The First-Price Shock and What Bid Shading Is

October 6, 2026

For most of programmatic’s life, display exchanges cleared at second price: you bid your value, the runner-up set the price, and the mechanism did the shading for you. In 2019, Google Ad Manager and the open exchanges moved to first price, and overnight every buyer was paying its own bid. Enormous engineering effort went into one question: how far below your value should you actually bid? That’s bid shading — and its theory is older, stranger, and prettier than the engineering around it. This entry is the whole thread: the equilibrium that nobody solves, the first-order condition everyone does solve, the methods catalogue, and the trap where shading quietly fights pacing.

Baseline: why first price isn’t trivial

Vickrey’s 1961 paper sets up the sealed-bid taxonomy and proves the second-price rule truthful. The first-price rule is older than Vickrey and, at first sight, worse: the winner pays their own bid, so bidding your value dominates nothing. Every bidder must form an expectation about other bidders’ bids and shade below value — the strategic content of the mechanism moves out of the mechanism and into the bidders’ heads.

Myerson’s optimal-auction framework makes the comparison precise. Under the standard assumptions — independent private values, risk-neutral quasilinear bidders, symmetric distributions, no reserve asymmetries — every efficient, individually-rational sealed-bid format yields the seller the same expected revenue. FPA and SPA are two points on one plateau. The revenue equivalence theorem is fragile, and the fragility is the whole applied story: every way display RTB violates the assumptions — asymmetric bidders with different data, hard budget constraints, repeated interaction, algorithmic play — picks a point off the plateau and determines which format serves whom. The exchanges’ motive for switching is legible in that fragility: truthfulness concentrates surplus with the buyers, and first price hands it to whoever models the field best.

The equilibrium nobody solves

Symmetric independent values, n bidders with values i.i.d. on cdf F over [0, v̄]: the unique strictly-increasing Bayesian Nash equilibrium bid function is

β(v) = E[Y₁ | Y₁ < v],   Y₁ = max of the other n−1 values

— bid the expected highest competing value, conditional on it being beaten. For uniform values this collapses to the charming textbook law

β(v) = (n−1)/n · v

and the reading I want you to keep: shading depth is a market-thickness statistic, not a policy. With ~30 effective competitors you bid ≈ 97% of value; with 2, ≈ 67%. The right shade factor is a function of the conditional win-rate curve w(b | x), not a per-market constant — one more argument for the landscape-estimation machinery existing at all.

Asymmetry makes the analysis genuinely hard, and there’s a bonus result worth knowing: with asymmetric bidders, Maskin & Riley show the second-price format generally has no equilibrium in pure monotone strategies while first price retains clean structure — and that sealed high-bid auctions screen better: equilibrium bidding discloses more of the weak bidder’s private information, giving the seller a post-auction informational edge. Milgrom & Weber’s linkage principle is the general language: formats rank by how much the winner’s payoff reveals about the common value. The RTB read-across is imperfect (values not costs, no resale market), but the screening logic reappears whenever an exchange decides whether to disclose clearing prices.

The first-order condition everybody does solve

Forget the game. Maximize expected surplus (v − b)·W(b) over the bid, where W(b) is the win probability and w = W′ its density:

v − b* = W(b*) / w(b*)

Surplus over the bid equals the inverse hazard rate of the market price at your bid. This condition is equilibrium-free — it needs only a calibrated win-rate curve. That single property is why shading became a product: a DSP doesn’t have to solve the game’s fixed point per request (which, as the pacing literature proves, is PPAD-hard in general anyway); it has to estimate one conditional CDF well.

Where does W(b|x) come from? The same bid-landscape estimators as everywhere else, with an FPA wrinkle on the labels: market price z = minimum winning bid (runner-up bid, or the floor if binding). Wins reveal your own payment but not z; losses reveal nothing unless the exchange is configured to fire loss-notice macros. Your shading ceiling is set by a counterparty’s logging choices — some exchange templates blank AUCTION_PRICE entirely — and that dependency never appears in a contract.

The methods catalogue, in the order the field discovered them

Uniform / order-statistics shading (α·v) survives every serious review because two independent facts back it: the symmetric equilibrium factor (n−1)/n is an order-statistics fact, and pacing equilibria in repeated first-price auctions with budgets are uniform multiplicative shading (Conitzer, Kroer, Sodomka & Stier-Moses — the componentwise-maximal budget-feasible multiplier vector, with the practitioners’ distrusted “uniform % shade” literally being the equilibrium play). Also reassuring: regret-optimal learners in repeated FPA (Han, Zhou & Weissman) converge to good average play with no knowledge of bidder counts or distributions, and regret-based learning provably approaches symmetric Bayes–Nash in the i.i.d. model (Danak & Mannor). But there’s a negative-space warning too (Ratan & Wen): certain adaptive adjustment rules converge to wrong — over- or under-shaded — points and lose against equilibrium play. Uniform shading deserves its status; naive heuristics tuned by vibes do not. Check your heuristic against the FOC.

