grindrewind

Tyler Cowen at The Marginal Revolution attributes Magnus Carlsen’s ascendance to chess world champion to his “nettlesomemess.” This doesn’t mean he picks his nose or makes annoying sounds while he plays. Rather, it means he plays moves that are tricky to answer correctly.

Carlsen has the highest nettlesomeness score by this metric, because his creative moves pressure the other player and open up a lot of room for mistakes.  In contrast, a player such as Kramnik plays a high percentage of very accurate moves, and of course he is very strong, but those moves are in some way calmer and they are less likely to induce mistakes in response.

Game theory has for the most part ignored chess, focusing as it does on situations where what you do depends on what the other guy does. In chess, in theory at least, you could just survey the position and play the best move without a care in the world for what your opponent might be thinking. According to this account, all the judgment and calculation that goes into making a chess move is just a technical process best left to the experts.

I’ve noticed that some game theorists seem to take a sort of pride in being uninterested in chess (and other games of complete information, like Go). If game theory doesn’t have anything interesting to say about these games, that’s more like a blind spot than a triumph. If we want to learn something meaningful about real world decision making, the process of figuring these things out certainly matters.

I’m excited about the idea of nettlesomeness as it applies to chess, as well as poker, which game theory has historically had more to say about. But I’ve been thinking about nettlesomeness as it applies to a less heralded game: tic-tac-toe.

If both sides play well, a game of tic-tac-toe should end in a tie. This is true no matter how Xs starts the game. No opening move is so good that it forces a win, nor is any so bad that it leads to a loss. From the traditional game theory perspective then all the moves are equally good. Xs can, however, control Os’ available paths to a tie. This is where nettlesomeness comes in.

The most “natural” first move is in the center. After that, Os has eight moves, although it’s effectively two moves: he can go in the corner or the edge. It turns out that going on the edge loses, while going in the corner ties.

Xs can also start on the edge. In that case Os can tie by going in the center, any corner, or the opposite edge. The only moves that lose are the edges next to the initial X.

Finally Xs can open in the corner. Then Os must go in the center to hold a tie. All other moves allow Xs to force a win.

This analysis suggests a definition of nettlesomeness.

Nettlesomeness = Losing moves / Possible moves

Let’s call this measure N. Going in the center has an N of .5, the edge has an N of .25, and the corner has an N of .875. According to this metric, even though the center seems to be the natural move, the corner is actually a much better move.

It’s far from a perfect measure of course. If our opponent agrees that going in the center is the most natural move, that’s exactly what he’s likely to do when we go in a corner. Even though he has only one move to save the game, it’s an obvious move. And since it’s tic-tac-toe we can hardly expect to win anyway unless we’re playing a child or an animal. But in more challenging games like chess, this principle – not only fewer saving moves, but less obvious saving moves – is a big deal.

Extra: if you want to work on your tic-tac-toe game xkcd has a nice map.

Edit: fixed edge diagram.