I first became aware of these in my schooldays, when my friend's mother raised some objection to our playing Chess on a Sunday. "It's a game of skill", we protested, "not a game of luck". "Well", she countered, "You're lucky if you win, aren't you?". It was at this point that I realised that (a) luck and chance are not the same thing, and (b) you are indeed lucky if in a Chess tournament you happen to be drawn against a weaker player, or one whose opening you have just been mugging up, or one who happens to be feeling unwell at the time. Of such chance factors, no more need be said.

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Aspects of skill


The best way to exercise control over the outcome of a game is to cheat. This pits maximum controllability on your behalf against baffling uncertainty on the part of your victim, who, if sufficiently gullible, may look upon your constant success as a form of magic. And why not - when you consider that cheating and magic are little more than differing interpretations of the same conjuring trick?

Memory - past and future

The importance of memory is obvious, but its ramifications are subtle and it would be impossible to outline them here without going into disproportionate detail.

Abrahams, in The Chess Mind, holds that memory is easily overestimated, especially if it is taken to imply remembering a number of standard openings. Very long retentiveness, indeed, is "often a concomitant of minds lacking in originality". More important than the consciously recollected "is that set of mental habits which smooths the action of the mind", a capability best described as "technique", and most relevant in the endgame. Of greatest significance, however, is what he refers to as "holding in [one's] mind a clear conception that is in part constituted by the memory of what will have happened, i.e. what has already happened as a mental event".

Here we find ourselves talking about the forward visualisation involved in combinatorial games like Chess. What do we mean by forward visualisation? At first sight we mean looking ahead to our next move and to the sequence of moves likely to result from it. This has been described as examining the branches of the strategy tree, and is something that computers are very good at. In human terms it seems like a form of memory, only in reverse, in that we are following a sequence forward into the future rather than backward into the past. I have always described this ability as mental projection, in that we are projecting ourselves into the future. Abrahams refers to it simply as vision. But in fact future recall or reverse memory is a pretty good term for it, as experiments have shown that exactly the same parts of the brain light up as when it is engaged in tracing backward memories. (See Jessica Marshall, "Future recall: your mind can slip through time", in The New Scientist, 24 March 2007, page 36-40.)


Against this, however, must be set the discovery that Chess masters do not normally go down this analytical route on a step-by-step basis. Capablanca, on being asked "How many moves do you see ahead?", is said to have replied "Only one - but it's always the best one". As Philip Ross observes ["The Expert Mind: Studies of the mental processes of chess grandmasters", Scientific American, August 2006]:

He thus put in a nutshell what a century of psychological research has subsequently established: much of the chess master's advantage over the novice derives from the first few seconds of thought. This rapid, knowledge-guided perception, sometimes called apperception, can be seen in experts in other fields as well. Just as a master can recall all the moves in a game he has played, so can an accomplished musician often reconstruct the score to a sonata heard just once. And just as the chess master often finds the best move in a flash, an expert physician can sometimes make an accurate diagnosis within moments of laying eyes on a patient.

How do they do this? They see total situations as in a photographic memory, to such an extent that it seems to an outside observer - or to their hapless opponents - more like a stroke of intuition than a process of cerebral analysis and future recall. Ross notes that it was in 1894 that Alfred Binet, co-inventor of the first intelligence test, hypothesised that Chess masters achieved an almost photographic image of the board, but he soon concluded that the visualisation was much more abstract, resembling, rather, the same kind of implicit knowledge that the commuter has of the stops on a subway line. The expert relies not so much on an intrinsically stronger power of analysis as on a store of structured knowledge, enabling him to reconstruct any particular detail at will by tapping a well-organized system of connections. A weaker player, confronted with a difficult position, may calculate for half an hour, often looking many moves ahead, yet miss the right continuation, whereas a grandmaster sees the move immediately, without consciously analysing anything at all. In brief, experts rely more on structured knowledge than on analysis.

Ross goes on to invoke the theory of information chunking developed by [Herbert A.] Simon and [William] Chase, of Carnegie Mellon University. Simon explained the masters' reconstruction of Chess positions with the aid of a model based on meaningful patterns called chunks, enabling them to manipulate vast amounts of stored information that would be expected to strain the working memory beyond its normal capacity to contemplate more than seven items at a time.