Market-price distribution models are the production-ML answer. Yahoo’s CIKM 2020 paper predicts a high quantile of z|x with a safety buffer and bids min(v, ẑ + ε); the 2021 KDD network models the entire distribution of market price rather than one quantile and then applies the inverse-hazard FOC exactly. Same conditional object as landscape forecasting, consumed at the payment side. The subsequent literature went end-to-end and robust: distributionally-robust shading over an ambiguity set around the price model, multi-slot end-to-end shading, concave shading policies via measure-valued proximal optimization, shading-factor search as a bandit (Tilli & Espinosa-Leal), and best-response iteration for shading factors (Fagandini & Dierickx) — the game-theoretic formalization of the uniform heuristic, which I find lovely: the heuristic everyone shipped with, made into a solvable fixed-point problem a decade later.

Shape constraints are the load-bearing plumbing. w(b|x) must be monotone in b or the FOC v − b = W/w is ill-posed and the shade is unstable. The econometric anchor is real, not folklore: Zhang (Economics Letters 2017) gives a shape-constrained nonparametric estimator of the FPA bidding function with consistency guarantees, and Curmei & Hall (Operations Research 2025) supply rates and a computable hierarchy for monotone/convex regression; isotonic regression (PAVA) is the monotone special case you’d reach for in code.

Interaction with pacing is the trap section, and the most transferable finding here. In FPA, shading and min-pacing act on the same multiplicative channel: bid = min(λ_budget, λ_ROS) · shade(x) · v — the dual structure survives the format change. An aggressive shade lowers effective spend, which a naive pacing controller misreads as budget slack-free pressure: λ drifts up, and the shade is silently undone. Attributing λ-movement to win-rate versus price requires loss-notice data again. When two controllers share one actuator, one of them is hallucinating; don’t cascade, min-compose — a law about shared actuators that generalizes far beyond advertising.

What the exchanges engineered around it

The 2019 migration (at flipped in OpenRTB; “first-price auctions with soft floors” is the honest description) created a new mechanism-design surface rather than a level field:

  1. Floors bite differently. With pay-your-bid, a floor f makes every auction “max(b, f) wins, pays own bid”; the option value of shading falls because shaded bids that used to clear a runner-up now collide with the floor. Yahoo’s floor-optimization study shows strongly non-monotone revenue effects mediated by bidder adaptation. The equilibrium bid function literally kinks at f, with mass points piling near it.
  2. The second-price illusion. Post-migration, exchanges kept reporting second-price-like quantities (AUCTION_PRICE macros, min-to-win), so shaded first-price payments looked like SPA outcomes in DSP reporting. The empirical record we can anchor: FPA adoption explained by soft floors and SSP competition (Despotakis, Ravi & Sayedi, JMR 2021); measured bidder responses — prices, and the inferred undershading that decays as bidders learn to shade — to the format change (Goke, Weintraub, Mastromonaco & Seljan 2021); and learning-agent simulation comparing FPA vs SPA revenue in display (Bichler, Gupta & Oberlechner, ISR 2026). The widely-quoted trade-press percentage deltas remained, in my reading, unanchored to any primary source.
  3. Cache risk changes. In SPA a cached bid pays second price whenever it fires; in FPA the cached bid pays itself at a later, possibly thinner auction — same trick, different price exposure. Under budgeted auto-bidding, moreover, SPA’s truthfulness is moot anyway (Balseiro, Deng, Mao, Mirrokni & Zuo; Liaw, Mehta & Perlroth), which is the theoretical footnote the trade press never printed when it explained the switch by “simplicity.”
  4. DSP countermeasures, which are engineering positions, I’d add, not literature: per-exchange p(z|x) models with floor decomposition z = max(f, z⁻); loss-notice-driven censoring repair; treating floor-policy changes as covariate drift rather than model damage; refusing the uniform-% default wherever the z-forecast is confident.

The synthesis

Every prior mechanism change — SPAs to FPAs, server-side to header bidding, cookies to signal loss — invalidated one subsystem at a time while competitors’ algorithms changed underneath, and every vendor rebuilt on the same public papers. Mechanisms commoditize the moment they’re standardized; advantage retreats into execution and data. The 2019 flip didn’t hand anyone an economic idea the theory crowd hadn’t published in 1961; it handed a temporary edge to whoever could estimate hazard rates on a billion heterogeneous auctions a day. That is an infrastructure advantage, not an idea advantage — which is, as far as I can tell, the only kind programmatic ever hands out.

