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One-Shot Pricing for Hands-Off-the-Wheel Advertising Markets, 2026, with Emerson Melo & Matthew Shum
Per-impression auctions have long served as the allocation mechanism in online advertising. We argue that in ``hands-off-the-wheel'' (HOTW) markets, where advertisers declare budgets and ``return-on-investment'' (ROI) targets and the exchange's ML models predict click values, auctions are no longer necessary: all information required for optimal pricing is already known to the exchange, which occupies the position of a monopolist pricing against a downward-sloping demand curve. A HOTW market is a Fisher market, whose competitive equilibrium can be computed via the convex program of Eisenberg and Gale, yielding market-clearing prices and allocations satisfying all budget and ROI constraints simultaneously. This competitive-equilibrium price is revenue-optimal for the exchange among all uniform-price mechanisms: avoiding the demand reduction problem in typical uniform-price multi-unit auctions. The resulting competitive equilibrium is moreover outcome-equivalent to sequential first-price auctions with pacing, with pacing multipliers computable ex-ante by the exchange. This one-shot approach replaces millions of individual auctions with one convex program, which is not only operationally simpler than dynamically evolving bidding strategies, but revenue-optimal for the exchange, while delivering the same equilibrium outcome.
Prophet Inequalities via Linear Programming, 2026, with Halil I. Bayrak and Mustafa C. Pinar
Prophet inequalities bound the expected reward that can be obtained in a stopping problem by the optimal reward of its corresponding off-line version. We propose a systematic technique for deriving prophet inequalities for stopping problems associated with selecting a point in a polyhedron. It utilizes a reduced-form linear programming representation of the stopping problem. We illustrate the technique to derive a number of known results as well as some new ones. For instance, we prove a 1/2-prophet inequality when the underlying polyhedron is an on-line polymatroid; one whose underlying submodular function depends upon the realized rewards. We also demonstrate a composition by the Minkowski sum property. If an r-prophet inequality holds for polyhedra P^1 and P^2 it also holds for their Minkowski sum.
Survey on Mechanism Design with Inspection (DRAFT), 2026
This survey examines the literature on mechanism design with inspection, organized around Ben-Porath, Dekel and Lipman (2014) (henceforth BDL): a principal allocates a good to one of several privately informed agents, cannot use transfers, but can inspect an agent's claim at a cost. BDL can be interpreted as an instance of Townsend's costly state verification when transfers are not permitted. I discuss this connection along with two further related literatures that consider superficially similar problems with different primitives: costly signaling and evidence and disclosure games. I then survey work that extends or varies the BDL model along nearly every dimension of its primitives: the inspection technology (perfect, partial, probabilistic, noisy, capacity-constrained), the punishment structure, the presence and structure of transfers, the timing of inspection, the number and divisibility of goods, the informativeness of agents' own private information, and the underlying objective (welfare, revenue, or collective choice).
Survey of Algorithmic Collusion (DRAFT), 2026
Our models of price competition make no distinction between prices set by humans as opposed to algorithms. So what new challenges arise when algorithms set prices? I argue that if human- and algorithm-set prices differ in kind, it is in what can be policed. When humans set prices, it is unclear what rule (if any) they follow. When a firm delegates pricing to an algorithm, it chooses a rule rather than a price. That rule exists in the code, which can, in principle, be read, certified, run against hypothetical rivals, and audited against a complete record of what it did. That makes a new instrument available: the law can prescribe the class of rules firms may deploy, the way those rules may learn, rather than inferring conduct after the fact. I survey the literature on algorithmic pricing to identify the challenges associated with policing algorithms to prevent supra-competitive prices.
The Complexity of Benchmark Testing, 2026, with Lance Fortnow
Benchmark testing of large language models (LLMs) is, we argue, the same problem that economists and statisticians studied decades ago under the name \emph{forecast testing}. Forecast testing has three agents: Nature, a Forecaster, and a Tester. The correspondence with LLM benchmarking is exact. The Forecaster is the LLM: both are algorithms that, given the history so far---which includes both the forecaster's own past outputs and Nature's realized outcomes---produce a probability distribution over the next token. Nature is the benchmark's distribution over questions and their correct answers. The Tester is the grading procedure that checks whether the benchmark questions were answered correctly and decides whether to certify the model. Benchmarks are used both to compare LLMs and to certify them; we focus on certification, where certifying the LLM is the Tester passing the Forecaster. The name ``forecast testing'' is historical, from the original motivation of testing weather forecasters; nothing requires the forecaster's outputs to be probabilities of rain---any token sequence works. Under this correspondence, Sandroni's theorem---for any test that passes or fails in a finite number of steps and passes every forecaster whose forecasts agree with Nature, there is a probabilistic forecaster that passes with high probability even when Nature is adversarial---says that there is a machine-learning model that will pass any such benchmark with high probability knowing only how the test works, without knowing the questions. The Fortnow--Vohra theorem---when forecasters and testers are required to be computationally efficient, there is a distribution of Nature under which passing the test ignorantly requires factoring integers---says that benchmarks built on verifiable computational hardness resist such gaming, and LLMs indeed have trouble factoring, for analogous reasons. We develop this dictionary in full, reinterpret hallucination and benchmark contamination as instances of a forecaster optimizing against the tester rather than against Nature, and derive concrete design lessons: the two known escapes from manipulability, computational hardness and counterfactual conditioning, correspond respectively to benchmarks built from freshly generated, verifiably hard instances and to benchmarks that score consistency across families of counterfactual question variants.
