SIGEST

Abstract

Suppose three people walk into a room from three different entrances. Before they enter, a hat is placed on each person. Each hat is either red or blue, but the wearer of the hat doesn't know its color. They then see each other and their hats for the first time, and need to guess simultaneously whether their own hat is red or blue. (No mirrors allowed!) Each person can either guess “red,” “blue,” or “pass.” They win a prize collectively if they all are correct, including at least one correct guess and the remainder passes. What is the optimal percentage of times they can win? There is an obvious strategy for 50%: two pass and the third says red (or blue). But there is a strategy that achieves 75% success (without any ordering of the three people). If you search “hat guessing games” online, you will find it along with some simple explanations of how achieving 75% success is feasible. Hat guessing games have attracted increased attention in the discrete mathematics community in the past decade. They are far more than fun parlor games—they have applications in a variety of areas. The most established is to coding theory, which deals with the design of efficient and error correcting methods for data transmission. Another application is to the design of deterministic auction mechanisms. Hat guessing games also have linkages to various important problems in theoretical computer science, including multiparty communication complexity. This issue's SIGEST selection, from the SIAM Journal on Discrete Mathematics, provides a brief review of the state of the art in analyzing hat guessing games as well as important new research results. Appropriately titled “Hat Guessing Games,” it is co-authored by Steve Butler, Mohammad Hajiaghayi, Robert Kleinberg, and Tom Leighton. The paper begins with the basic problem, where there are n players, k hat colors, and a deterministic guessing strategy known to both the players and the “adversary” placing the hats on each person. The problem is to find the deterministic guessing strategy that maximizes the percentage of persons guessing their own hat color correctly. (Note that this version of the problem is different from the collective success in the initial example described above, and hence there is no “pass” guess.) The paper summarizes the state of the art for this problem and then considers several variants, including the version where people see only a subset of the other participants (which has applications to index coding problems). It also shows an interesting and perhaps intuitive result: under certain conditions, optimal strategies are unbiased, meaning that each player guesses each color had with equal probability. It concludes with a nice discussion of open research questions. “Hat Guessing Games” provides a very nice and understandable glance into an important area of discrete mathematics, in a way that should be accessible to most SIAM Review readers. The mathematics, which comes from graph theory and combinatorics, is widely accessible, and the exposition is particularly nice. And besides, it may provide you with a new idea for your next party!