How the Market for Unloved Tasks Finds Equilibrium Without a Conductor ✛ ✎

A hundred agents, two thousand tasks, less than a hundred iterations, and the market for unloved chores settled into equilibrium all on its own. No one was directing things from the center; each participant only responded to the nearest signal. That’s how multiplicative tatonnement works — a mechanism like the synchronized dance of a flock without a leader.

Economists call such tasks “chores.” Doing the dishes, taking out the trash, reviewing papers — demand for these is negative: people want to avoid them. Classic tatonnement, an auctioneer groping for prices, breaks down here. If nobody takes a job, you need to raise the pay, not lower it. The logic flips. Recently, relative tatonnement emerged to restore convergence, but it required global coordination: every price depended on all the others. In the real world, a job manager doesn’t see the big picture: they only see the imbalance for their own task.

Multiplicative tatonnement changes the rule: the job manager updates the price by multiplying the current value by \(1 + \eta_k \times (\text{excess supply})\). That’s a purely local signal. Each job is like a bird in a flock, adjusting its position based only on its immediate neighbors. No leader needed, just a simple signal on the spot. Mathematically, the process turns out to be mirror descent with respect to relative entropy. The updates don’t follow the shortest Euclidean path but the geodesics of the probability simplex — the natural geometry for shares and prices. This nonlinear dynamics keeps prices positive without any forced projection.

The results of numerical experiments are impressive. The algorithm converges within 100 iterations even when the “heaviness” of tasks varies by a factor of 100,000—roughly the difference between washing a spoon and building a house. Relative tatonnement, with its global view, takes tens of thousands of steps. Local feedback turned out to be many times faster than global. The reason is the larger permissible step size. The multiplicative approach allows a step up to \(d_{min}^{-1}\) times bigger, which really shows on sharply heterogeneous markets.

This discovery changes how we approach automatic task distribution. Decentralized coordination without a global arbiter is now possible even for markets with negative utility. Robotic warehouses, conference review assignment, multi-agent systems — anywhere distribution matters, multiplicative tatonnement could become a standard protocol. The next step: markets with mixed goods (both goods and chores) or operating in noisy conditions, like a flock navigating fog. The method’s robustness promises new ideas for distributed optimization and game theory.

When thousands of birds launch into the sky and weave complex patterns with no apparent control, it looks like magic. Multiplicative tatonnement lifts the curtain: behind it are a local multiplication rule, mirror descent on entropy, and Walras’s Law. The market for unloved tasks finds its equilibrium the same way — without a conductor, but with mathematical precision.