References

  1. Vickrey, W. (1961). “Counterspeculation, Auctions, and Competitive Sealed Tenders.” Journal of Finance 16(1). doi:10.1111/j.1540-6261.1961.tb02789.x
  2. Myerson, R. (1981). “Optimal Auction Design.” Mathematics of Operations Research 6(1). doi:10.1287/moor.6.1.58
  3. Maskin, E. & Riley, J. (2000). “Asymmetric Auctions.” Review of Economic Studies 67(2). doi:10.1111/1467-937X.00137; and “Equilibrium in Sealed High Bid Auctions.” doi:10.1111/1467-937X.00138
  4. Milgrom, P. & Weber, R. (1982). “A Theory of Auctions and Competitive Bidding.” Econometrica 50(5). doi:10.2307/1911865
  5. Gligorijevic, D. et al. (2020). “Bid Shading in The Brave New World of First-Price Auctions.” CIKM. doi:10.1145/3340531.3412689 (preprint arXiv:2009.01360)
  6. Zhou, T. et al. (2021). “An Efficient Deep Distribution Network for Bid Shading in First-Price Auctions.” KDD. doi:10.1145/3447548.3467167, arXiv:2107.06650
  7. Conitzer, V., Kroer, C., Sodomka, E. & Stier-Moses, N. (2019/2022). “Pacing Equilibrium in First-Price Auction Markets,” EC, doi:10.1145/3328526.3329600; “Multiplicative Pacing Equilibria in Auction Markets,” Management Science 70(2). doi:10.1287/mnsc.2022.4310, arXiv:1811.07166
  8. Zhang, Y.Y. (2017). “A Shape Constrained Estimator of Bidding Function of First-Price Sealed-Bid Auctions.” Economics Letters. doi:10.1016/j.econlet.2016.11.001
  9. Curmei, D. & Hall, D. (2025). “Shape-constrained regression using sums-of-squares of polynomials.” Operations Research. doi:10.1287/opre.2021.0383
  10. Han, D., Zhou, T. & Weissman, T. (2021/24). “Optimal No-Regret Learning in Repeated First-Price Auctions.” Operations Research. doi:10.1287/opre.2020.0282
  11. Danak, A. & Mannor, S. (2012). “Approximately Optimal Bidding Policies for Repeated First-Price Auctions.” Annals of Operations Research. doi:10.1007/s10479-012-1148-8
  12. Ratan, A. & Wen, Q. (2016). “Does Regret Matter in First-Price Auctions?” Economics Letters. doi:10.1016/j.econlet.2016.03.021
  13. Balseiro, S., Besbes, O. & Weintraub, G. (2015). “Repeated Auctions with Budgets in Ad Exchanges: Equilibria and Design.” Management Science 61(4):864–884. doi:10.1287/mnsc.2014.2022
  14. Balseiro, S., Deng, Y., Mao, J., Mirrokni, V. & Zuo, S. (2021). “The Landscape of Auto-Bidding Auctions: Value versus Utility Maximization.” ACM EC. doi:10.1145/3465456.3467607
  15. Despotakis, S., Ravi, R. & Sayedi, A. (2021). “First-Price Auctions in Online Display Advertising.” Journal of Marketing Research. doi:10.1177/00222437211030201
  16. Goke, S., Weintraub, G.Y., Mastromonaco, R. & Seljan, S. (2021). “Bidders’ Responses to Auction Format Change in Internet Display Advertising Auctions.” arXiv:2110.13814
  17. Bichler, M., Gupta, V. & Oberlechner, M. (2026). “Revenue in First- and Second-Price Display Advertising Auctions: Understanding Markets with Learning Agents.” Information Systems Research. doi:10.1287/isre.2025.2160
  18. Alcobendas, M., Ji, J., Gokulakannan, H., et al. (2023). “Optimizing Floors in First Price Auctions: An Empirical Study of Yahoo Advertising.” arXiv:2302.06018
  19. Fagandini, A. & Dierickx, I. (2023). “Computing profit-maximizing bid-shading factors…,” Computational Economics. doi:10.1007/s10614-022-10321-y (corrigendum doi:10.1007/s10614-022-10343-6); Tilli, P. & Espinosa-Leal, C. (2021). “Multi-Armed Bandits for Bid Shading in First-Price Real-Time Bidding Auctions.” J. Intell. Fuzzy Systems. doi:10.3233/JIFS-202665
  20. IAB Tech Lab, OpenRTB 2.6, §3.2.1 (at field), §4.4 (AUCTION_PRICE macros). https://github.com/InteractiveAdvertisingBureau/openrtb2.x/blob/main/2.6.md
  21. Google Ad Manager first-price migration (2019); PubMatic, “First Price Auctions & Auction Dynamics.” https://pubmatic.com/blog/first-price-auctions-auction-dynamics/