Equivalence of equilibrium markets and optimal central control

Prove that, under the preconditions (i) agents act competitively (price-taking), (ii) a market equilibrium exists, and (iii) the systems performance index can be written as a linear combination of local contributions, multi-agent equilibrium markets produce optimal decentralized solutions to distributed resource allocation problems that match the quality—measured by the given overall systems performance index—of solutions produced by the optimal central multi-input, multi-output systems controller with access to all relevant local data (i.e., the total system state and control vectors).

Background

The paper compares multi-agent market mechanisms with conventional centralized control for building climate regulation, developing both improved control schemes and a decentralized market design (MARKET-B) that achieves performance comparable to centralized control without relying on global information. This leads to the qualitative conclusion that local data combined with market communication can yield global control.

Building on these results, the authors propose a broader claim beyond the specific application: that equilibrium markets can provide optimal decentralized solutions equivalent in quality to those from optimal central controllers, given specific preconditions. They explicitly frame this broader claim as a conjecture and call for a rigorous mathematical proof grounded in dynamic systems and optimal control theory.