text: Several different "hat games" have recently received a fair amount of attention. Typically, in a hat game, one or more players are required to correctly guess their hat colour when given some information about other players' hat colours. Some versions of these games have been motivated by research in complexity theory and have ties to well-known research problems in coding theory, and some variations have led to interesting new research. In this paper, we review Ebert's Hat Game, which garnered a considerable amount of publicity in the late 90's and early 00's, and the Hats-on-a-line Game. Then we introduce a new hat game which is a "hybrid" of these two games and provide an optimal strategy for playing the new game. The optimal strategy is quite simple, but the proof involves an interesting combinatorial argument.
  @cite_7: We consider the problem of approximating a Boolean functionf∶ 0,1 n → 0,1 by the sign of an integer polynomialp of degreek. For us, a polynomialp(x) predicts the value off(x) if, wheneverp(x)≥0,f(x)=1, and wheneverp(x)<0,f(x)=0. A low-degree polynomialp is a good approximator forf if it predictsf at almost all points. Given a positive integerk, and a Boolean functionf, we ask, “how good is the best degreek approximation tof?” We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the casef is parity. Minsky and Papert [10] proved that a perceptron cannot compute parity; our bounds indicate exactly how well a perceptron canapproximate it. As a consequence, we are able to give the first correct proof that, for a random oracleA, PP A is properly contained in PSPACE A . We are also able to prove the old AC0 exponential-size lower bounds in a new way. This allows us to prove the new result that an AC0 circuit with one majority gate cannot approximate parity. Our proof depends only on basic properties of integer polynomials.
summary: A few years prior to the introduction of Ebert's Hat Game, in 1994, a similar game was described by Aspnes, Beigel, Furst and Rudich @cite_7 . In their version of the game, players are not allowed to pass, and the objective is for a majority of the players to guess correctly. For the three-player game, it is easy to describe a strategy that will succeed with probability @math , just as in Ebert's game: Alice votes the opposite of Bob's hat colour; Bob votes the opposite of Charlie's hat colour; and Charlie votes the opposite of Alice's hat colour.