Given the first component x ∈ [−1, 2]^4m of Alice’s strategy, we define two decoding functions Dv, Dw :[−1, 2]^4m → {0, 1}ⁿ as follows. Let Rv(x) ∈ {0, 1}^m be the rounding of the first m-tuple of coordinates of x to {0, 1}^m; let Dv(x) = E⁻¹(Rv(x)) ∈ {0, 1}^2n+log(n+1), where E⁻¹ denotes the decoding of the error-correcting code from Subsection 3.5.3. We define Dw(x) ∈ {0, 1}^2n+log(n+1) analogously with respect to the second m-tuple of coordinates of x. The second component of Bob’s utility is now given by iff he guesses correctly the vertex D_v(x), and the corresponding β operation on this vertex. Namely, if vB = Dv(x) and , and otherwise. We similarly define Bob’s third component with respect to Dw(x).

      Note that Bob knows the indexes (for every v); thus to achieve Bob needs to guess correctly only the vertices Dv(x), Dw(x) and announce the corresponding triplet of β indexes.

      Going back to Alice, the second component of her utility is given by iff she guesses correctly the triplet I(vB) = (T(vB), S(vB), P(vB)) when the calculation of T, S, P is done by the decomposition of α(βB). Namely, = 1 if , and otherwise. We similarly define Alice’s third component .

      Finally, the first component of Bob’s utility is given by: