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.