Sufficient condition for budget balance under the Symmetric Protocol

Determine a concrete sufficient condition on normalized incentive-compatible double auction rules used in all markets that guarantees the supply chain mechanism produced by the Symmetric Protocol is budget-balanced for every bid profile, analogous to the Pivot Protocol’s sufficient condition that the pivot-market buyer price be at least the (q+1)-st seller bid.

Background

The paper introduces two mechanisms for coordinating double auctions along a linear supply chain: the Symmetric Protocol and the Pivot Protocol. The Pivot Protocol admits a clear sufficient condition ensuring budget balance: if, for every supply and demand curve, the pivot-market buyer price PB is at least the (q+1)-st seller bid S_{q+1}, then the resulting supply chain mechanism is budget-balanced.

For the Symmetric Protocol, which runs a double auction independently at each market after propagating supply and demand curves, the authors state that they could not provide an analogous sufficient condition guaranteeing budget balance. Establishing such a condition would close this gap and align the understanding of budget-balance guarantees across both protocols.