Understanding Mutual and Common Knowledge with the Hats Riddle

A state space Ω lists every possible outcome of a situation, such as the result of a die roll or the color pattern of three hats. Each player’s information partition Kᵢ groups the states into cells that the player cannot distinguish. The player never learns the exact state; instead, the player learns which cell the true state occupies.

Information Partitions

The partition Kᵢ assigns every state ω to a cell Kᵢ(ω). If the die shows 4, a player might only know that the result lies in the cell {4,5,6}. The player’s knowledge is limited to the cell, not the precise number. As Ian Ball explains, “The interpretation is that a player, player i, does not learn the actual state. They don't actually see the roll of the die. But what they do learn is which cell the state lies in.”

Defining Knowledge

An event E is any subset of Ω. A player knows E in state ω when the entire cell Kᵢ(ω) fits inside E. Formally, the set of states where player i knows E is Kᵢ(E) = { ω ∈ Ω | Kᵢ(ω) ⊆ E }. This yields the fundamental property Kᵢ(E) ⊆ E: a player can only know an event that is true. For example, in state 4, E′ has occurred, but the player i doesn’t know it; in state 5, E′ has not occurred, and the player doesn’t know it.

Interactive Knowledge

When every player knows E, the situation is called mutual knowledge: K(E) = ⋂ Kᵢ(E). Higher‑order mutual knowledge adds layers: K²(E) = K(K(E)) means everyone knows that everyone knows E, and so on. Common knowledge CK(E) is the infinite intersection of all orders, CK(E) = K(E) ∩ K²(E) ∩ K³(E) ⋯. As the lecture notes, “Mutual knowledge sounds pretty strong, but it's actually not quite as strong as we might think.”

The Hats Riddle

Three students wear either red or white hats. Each sees the other two hats but not their own. The teacher announces, “At least one hat is red.” This public announcement converts an event that was already mutual knowledge into common knowledge. By eliminating the state where all hats are white, the announcement prunes the state space, reshaping every player’s partition. Subsequent questioning splits cells further, allowing the students to deduce their own hat colors. The mechanism mirrors the formal definition: public announcements act as a knowledge‑verification step that updates Kᵢ for all i, potentially triggering a cascade of higher‑order knowledge.

Mechanisms in Action

To verify knowledge, check whether the entire cell Kᵢ(ω) lies inside the event E. If any part of the cell falls outside E, the player does not know E. Public announcements prune the state space by ruling out impossible states, thereby refining each Kᵢ. This refinement can create common knowledge when the eliminated states were the only obstacle to infinite mutual awareness.