Diffuse

One item, sold every round, to whoever pays the most. Every bidder paces its own budget the same simple way real ad platforms do. Turn the population dial and watch whether the pacing settles down.

What this actually is, honestly

The paper behind this toy proves that computing an autobidding equilibrium is worst-case intractable with a handful of bidders (an "atomic" market), but that the hardness disappears once bidder values are drawn from a large, diffuse population — and it hands over a solver with a proven geometric convergence rate for that regime.

This page does not run that solver. It runs the ordinary multiplicative pacing heuristic that real ad platforms use — no monotone-equilibrium guarantee, no proof. What you're watching is whether that ordinary heuristic happens to echo the paper's qualitative signature (a stalling, cycling plot at low population that straightens into a clean decaying line at high population), not a demonstration that the guarantee applies to it. It usually doesn't, and the page isn't claiming otherwise.

Run the market

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How this market works

Every round, all N bidders draw a value for the one item on offer and bid αi · vi, where αi is bidder i's own pacing multiplier. Highest bid wins and pays the second-highest (a sealed-bid second-price auction). After each round every bidder — winner or not — nudges its own multiplier:

α_i ← clip( α_i · exp( −η · (spend_i − ρ_i) ), 0.02, 1 )

where spend_i is a running average of what i has actually paid and ρ_i is its target spend rate per round. Each bidder's target scales as 1/N — a fair share of the one item on offer shrinks as the crowd grows, which is what makes "population size" the same knob as the paper's "diffuse" framing rather than just a bigger loop.

The plot's y-axis is the root-mean-square size of that round's α update across every bidder, on a log scale. When bidders stop adjusting, that line hits the floor and stays there — read that as "locked in". A line that keeps climbing back up hasn't settled — read that as "still moving", and possibly cycling: a couple of large bidders can keep out-jostling each other indefinitely, which is exactly the atomic case the paper says is hard.

fastest to lock in

Submit how many rounds your last run took to lock in. Lower is better — you're racing to find a population size (and luck of the draw) that settles fast.