Set It and Forget It

In a power-diagram auction, advertisers declare three numbers: center, sigma, and bid. The bid is the hard part. Guessing margin × P(conversion) requires knowing your conversion rate, which requires running ads, which requires a bid. Circular.

With verified conversions (cryptographic coupons), the bid collapses to margin. But verified conversions do something else. They make sigma tunable from data.

The Guessing Game

Sigma controls how broad or narrow the advertiser’s reach should be. A climbing gym that sets sigma too wide wastes money on irrelevant impressions. Set it too narrow and they miss customers who would have converted. The advertiser has to guess how far their relevance extends in embedding space. Nobody knows that before running ads.

The data exists, after. The publisher reports aggregated distance histograms: how many coupons were distributed at each distance bin from the advertiser’s center. No individual embeddings, no timestamps, no point cloud to fingerprint. Just a histogram.

This isn’t optional. de Montjoye et al. (2013) showed that 4 spatio-temporal data points re-identify 95% of individuals in a 1.5 million person dataset. The FTC has brought five enforcement actions against location data brokers since 2022 for claiming “anonymized” data that wasn’t. CMS requires a minimum cell size of 11 before any count can be reported. Google’s own Ads Data Hub requires 50 users minimum per query. Individual embedding points are location data with extra dimensions. Histograms with minimum bin sizes satisfy these thresholds by design.

Pair the histograms with verified conversions, and the exchange observes conversion rates at different distances. It estimates the Gaussian decay curve. That curve is sigma. The exchange learns it from data the advertiser never had to provide.

The advertiser can override sigma anytime. But the default improves with every conversion.

The Control Loop

Sigma is continuously adjusted. Competitors enter. Seasons shift the audience. A new creative performs differently at the edges. A static sigma drifts out of calibration.

A PID controller keeps sigma tracking reality. The target is the conversion rate the advertiser’s margin can sustain. The error is the gap between observed and target rates at the boundary.

• Proportional: conversions at the current boundary are below target. Sigma is too wide. Pull it in.
• Integral: sigma has been slightly too wide for three weeks. The accumulated error forces a correction the proportional term alone wouldn’t catch.
• Derivative: conversions are dropping fast, say a competitor just entered the adjacent territory. Adjust now instead of waiting for the error to accumulate.

Overshoot, correct, settle.

The Upsell

The exchange offers to tune sigma for a fee. The algorithm is public, the inputs (distance histograms, conversion counts) are on the public ledger, and the output is deterministic. Anyone can replay the computation. The exchange can’t widen sigma beyond what the data justifies without contradicting the ledger.

The advertiser who wants control keeps it. The advertiser who wants convenience pays for it and audits anytime. Aligned greed.

Three numbers, a margin, and a feedback loop. Set it and forget it.

Part of the Vector Space series.