Their work was based in part on a complex theorem by Kochen and Ernst Specker in 1967 that is a variation on Bell's Theorem. The Kochen-Specker Theorem puts limits on "hidden variables" that could make quantum mechanics a deterministic theory.

Albert Einstein was convinced that the perfectly correlated measurements between two observers Alice and Bob measuring the spins of entangled was the result of the spins being pre-determined. If everything in physics is similarly pre-determined, free will is an illusion.

Although Conway and Kochen do not claim to have proven free will in humans, they assert that should such a freedom exist, then the same freedom must apply to the elementary particles.

Conway and Kochen are really describing the indeterminism that quantum mechanics has introduced into the world. While indeterminism is a precondition for human freedom, it is in no sense free will. They qualify their claim:

To be more precise, what we shall show is that the particles’ response∗ to a certain type of experiment is not determined by the entire previous history of that part of the universe accessible to them. The free will we assume is just that the experimenter can freely choose to make any one of a small number of observations.

∗More precisely still, the universe’s response in the neighborhood of the particles.

Conway and Kochen assume three axioms, which they call "SPIN", "TWIN" and "FIN". The spin and twin axioms can be established by entanglement experiments. Fin is a consequence of relativity theory.

1. SPIN: Measuring the square of the component of spin of certain elementary particles of spin one, taken in three orthogonal directions, results in a permutation of (1,1,0).

2. TWIN: It is possible to "entangle" two elementary particles, and separate them by a significant distance, so that they give the same answers to corresponding questions. The squared spin results are the same if measured in parallel directions. If the first experimenter A (on Earth) performs a triple experiment for the frame (x, y, z), producing the result x → j, y → k, z → l while the second experimenter B (on Mars, at least 5 light minutes away) measures a single spin in direction w, then if w is one of x, y, z, its result is that w → j, k, or l, respectively.

3. FIN: There is a finite upper bound to the speed with which information can be effectively transmitted. Conway and Kochen say this is a consequence of "effective causality."

[But the collapse of the probability amplitude wave function is instantaneous and not so limited. ]

If the choice of directions in which to perform spin 1 experiments is not a function of the information accessible to the experimenters, then the responses of the particles are equally not functions of the information accessible to them.

Why do we call this result the Free Will theorem? It is usually tacitly assumed that experimenters have sufficient free will to choose the settings of their apparatus in a way that is not determined by past history. We make this assumption explicit precisely because our theorem deduces from it the more surprising fact that the particles' responses are also not determined by past history.

Thus the theorem asserts that if experimenters have a certain property, then spin 1 particles have exactly the same property. Since this property for experimenters is an instance of what is usually called "free will," we find it appropriate to use the same term also for particles.

The theorem states that, given the axioms, if the two experimenters in question are free to make choices about what measurements to take, then the results of the measurements cannot be determined by anything previous to the experiments.

[See the discussion of the EPR experiments to see that what determines the second experimenter's results is simply the first experimenter's measurement, which instantaneously collapses the superposition of states into a particular state.]

Since the theorem applies to any arbitrary physical theory consistent with the axioms, it would not even be possible to place the information into the universe's past in an ad hoc way. The argument proceeds from the Kochen-Specker theorem, which shows that the result of any individual measurement of spin was not fixed independently of the choice of measurements.

Conway and Kochen describe new bits of information coming into existence in the universe, and we agree that this is the key to understanding both EPR entanglement experiments and human free will. They say

...there will be a time t₀ after x, y, z are chosen with the property that for each time t < t₀ no such bit is available, but for every t > t₀ some such bit is available.

But in this case the universe has taken a free decision at time t₀, because the information about it after t₀ is, by definition, not a function of the information available before t₀!

In "The Strong Free Will Theorem," Conway and Kochen replace the FIN axiom with a new axiom called MIN, which asserts only that two experimenters separated in a space-like way can make choices of measurements independently of each other. In particular, they are not asserting that all information must travel finitely fast; only the particular information about choices of measurements made by the two experimenters.