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A task in which separated players receive local questions and return local answers, with a joint winning condition and no communication after questions are received. A strategy may use pre-shared randomness, or pre-shared quantum entanglement when quantum resources are allowed. An average winning probability requires a specified input distribution; winning with certainty on every promised input is a separate worst-case requirement. The GHZ game gives a finite example with a perfect quantum strategy and no perfect classical strategy.

For deterministic classical responses, denote the two possible local answers by $x_0,x_1$, $y_0,y_1$ and $z_0,z_1$. Certainty in the GHZ game would requireTaking the exclusive or of all four equations gives zero on the left, since every answer occurs twice, but one on the right. This contradiction rules out deterministic certainty. Pre-shared randomness cannot cure it: conditioning on the random seed fixes a deterministic response table, and none succeeds on every allowed input. Thus noncommunicating classical players cannot win with certainty. No uniform distribution on the referee's questions is needed for this worst-case claim.

For the quantum strategy, share the GHZ state and measure the Pauli X gate observable when the local question is 0, or the Pauli Y gate observable when it is 1. Return bit 0 for measurement eigenvalue $+1$ and bit 1 for eigenvalue $-1$. The four relevant operator identities areFor the first identity, the three bit flips exchange $|000⟩$ and $|111⟩$. For each remaining identity, the two Y actions contribute $i^2=-1$ on $|000⟩$ and $(-i{)}^2=-1$ on $|111⟩$, so they exchange the terms with a minus sign. Local measurements on separate systems commute, and the product of their observed eigenvalues is therefore fixed by the displayed joint eigenvalue. That product equals $(-1{)}^{x+y+z}$, giving exactly the required output parity for each promised input. Hence the quantum strategy wins with certainty without communication. This is the GHZ theorem in nonlocal-game form; it does not let a player control another player's individual outcome.