## Sunday, January 1, 2017

### 2016 Holiday Challenge: Results for Challenge 1, Rock Paper Scissors

I'm not sure what the psychology of this is, but my readers mostly threw Rock, and that means I lost bigly, because my throw was Scissors!

The first person to beat me on their first try was reader Bill M.—but I have reason to know that Bill already owns a copy of Word Games 5, so the prize for Challenge 1 goes to reader Joanie, who threw Rock just an instant after Bill did. Congratulations, Joanie! A copy of Word Games 5 is on its way.

For what it's worth, I chose my throw randomly, with probabilities ⅓—⅓—⅓. According to classical game theory, this the optimal strategy (the so-called "Nash equilibrium"). Of course, the theory assumes that both players are rational, whereas in real life this is seldom the case. You'd be leaving a lot of money on the table if you played ⅓—⅓—⅓ against a little kid.

In Challenge 1, we only played a single round. But rock/paper/scissors is usually played as a series of multiple rounds, creating a so-called "supergame." The mathematical theory of supergames is much more complicated than for single games, especially when the underlying game is non-zero-sum. This is due to the presence of such factors as signaling. You might have heard of the famous Prisoner's Dilemma tournament of 1980, created by political scientist Richard Axelrod. In Axelrod's tournament, game theorists and others submitted computer programs that repeatedly played Prisoner's Dilemma against one another. The winning strategy, submitted by Anatol Rapoport, was TIT-FOR-TAT. This profound strategy makes an initial friendly move and thereafter simply repeats whatever its opponent does. These matters are still lively topics of discussion in evolutionary biology (reciprocal altruism), economics, and other fields. Here is a bit more on TIT-FOR-TAT.

Bennington College held a tournament like Axelrod's when I was there, with each science professor advising a team of students. A team was allowed to submit more than one entry to the tournament. My team won the tournament by using the dastardly meta-strategy of coordinating our entries. We submitted one primary strategy, plus several copies of a secondary strategy. We pictured the primary strategy as being like an ant, and we pictured the secondary strategies as being like aphids, whose purpose was to be milked by the ant. Aphids were programmed to play a predetermined initial sequence of moves that would identify them to the ant. At first, until the ant was confident that it was dealing with an aphid, the ant played TIT-FOR-TAT. But as soon as the ant was confident that it was dealing with an aphid, the ant started milking it by playing an insanely aggressive strategy, which the aphid would do nothing to counter.  Of course, only the ant knew that there were aphids in the tournament available for milking, so only the ant racked up those easy points. The other competitors were understandably annoyed when we revealed how we'd won, but they had to acknowledge that we were within the rules (which hadn't anticipated coordination of strategies). So the trophy—an ant farm, as it turns out—went to our team.

My kids won the 2014 Holiday Challenge, which was a game of Hangman. This year, they wanted to play rock/paper/scissors, so I wrote a computer program they could play it on. The program is a simple Bayesian updater. The more you play, the better the program learns your tendencies. At each stage, it uses what it knows to choose the best throw. Here is an animation showing how the computer makes sharper and sharper predictions over time for what your next throw will be.

I would guess that over the long run, the best you can do against this program is to play ⅓—⅓—⅓ resulting in a stalemate. But the computer probably has exploitable weaknesses during a run of five to ten games.

I'll end with some "news you can use"—an article called "How to Beat Anyone at Rock Paper Scissors," published by the World RPS Society. (There had to be a society.) The article has some pretty good tips! In any case, the theory of RPS seems to be better documented online than the theory of Hangman. You could also check out this article summarizing the findings of an experiment in which people played rock/paper/scissors for money.

P.S. And in the category of "anything can be exciting," I give you this 4-minute video of the final round of the 2007 US Championships of Rock/Paper/Scissors.