Suppose you have two hat color assignments. These are maps U,V from eg the natural numbers (countably many players) to C, where C in this example is all ration

Suppose you have two hat color assignments. These are maps U,V from eg the natural numbers (countably many players) to C, where C in this example is all rationals in [0,1]. On some inputs, U and V may output the same value, and on other inputs they may output different values. Call this latter set Diff(U,V) := { x in Naturals such that U(x) != V(x) }. U and V are "almost the same" if Diff(U,V) is a finite set. This is an equivalence relation because if Diff(U,V) is finite, and Diff(V,W) is finite, then Diff(U,W) is also finite: it's a subset of the union of the other two. The same relation exists, and is an equivalence relation, for any set of players and for any set C of hat colors. It even exists if there are only finitely many players, but it isn't interesting: in that case all assignments are equivalent.