An alternative explanation is offered by Lois D Isenman in her paper on Intuition. [Toward an understanding of intuition and its importance in scientific endeavour, Lois D. Isenman, 1997.]

Intuition as a bridging function brings the power of the unconscious into conscious thought. Through intuition, the unconscious with its vast memory banks, its associative accessing system, its speed, and its ability to process multiple items in parallel, greatly enriches the ability of conscious mental activity to manipulate logic and construct empiric tests... As dreams demonstrate, the unconscious frequently communicates in the language of symbols. In symbol formation, each object can be represented by multiple associative categories. In symbolic expression, any one of these aspects can stand for the whole item; however, symbols often simultaneously encode a number of different levels, presenting a richly textured and very often surprising understanding of the object under consideration. In the unconscious, in effect, each item is categorized by all its different component parts, as well as its descriptive, situational, and affective associations [5]. Intuitions very frequently come through to awareness in symbolic form and tend to share in the rich and unexpected quality that characterizes unconscious mental processes... Associative processing, an important component of symbol formation, plays a central role in intuition whether or not intuition is expressed in consciousness in symbolic form.

Intuition may be closely allied to the skills of deduction and inference required of intelligent card games. Does it also shade into elements of extra-sensory perception as might be the case in the game of Pelmanism, also known as Memory? If so, are we also encroaching on the borders of what some might categorise as "magic"?

Inference and deduction

When I was a teacher in my early twenties I played a lot of Chess. At one school most of the staff played Kriegspiel, even those members who did not normally play Chess. I soon discovered, to my surprise, that I was better at Kriegspiel than at Orthochess. Then I noticed that the annual Kriegspiel tournament was invariably won not by the strongest Chess players but by the strongest Bridge players. It was obvious that one of the skills demanded by Kriegspiel was that of deduction or inference as to the positions of the playing pieces; that this was a skill particularly demanded of card-players; and that it might therefore be appropriate for me go in for card games rather board games.

In reading the literature on games I also became aware that Chess players tend to look down on card-playing on the grounds that cards are not games of perfect information, with the further implication that games of perfect information require more skill than games of imperfect information, which are therefore to be equated with games of chance. Mortimer Collins, in Attic Salt (1887), writes:

There are two classes of men, those who are content to yield to circumstances, and who play Whist; and those who aim to control circumstances, and play Chess.

But of course this is nonsense. Information is not absent from strategic card games: rather, it is released gradually as cards are played or announcements made, and much of the information that has not yet been revealed is to be deduced or inferred - or even "intuited" - from that which has. The acquisition of information is as much the goal of strategy in strategic card games as the positional moves made as a result of the knowledge acquired. Indeed, in the higher trick-taking games positional moves may be made specifically for the purpose of acquiring information, even at the expense of loss of material - a device equivalent to the gambit at Chess.

Specific and peculiar skills

Another question from personal experience. My favourite abstract board game is the game of Pentominoes originally proposed by Solomon Golomb and sometimes referred to as Golomb's Game. Why do I enjoy this game so much more than games of the Chess / Draughts variety? I would say that it involves what I call the packing skill. It's interesting that I happen to be very good at efficiently packing suitcases, loading excessive amounts of luggage into the car when going on holiday, and finding new ways of rearranging my expanding collection of books and games without taking up much more space than they did when I started.

In what way does Pentominoes differ from (say) Chess or Draughts? My first observation is that it is a game of placement rather than movement. It starts with an empty board and play proceeds with each in turning placing a piece but not thereafter moving it. The same applies to another game I enjoy, namely Reversi (Othello). On these grounds, I often wonder whether I might have had some aptitude for Go, also a game of placement rather than movement, had I only discovered it early enough in life.

This is one example of how the classification of games may relate to a classification of different types of skill involved in playing them, and perhaps also taste and temperament. It is not surprising that some people will play only word games, and some only war games or fantasy games.

