How the Market for Unloved Tasks Finds Balance on Its Own ✛ ✎

A hundred people, two thousand tasks, and fewer than a hundred steps to perfect order. No one gives commands, like a flock of birds without a leader, the market for unloved tasks finds its own equilibrium. That’s how multiplicative tâtonnement (a process of groping for prices by multiplication) works.

Classic tâtonnement, where an auctioneer calls out prices, breaks down in the market for chores. If no one wants to wash dishes, you need to raise the payment, not lower it. The new rule works differently. Each job’s manager sees only its own surplus or shortage of takers. It multiplies the current price by (1 + a small step × that imbalance). No central overview. It’s like a bird in a flock: it adjusts its flight by looking only at its neighbors. Birds don’t need a flight plan — a simple local signal suffices.

Mathematically, this process is mirror descent on relative entropy (a measure of divergence between probability distributions). Instead of moving in a straight line, the updates glide along the natural geometry of prices, always staying positive. This nonlinear path keeps prices from falling to zero without artificial constraints. Results from numerical experiments showed: even when tasks differ by a factor of a hundred thousand, from washing a spoon to building a house, equilibrium is reached in 100 steps. A global method that knows the whole picture takes tens of thousands. On-the-spot feedback proved to be many times faster.

The discovery paves the way for distributed control of warehouse robots, automatic assignment of reviewers at conferences, and any system where hundreds of participants share unloved tasks. The next step is mixed markets with both goods and chores, or working amid noise, like a flock in fog. The method’s robustness promises new ideas for game theory and distributed optimization.

Thousands of birds take flight and weave patterns in the sky with no visible control. Behind this lies a simple local rule of multiplication, mirror descent on entropy, and Walras’s Law. The market for unloved tasks finds equilibrium the same way — without a conductor, yet with mathematical precision.