Why use the entanglement in Conway's free will theorem?

Question

In Conway's Strong Free Will Theorem paper, the proof consists of two parts. In the first part, they proved the Specker-Kochen theorem. In the second part, they constructed two experimenters that are space-like separated, each possessing one spin-1 particle from an entangled pair, then they each made a measurement.

I don't see the point of making up the entanglement. From what I understood, the proof is like this:

• Suppose particles have no free will, but experimenters have, then there are two functions $\theta_a(x, y, z), \theta_b(w)$, such that $\theta_a(x, y, z)\in \{(110),(101),(011)\}$, and $\theta_b(w)\in\{0, 1\}$.
• By entanglement, we have $\theta_a(x, y, z) = (\theta_b(x), \theta_b(y), \theta_b(z))$.
• So $\theta_b$ satisfies the 101 property, which is impossible.

However, this just looks like the Specker-Kochen theorem with extra steps. Why is it necessary to show that $\theta_b$ has the 101 property by detouring through $\theta_a$? Is it supposed to squeeze out any possible contexuality?

Perhaps, phrased in another way, I'm asking this: suppose the universe has exactly one spin-1 particle, then is it possible that $\theta_b$ does not have the 101 property?

Answer 1

I figured it out: it's to close the contextuality loophole. In the 2006 free will theorem paper (which the 2009 paper referred to), it's said here:

One way of blocking no-go theorems that hidden variable theories have proposed is “contextuality”– that the outcome of an experiment depends upon hidden variables in the apparatus. For the triple experiment in SPIN, contextuality allows the particle’s spin in the z direction (say) to depend upon the frame (x, y, z). However, since the particle’s past history includes all its interactions with the apparatus, the Free Will theorem closes that loophole.

What it means in my notation is that, if we had only one spin-1 particle a, then the function $\theta_a(x, y, z)$ might not be of form $\theta_a(x, y, z) = (\theta_b(x), \theta_b(y), \theta_b(z))$, due to "contextuality", meaning that the squared-spin in direction x can be affected by the other two directions the experimenter will measure.

And as for why $\theta_b$ alone is not enough, I suppose it is also due to contextuality: if we only have $\theta_b$, we can't show that it has the 101 property without the help of $\theta_a$. Suppose we choose direction $w$, and measure the squared-spin of particle b along $w$. That gives $\theta_b(w)$, but now if we want to measure some other direction $w'$, we would be using $\theta_b'(w')$, which is not $\theta_b$, and thus we cannot show that $\theta_b$ itself has 101 property.