Let’s take a generic game, and ask the natural question before embarking on what may become a very long family game night.
How complex is this game?
Mathematically, there are two basic ways to answer. First, you might give the size of the state space: the number of possible arrangements of pieces that can arise during gameplay. In tic-tac-toe, for example, we envision every possible board scenario, and count them up.
Second, you might give the size of the game tree: the number of different sequences of moves that can unfold from the first turn to the last. (Since most board scenarios appear multiple times on the game tree, this is much larger than the state space.)
Now, let’s swap in that generic game for a specific one: chess.
I am told by the experts that there are, crudely speaking, two ways to play.
First, positional play: focus on maneuvering yourself into an advantageous position, with your pieces controlling the center, and your king tucked safely away. Try to wind up with more material. Don’t worry too much about specific checkmates.
Second, combination play: seek an elegant and tightly planned sequence of moves. (“A series of staggering blows,” as Irving Chernev put it). Don’t worry too much about material. Sacrifice as needed to set up specific checkmates and devastating attacks.
I realized recently that the two approaches to chess roughly correspond to the two measures of complexity.
Positional players seek out sweet spots in state space, arrangements that they evaluate as generally advantageous. Combination players, meanwhile, search for ideal branches in the game tree. Each way of playing seems to embed a style of thinking, a high-level model of gameplay itself.
I find that these lessons echo far beyond chess.
Writing is a game of combining words. Are you a positional-type writer, focused on identifying and gathering good material without worrying too much how it will fit together? Or are you a combination-type writer, polishing and honing sequences of words to achieve the perfect effect?
(I grew up as a combination-type writer. But this can manifest as a kind of inefficient perfectionism. I’ve spent recent years trying to become more positional.)
Or what about cooking? Perhaps you’re a positional-type cook, filling your fridge with tasty ingredients and trusting that they’ll come together into something delicious. Or perhaps you’re a combination-type cook, following precise and planful recipes.
(I’m a positional cook by nature. But I know good recipes achieve effects that my hackish improvisations cannot, so I’m trying to get better at combination play.)
And then there’s hosting a party—a game of combining guests. Are you a positional-type planner, inviting a bunch of interesting people and letting the sparks fly? Or are you a combination-type planner, paying careful attention to specific interpersonal dynamics such as whether Ms. X gets along with Dr. Y in the presence of Prof. Z?
(I, uh… haven’t been doing a lot of hosting lately… but maybe soon!)
Any combinatorial world is too vast to explore in its entirety. But I find it helpful to catalogue the different ways to play.
Great article.
You’ve pretty well got it right on the chess except that winning material probably belongs under the combinational header rather than positional. A typical book on positional chess might have chapter names like the centre, king safety, outposts, weak pawns, open files, half open files, black and white squares, space,… I’m very much a positional player. The positional player’s credo is that if you reach a position where you have strengths and your opponent has weaknesses, the combinations will suddenly appear as if by magic, with t( combinations resulting in things like checkmates and material gains.
Oh, and in your extension to other fields, the combinatorial approach to writing is probably what we call poetry.
Ah, good to know! I like that credo. The point of “position play” (in any discipline) is that from a sufficiently good position, there are lots of winning combinations.
I do think poetry lends itself to “combinatorial play,” though I can imagine a more “position play” kind of poet (who just gathers lots of lovely phrases and images without worrying about precisely how they’ll fit together).
I really like this generalized way of thinking how type of player game styles can reflect in other life choices. I teach a class on Ethics for Professional Analysts and a discussion of this type could generate lots of good discussion as we explore views about situational ethics. I particularly relate this to work by game theorists Professors Steven Brams and Alan Taylor wrote on Fair Division. Any thoughts?
I’m not familiar with the book – sounds interesting! My memory is that fair division procedures for cake-cutting get very complicated, very fast; with two players it’s “I cut, you choose,” with three players it’s something kind of complicated, and with four players it’s prohibitively complicated. But I’m sure there are good practical heuristics, which I suppose is what the book discusses.
What a beautiful article! It reminds me of “Goals versus Systems” blog-post of Scott Adams. Analogies are often imperfect, but a positional-type person (in any endeavor) sounds like a systems-oriented person and a combination-type like a goal-oriented one. I have sometimes struggled with choosing between the two. When not too pressed for time, I have tended to be a positional-type person. I sometimes wonder if a combination-type person would turn a positional-type person if the “clock” is removed (even in a game of chess)!
This game is call tic tac toe, right?
A tic tac toe game to be exact.
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Each game type tends to instill a thinking style and high-level gaming paradigm.