Proposition 2: coordinatewise-boundary reading of Model B

Proposition 2 uses the following reading of its Model B response rule:

Move every coordinate toward its ideal, up to its coordinatewise boundary.

For an L∞ neighborhood of radius r_t, every non-tied coordinate therefore moves by exactly r_t toward the corresponding coordinate of the sampled voter’s ideal. A tied coordinate does not move. This is the coordinatewise rule used by the paper’s proof of Proposition 2, which says to treat each dimension separately and describes the coordinate as increasing or decreasing by r_t according to the side of the voter’s ideal (Appendix C.7).

Relation to the general Model B formula

Earlier in the paper, Model B is written as movement along the utility-gradient ray, normalized by the neighborhood norm. For an L∞ neighborhood, a literal reading gives coordinate increment

r_t * (partial_m f_v(x) / ||partial f_v(x)||_infinity).

That expression reaches the boundary in the coordinate with largest derivative magnitude. It reaches every active coordinate boundary only when the normalized active-coordinate magnitudes agree. Proposition 2’s proof instead uses only the sign of each coordinate derivative and moves every active coordinate the full radius. The formalization makes that proof convention explicit rather than inferring it from the more general normalized-gradient-ray display.

The distinction is mathematically observable. For example, on [-1,1]^2, let the ideal Y be uniform and define the decomposable concave utility

f_y(x) = - a(y_1) |x_1-y_1| - 2 |x_2-y_2|,

a(y_1) = 1     if y_1 >= 0,
         1/2   if y_1 < 0.

Under the literal normalized-gradient-ray rule, the first-coordinate drift vanishes at x_1 = 1/4, not at the ordinary median x_1 = 0. Thus that literal variant targets a weighted median. Under the coordinatewise-boundary rule used in the Proposition 2 proof, the positive coordinate weights do not affect the step: each coordinate moves by its sign, so the ordinary coordinatewise median is the correct target.

Coordinatewise feasible geometry

On product domains, each coordinate can be varied independently and the limit is a coordinatewise median. On a general convex feasible set, those medians may be infeasible. The checked constrained conclusion instead minimizes expected coordinatewise absolute distance over the feasible set. The constrained-domain explanation gives the geometry and an example.

The normalized-gradient-ray rule is a distinct model whose coordinate weights can change the limit, as the example above shows. Proposition 2’s formalization uses the coordinatewise rule described by the source proof.