Optimization and Equilibrium Seminar “From Pairwise Trades to Market Equilibrium: A Convergence Theorem”

Abstract: 
We study a discrete dynamical system of pairwise trades among m agents exchanging n+1 divisible goods, each holding a strongly concave  utility. Two agents trade a single good for money whenever gradient-based price thresholds — ratios of partial utility derivatives — admit a mutually improving exchange within a prescribed bid-ask spread. Using convex analysis, we characterize the fixed points of this dynamics: an allocation is invariant, at every spread level, exactly when it satisfies the normal-cone (first-order) conditions of a Walrasian equilibrium. The key technical result is a quantitative descent lemma showing each trade raises the two agents' utilities by an amount bounded below by the square of the spread, which precludes stagnation and forces termination after finitely many trades at any fixed spread. Letting the spread shrink to zero yields a sequence of such approximate equilibria that, by a strict-concavity argument, converges to a unique limit rather than merely a cluster point. Numerical experiments illustrate convergence for economies with up to 100 goods and 10 agents.

Speaker: Alejandro Jofré (with J. Deride y  R. T. Rockafellar)