Finite-dimensional approach to optimal dynamic information acquisition, 2026
Zhong (2022) characterizes, in continuous time, how a decision maker should acquire information before acting when the state is binary. For more than two states, his characterization assumes that an associated partial differential equation has a solution with suitable regularity and shape. I study the same problem with time in discrete periods and beliefs restricted to a finite grid. The decision maker's problem is then a finite linear or concave program, and each property of its solution follows from an elementary fact about such programs: duality, the number of positive entries in a basic solution, the greedy solution of a knapsack problem, and the mediant inequality. For two states the discrete model reproduces every qualitative feature of Zhong's characterization: the value function is quasi-convex, learning is confirmatory, effort rises with the value, targets move toward the current belief as that value rises, and the decision maker acts immediately after a breakthrough. For any number of states, an optimal strategy exists; it uses at most one more posterior belief than there are states, effort rises with the stakes, and the value is no lower at beliefs between the current belief and the posteriors it moves to. The last is the discrete form of Zhong's result on the direction of learning, which he obtains for more than two states assuming the existence of a desired solution for the associated partial differential equation.
Confidence and conformal sets are used to inform decisions. To employ them, a decision maker must give it a probabilistic interpretation, and that interpretation determines an ambiguity set: the probability distributions over states she regards as possible. This paper characterizes when a proffered confidence/conformal set admits a coherent probabilistic interpretation, a property we call \emph{rationalizability}. We then study decision making over a rationalizable confidence/conformal set. A common practice assigns zero probability to states outside the proffered set. This replaces the original coverage requirement with one that can become incompatible with additional probabilistic information. The rationalizable set, however, preserves the original guarantee and remains compatible with information refinements. As information accumulates and the ambiguity set shrinks toward a singleton, the associated robust decision rule converges to Bayesian expected-loss minimization. This consistency under information refinement is our normative justification for the coherent interpretation.
Posterior Envelopes and Bayesian Feasibility, 2026, with Thanh Nguyen.
We study Bayesian updating when actions reveal feasible sets of posterior beliefs rather than exact beliefs. In market segmentation, for example, demand curves are unobserved, but optimal prices reveal information about latent demand. With exact posterior beliefs, the splitting lemma requires only that the prior equal the average posterior. With posterior envelopes, average consistency is not enough: each interval of types must contain sufficient prior mass to support gaps between posterior bounds. Applied to segmentation, these restrictions characterize aggregate demand and yield bounds on price sensitivity, welfare, and counterfactual revenue. The framework also applies to auctions, admissions, and classification design.
Randomization and the Robustness of Linear Contracts, 2025, with Ashwin Kambhampati, Bo Peng, Zhihao Gavin Tang & Juuso Toikka
We consider contract design by an uncertainty-averse principal who does not know the production technology available to an agent. For such a principal, randomizing the choice of contract provides a hedge against uncertainty. We show that it is optimal to only use linear contracts and randomize the principal's share according to a log-uniform distribution. Thus, an optimal response to uncertainty may generate rich heterogeneity in contracts offered to observationally identical agents. Our saddle-point characterization of the optimal contract implies that the gain from randomization can be arbitrarily large; optimal randomization does not require commitment; and that asking the agent to report the technology cannot improve the principal's payoff.
Signaling Design, 2025, with Matteo Camboni, Mingzi Niu & Mallesh Pai.
We revisit the classic job-market signaling model of Spence(1973), introducing profit-seeking schools as intermediaries that design the mapping from candidates' efforts to job-market signals. Each school commits to an attendance fee and a monitoring policy. We show that, in equilibrium, a monopolist school captures the entire social surplus by committing to low information signals and charging fees that extract students' surplus from being hired. In contrast, competition shifts surplus to students, with schools vying to attract high-ability students, enabling them to distinguish themselves from their lower-ability peers. However, this increased signal informativeness leads to more wasteful effort in equilibrium, contrasting with the usual argument that competition enhances social efficiency. This result may be reversed if schools face binding fee caps or students are credit-constrained.
Persuasion Made Transparent, 2024
This note highlights the fact that a number of results in information design can be obtained without fuss or muss via linear programming.
The Network Effect of Agency Conflicts, 2019 (with Yiqing Xing and Wu Zhu).
We argue that the nature of firm-level agency conflicts counters the role of network structure in the propagation of shocks. These conflicts can have an effect on system-wide behavior that is both significant and different from those predicted based on network structure alone. This implies that corporate governance can play an important role in macro fluctuations. We consider a collection of firms linked through equity cross-holdings whose managers can take investment decisions in response to an exogenous shock. Prior work concludes that more integrated networks amplify shocks. We find that if managers are subject to default costs or limited liability, this effect is reversed because their investment decisions mitigate the spread of an initial shock. In the face of moral hazard, however, their investment choices amplify an initial shock. In particular, when the network is fully diversified the aggregate effect of idiosyncratic shocks is not small as received wisdom would suggest.
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