Types of intelligence

I have long been seeking a classification of mental skills that might be applicable to games classification. In other words, can we classify games by reference to the skills required in playing them rather than directly by reference to the contents of the games themselves? Eric Solomon tells me that this was the basis of a booklet he was planning some years ago, but found too difficult to follow through. The most promising line of enquiry that I have discovered derives from Howard Gardner's concept of multiple intelligences.

Gardner distinguishes seven types of intelligence as follows:

I will comment on these in a different order, which I think more relevant to their application to games. (From Intelligence Reframed: Multiple Intelligences for the 21st Century by Howard Gardner, published by Basic Books, 1999.)

Spatial intelligence: the potential to recognise and manipulate the patterns of wide space (those used, for instance, by navigators and pilots) as well as the patterns of more confined areas (such as those of importance to sculptors, surgeons, chess players, graphic artists, or architects).

Comment.Obvious in most board games, with the possible exception of Mancala; irrelevant to most card games (even Patience) except those designed to imitate board-game activity.

Logical-mathematical intelligence: the capacity to analyse problems logically, carry out mathematical operations, and investigate issues scientifically.

Comment. Is this involved in the type of forward thinking relevant to most abstract board games? Does it relate to the deduction/inference skill of most card games? If not, how does deduction/inference fit into Gardner's schema? Should we not also posit some kind of creative intelligence?

Musical intelligence: skill in the performance, composition and appreciation of musical patterns.

Comment. I'm not personally acquainted with any musical games, but it is worth noting the frequent association of skill at Chess with advanced musical skills. Many good musicians are good Chess-players, and vice versa. In medieval universities, music and mathematics were closely linked.

Linguistic intelligence: sensitivity to spoken and written language, the ability to learn languages, and the capacity to use language to accomplish certain goals.

Comment. At first sight this might appear to be relevant only to word games. But consider what the essence of language is: it is the ability to encode our multi-dimensional experience of the world and our interaction with it into a linear stream of a limited number of discrete sounds, from 20 to 50 according to the language we use. This seems to me clearly related to both the logical-mathematical and the musical intelligence.

Interpersonal intelligence: a capacity to understand the intentions, motivations and desires of other people and, consequently, to work effectively with others.

Comment. In other words, a theory of mind. If we convert the phrase "work effectively with others" into "compete effectively against others", we find this obviously fundamental to the skills required to play any strategic game against a live opponent. An interesting sidetrack in this day and age, of course, is how relevant this ability is to playing against computer software. Interpersonal intelligence is especially relevant to all intelligent card games.

Intrapersonal intelligence: a capacity to understand oneself, to have an effective working model of oneself - including one's own desires, fears and capacities - and to use such information effectively in regulating one's own life.

Comment. Poker. Need I say more?

Bodily-kinaesthetic intelligence: the capacity to use one's whole body or parts of the body (like the hand or the mouth) to solve problems or to fashion products.

Comment. Obviously relevant to outdoor sports and to games of dexterity, manipulation, and hand-eye coordination. It would appear to be related to spatial intelligence.

Creative intelligence?

There must be such a thing as creative intelligence, but I'm not quite sure where it fits into Gardner's scheme of things. I would take it to be at least related to, if not a form of, the skill of inference already mentioned. Abrahams remarks, in Brains in Bridge:

What distinguishes the player of any of the best-known card games from the player of one of the main board games is that the former frequently analyses, whereas the latter always synthesises...

Synthesising is a form of creativity, and inference a form of inductive reasoning, classically tested in Abbott's celebrated game of Eleusis - which, pace Abrahams, is not a board game but a card game. (See also Robert Harris, An Introduction to Creative Thinking).

Intelligence is not the same as skill

Interesting as the concept of multiple intelligences may be, I feel that it doesn't entirely answer my enquiry into the skills involved in game-playing; for a skill is not the same as an intelligence: rather, it is the application of an intelligence, which is not only a skill in itself but also introduces another potential element of differentiation into the subject.

In brief, I have still not been able to track down a classification of mental skills as distinct from the mere intelligences on which they may be based.


Arthur C Clarke's Third Law of Prediction states "Any sufficiently advanced technology is indistinguishable from magic". Might we not say the same of any sufficiently advanced intelligence?

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