Jiangang HanNotes on what I've learned and built, and what I'm learning and building now.https://jianganghan.github.io/2026-09-20T00:00:00ZJiangang HanBidding in a Single Auctionhttps://jianganghan.github.io/posts/bidding/1-auction.html2026-09-20T00:00:00Z2026-09-20T00:00:00ZIn an auction, all you do is name one number. To you, all your competitors combined are just one random "market price": from it you can work out your win rate…Bidding Under a Budgethttps://jianganghan.github.io/posts/bidding/2-budget.html2026-09-20T00:00:00Z2026-09-20T00:00:00ZThere are thousands of auctions a day but only one budget. A Lagrange multiplier compresses that constraint into a single number λ, and every bid becomes "valu…Budget Pacinghttps://jianganghan.github.io/posts/bidding/3-pacing.html2026-09-20T00:00:00Z2026-09-20T00:00:00ZThe optimal λ can't be computed in advance: at 9 a.m. you have no idea what tonight's market will look like, so you have to adjust as you spend.Pricing a Single Producthttps://jianganghan.github.io/posts/pricing/1-single.html2026-09-20T00:00:00Z2026-09-20T00:00:00ZOne number to set. Raise the price and you make more on each sale, but fewer people buy — so profit traces an inverted U.Pricing Many Items Under Constraintshttps://jianganghan.github.io/posts/pricing/2-constraints.html2026-09-20T00:00:00Z2026-09-20T00:00:00ZGoing from one item to N doesn't make the problem harder — it splits cleanly into N independent small problems.Pricing Substitutable Productshttps://jianganghan.github.io/posts/pricing/3-substitution.html2026-09-20T00:00:00Z2026-09-20T00:00:00ZRaise the price of one product and another one sells more — they're competing for the same demand.Pricing at Scalehttps://jianganghan.github.io/posts/pricing/4-scale.html2026-09-20T00:00:00Z2026-09-20T00:00:00ZAt scale, closed-form solutions stop working. This post trades a little accuracy for tractability — linearize the demand model and hand it to a solver — and, j…Von Neumann, 1952: Majority Voting, Thresholds, and Multiplexinghttps://jianganghan.github.io/posts/reliability/1-von-neumann.html2026-09-20T00:00:00Z2026-09-20T00:00:00ZCan you build a machine that almost never fails out of parts that fail all the time? Von Neumann's answer was yes, on two conditions: each part's error rate ha…Why More Checks Stop Helping: Correlated Verifiers and the Reliability Ceilinghttps://jianganghan.github.io/posts/reliability/2-verifier-cascades.html2026-09-20T00:00:00Z2026-09-20T00:00:00ZHave a model generate an answer, have a model check it, and start over if it fails: that loop is one of the most common patterns in AI systems today.