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# Game-theoretic Approach to Tradition

Imagine a simple one-shot coordination game between two players who don't know each other and can't communicate with each other. Each can pick one of two cards, either blue one or green one. If both choose the same card they each get \$100. If they pick different cards they get nothing.

Without being able to make a deal (the players can't communicate), to guess each other's likes and dislikes (the players don't know each other), without playing the game repeatedly (the players play one game and never see each other again) and without an obvious Schelling point (the blue card isn't more conspicuous than the green one or vice versa) the players are expected to do no better than chance. In other words, they are going to win or lose with 50% probability.

Schelling point, also called focal point, is a strategy that happens to be conspicuous for whatever reason. If two people want to meet in New York but are unable to let each other know about the exact meeting place, waiting at Grand Central Station is a good guess. Grand Central Station is the Schelling point in the game of meeting in New York. If we change our game to use a bright red card and a plain white card instead of the blue and the green one the players, with no better information to work with, will reason that the red card is much more conspicuous than the white one and thus the other player is more likely to chose it. Therefore, they will chose the bright red card themselves.

Now let's modify the game.

We'll use the blue and the green card again. However, this time the entire game will be a TV show. Each week, two new contestants will be chosen, they will play the game and they will walk away either with \$100 each or with nothing.

Would the act of watching the show influence strategies used by the players?

At first sight no, nothing has changed. For the particular pair of players it's still a one-shot game. They still can't communicate in advance. They still don't know each other. There's still no obvious Schelling point… or, wait a second, is there?

Well, the fact that they both players watched the previous episodes provides a way to establish one. Let's say in the previous 10 episodes of the TV show both parties chose blue and won. Blue card can now become a focal point. If you are to pick a card, you can assume that your peer have seen the previous episodes and will think of choosing the blue card as of the winning strategy. Therefore, you should pick the blue card as well.

(If an actual experiment was to be performed, we could simplify the game by pre-recording the 10 episodes where people chose blue and win. All the tests subjects would watch the same 10 episodes and make their choice based on that.)

It's not hard to see that what I am trying to do here is to model "tradition" in game-theoretic way. It's not hard to see that if the TV show existed for real, some kind of tradition, either that of picking the blue card or that of picking the green card, would emerge. That way, participants would be able to win every time. If there was no tradition, they would win only half of the time. By the way, it's also not hard to see that the tradition is arbitrary: If the show was run in two countries, one could settle up on the strategy of picking the blue card, while the other one could very well decide to go with the green card.

The important question, however, is: Is the Schelling point established by tradition different from other types of Schelling points? If the two are experimentally indistinguishable, this entire exercise would be just a matter of linguistics with no relation to the real world.

So here's the experiment to test relevance of this concept:

Pairs of experimental subjects will be asked to play the above game. Half of the pairs will encounter externally imposed Schelling point (e.g. red vs. white card), the other half will use dull blue/green cards but they will have a Schelling point established by tradition (i.e. they will each watch 10 episodes of the show in which picking the blue card was the winning strategy). Moreover, the experimenter will try to bribe one of the players with \$150 to choose the non-Schelling-point card. Will the players in tradition-based game show more reluctance to accepting the bribe?

And my prediction is that yes, they will.

Note that this prediction is at odds with a naive game-theoretic approach: Accepting the bribe raises expected reward from \$100 to \$150. Therefore, subjects should chose to accept the bribe.

The reason why I believe there will be measurable difference is that people are wired-up to play along with the tradition. This was shown in 1950's by Asch conformity experiments. The willingness to conform is by no means absolute but it's definitely measurable. Around 25% of the test subjects in Asch's experiment conformed with what was obviously a wrong answer.

Thus, subjects having to chose between red and white card could do so without feeling the peer pressure. However, those who watched the previous episodes would feel the pressure to conform.

Further, I predict that following modifications to the experiment would make subjects reject the bribe more often: Understanding that they are being filmed and will appear on the next episode of the show. If the subjects watch the previous 10 episodes together and are allowed to talk about it. If the subjects are made aware that they will have to interact with other players of the game in the future.

The reason for this prediction is that a player who accepts the bribe not only breaks the convention but also makes winning in the future games less probable. If the tradition says you should pick the blue card and you pick the green one you introduce uncertainty about the Schelling point. If, for example, there was a newcomer to the game who have seen only the last episode, they would have no idea whether the tradition prefers the blue or the green card. And once again, if the tradition is broken the chance of winning in the future games drops from 100% to 50%. The double crime you've just committed was vandalizing a public property (the established Schelling point) and corrupting the youth. And the prospect of the society disapproving of your action will make the strategy of accepting the bribe look much less desirable.

Assuming that my predictions will prove to be true, what's all of that good for?

First, some real-world moral conventions look like they may have been ossified traditions in the above sense. They seem to be completely arbitrary. But imagine the TV show going on for generations and everyone picking the blue card. Surely, picking of the blue card would be seen as the proper thing to do. If all my forebears have chosen the blue card who am I to do otherwise? By picking the green card I would spit on their memory and on all they ever stood for. And so on. The framework introduced above could provide a way to look at such conventions from game-theoretic perspective.

More importantly though, by experimentally testing the measures to establish a tradition (e.g. watching the previous shows together, complaining about the offenders etc.) we can get at least some insight on how the social trust is built. What, after all, is social trust other than an expectation that a stranger on the street will behave according to a specific set of norms? And understanding social trust is crucial for our survival in today's ever more fractured social environment.

January 24th, 2017