Price of Privacy in the Keynesian Beauty Contest

Last year, Hadi Elzayn and I took a course in the Economics department about beliefs and learning in game theory with Annie Liang.
At the end of the class, we had to write a paper, and Hadi and I decided to look at what happens in a game when the players are worried that their action reveals some sensitive private information. If players’ response to this fact is to inject some random noise into their action, this will affect the players’ utilities and the equilibrium of the game. We wrote our paper and submitted it to EC and it was accepted! This is my first paper with only student authors and checks off an academic bucket list item of publishing a course project (I guess this was kind of my last opportunity, since I’m also done with coursework!).

The game we look at is a variation of the Keynesian Beauty Contest. In his General Theory, Keynes describes a game by way of analogy: a newspaper runs a beauty contest and prints pictures of the faces of 100 women (of course). Readers of the newspaper mail in with a selection of six faces. The reader who wins is the one whose six choices were the most popular choices. Therefore, in playing this game, an entrant needs to think about which faces they find the most beautiful, but also which faces everyone else might find the most beautiful. This is called the first-order belief about other players. The player also needs to wonder about everyone else’s first-order beliefs. That is, what does everyone think everyone thinks about the faces. These are, naturally, called the second-order beliefs. Keynes posits that people don’t generally go past the second-order of beliefs, but maybe the most clever among us might go up to a third- or fourth-order. However, in theory, we do need to worry about these things and have to consider equilibria consistent with this infinite hierarchy of beliefs.

In the formalism of the game we work with, in the absence of the privacy concern, there are some nice results about the equilibria. Therefore, our task isn’t to derive these from scratch but rather to see whether these results still exist in the modified setting. It’s possible that they don’t, and by adding a little bit of noise, players can knock the original game off equilibrium in a really bad way. Fortunately, we show that this isn’t the case. We show that the trade-off from adding noise (gaining some privacy, losing some utility in the original game) has a nice form and there is an equilibrium where players add a particular amount of noise to their private information and then play as if that were their true value. Since everyone knows that everyone knows that everyone knows that (and so on) people are doing this, we can easily relate our equilibrium to the one in the original game.

As a bonus, we give derivations for all the results in the case where there are finitely many players, since the existing results (as is typical in economics) only exist where there are infinitely many agents. You can check out the paper on the arXiv here.