Jacobi algorithm in full assignment non-transferable utility models

Determine conditions that ensure responsiveness—and hence the existence of coordinate updates and well-definedness of the Jacobi iteration—for the full assignment, non-transferable utility matching model (no outside option, with normalization p_{y0} = π) whose equilibrium system is Q(p) = 0 with components Q_x(p) = ∑_y min{exp(p_x + α_{xy}), exp(γ_{xy} − p_y)} − n_x and Q_y(p) = −∑_x min{exp(p_x + α_{xy}), exp(γ_{xy} − p_y)} + m_y. Specifically, ascertain structural assumptions on (α_{xy}, γ_{xy}) or equivalent primitives that guarantee, for every coordinate z and fixed p_{−z}, the equation Q_z(π, p_{−z}) = 0 has a solution, thereby making the Jacobi sequence well-defined in this full assignment NTU setting.

Background

In the full assignment case of the non-transferable utility (NTU) matching model with no outside option, the authors derive an equilibrium system Q(p) = 0 where each side’s equation involves a minimum over two exponential terms. This structure makes Q a Z-function with isotone aggregates (thus an M0-function), but also introduces a potential failure of the responsiveness condition typically used to guarantee that Jacobi coordinate updates exist.

Responsiveness is required to define each Jacobi update p_z^{t+1} by solving Q_z(π, p_{−z}^t) = 0. Because the model’s Q includes min operations, Q_z(·, p_{−z}) may not cross zero for some p_{−z}, so an update may not exist. The authors explicitly state that it is unclear whether the Jacobi sequence is well-defined in this setting and identify as an open problem the task of finding conditions under which the Jacobi algorithm works in the full assignment NTU case.