Dirk Bergemann Publications

We develop a framework for the optimal pricing and product design of LLMs in which a provider sells menus of token budgets to users who differ in their valuations across a continuum of tasks. Under a homogeneous production technology, we show that users’ high-dimensional type profiles are summarized by a scalar index, reducing the seller’s problem to one-dimensional screening. The optimal mechanism takes the form of committed-spend contracts: buyers pay for a budget that they allocate across token classes priced at marginal cost. We extend the analysis to environments with multiple differentiated models and to competition between a proprietary leader and an open-source fringe, showing that competitive pressure reshapes both the intensive and extensive margins of compute provision. Each element of our theory (token-budget menus, maximum- and minimum-spend plans, multi-model versioning, and linear API pricing) has a direct counterpart in the observed pricing practices of providers such as Anthropic, OpenAI, and GitHub.

We study how market segmentation affects consumers when a monopolist can adjust both prices and product qualities across segments, engaging in second- and third-degree price discrimination simultaneously. We characterize the consumer-optimal segmentation and show that it has a striking structure: consumers with the same value receive the same quality in every segment, though prices differ. Under mild conditions, any segmentation harms consumers if and only if demand is more elastic than a cost-determined threshold. Hence, potential benefits for consumers depend critically on cost and demand elasticities. These findings have implications for regulatory policy regarding price discrimination and market segmentation.

We analyze a nonlinear pricing model where the seller controls both product pricing (screening) and buyer information about their own values (persuasion). We prove that the optimal mechanism always consists of finitely many signals and items, even with a continuum of buyer values. The seller optimally pools buyer values and reduces product variety to minimize informational rents. We show that value pooling is optimal even for finite value distributions if their entropy exceeds a critical threshold. We also provide sufficient conditions under which the optimal menu restricts offering to a single item.

We develop an integrated framework for information design and mechanism design in screening environments with quasilinear utility. Using the tools of majorization theory and quantile functions, we show that both information design and mechanism design problems reduce to maximizing linear functionals subject to majorization constraints. For mechanism design, the designer chooses allocations weakly majorized by the exogenous inventory. For information design, the designer chooses information structures that are majorized by the prior distribution. When the designer can choose both the mechanism and the information structure simultaneously, then the joint optimization problem becomes bilinear with two majorization constraints. We show that pooling of values and associated allocations is always optimal in this case. Our approach unifies classic results in auction theory and screening, extends them to information design settings, and provides new insights into the welfare effects of jointly optimizing allocation and information.

We study mechanism design in environments where agents have private preferences and private information about a common payoff-relevant state. In such settings with multi-dimensional types, standard mechanisms fail to implement efficient allocations. We address this limitation by proposing data-driven mechanisms that condition transfers on additional post-allocation information, modeled as an estimator of the payoff-relevant state. Our mechanisms extend the classic Vickrey–Clarke–Groves framework. We show they achieve exact implementation in posterior equilibrium when the state is fully revealed or utilities are affine in an unbiased estimator. With a consistent estimator, they achieve approximate implementation that converges to exact implementation as the estimator converges, and we provide bounds on the convergence rate. We demonstrate applications to digital advertising auctions and AI shopping assistants, where user engagement naturally reveals relevant information, and to procurement auctions with consumer spot markets, where additional information arises from a pricing game played by the same agents.

How should a buyer design procurement mechanisms when suppliers’ costs are unknown, and the buyer does not have a prior belief? We demonstrate that notably simple mechanisms—those that share a constant fraction of the buyer utility with the seller—allow the buyer to realize a guaranteed positive fraction of the efficient social surplus across all possible costs. Moreover, a judicious choice of the share based on the known demand maximizes the surplus ratio guarantee that can be attained across all possible (arbitrarily complex and nonlinear) mechanisms and cost functions. Results apply to related nonlinear pricing and optimal regulation problems.

As businesses increasingly rely on granular consumer data, the public has increasingly pushed for enhanced regulation to protect consumers’ privacy. We provide a perspective based on the academic marketing literature that evaluates the various benefits and costs of existing and pending government regulations and corporate privacy policies. We make four key points. First, data-based personalized marketing is not automatically harmful. Second, consumers have heterogeneous privacy preferences, and privacy policies may unintentionally favor the preferences of the rich. Third, privacy regulations may stifle innovation by entrepreneurs who are more likely to cater to underserved, niche consumer segments. Fourth, privacy measures may favor large companies who have less need for third-party data and can afford compliance costs. We also discuss technology platforms’ recent proposals for privacy solutions that mitigate some of these harms but, again, in a way that might disadvantage small firms and entrepreneurs.

A municipality (social planner) is seeking to establish a renewable energy community paying the initial investment costs, while also identifying the optimal management framework. In this context, two distinct modes of governance are analyzed: the private and the public one. In the first case, a private (or profit) aggregator oversees the energy community with a monopolistic behavior, while in the other the aggregator is a public owned, or controlled, company following the social approach advocated by the promoter, i.e the municipality. In both scenarios, the effective functioning of the community requires the collection of private data on members’ energy consumption. This process allows for optimal management of the community, but also results in a loss of privacy for members. The model incorporates this as a dis-utility, assuming that the members address the portion of their energy needs not covered by the community’s production by purchasing energy from the manager at a price determined on the basis of the information collected. In addition, the aggregator is allowed to sell the collected data to third parties for financial gain. By integrating the members’ energy valuation and incorporating uncertainty regarding the investment cost, we examine policy recommendations aimed at establishing a community size closer to the social optimum.

We present a model of digital advertising with three key features: (1) advertisers can reach consumers on and off a platform, (2) additional data enhances the value of advertiser–consumer matches, and (3) the allocation of advertisements follows an auction-like mechanism. We contrast data-augmented auctions, which leverage the platform’s data advantage to improve match quality, with managed-campaign mechanisms that automate match formation and price-setting. The platform-optimal mechanism is a managed campaign that conditions the on-platform prices for sponsored products on the off-platform prices set by all advertisers. This mechanism yields the efficient on-platform allocation but inefficiently high off-platform product prices. It attains the vertical integration profit for the platform and the advertisers, and it increases off-platform product prices while decreasing consumer surplus, relative to data-augmented auctions.

In digital advertising, auctions determine the allocation of sponsored search, sponsored product, or display advertisements. The bids in these auctions for attention are largely generated by auto-bidding algorithms that are driven by platform-provided data.

We analyze the equilibrium properties of a sequence of increasingly sophisticated auto-bidding algorithms. First, we consider the equilibrium bidding behavior of an individual advertiser who controls the auto-bidding algorithm through the choice of their budget. Second, we examine the interaction when all bidders use budget-controlled bidding algorithms. Finally, we derive the bidding algorithm that maximizes the platform revenue while ensuring that all advertisers continue to participate.

A soft-floor auction asks bidders to accept an opening price to participate in an ascending auction. If no bidder accepts, lower bids are considered using first-price rules. Soft floors are common despite being irrelevant with standard assumptions. When bidders regret losing, soft-floor auctions are more efficient and profitable than standard optimal auctions. Revenue increases as bidders are inclined to accept the opening price to compete in a regret-free ascending auction. Efficiency is improved since having a soft floor allows for a lower hard reserve price, reducing the frequency of no sale. Theory and experiment confirm these motivations from practice.

We study mechanism design when agents have private preferences and private information about a common payoff-relevant state. We show that standard message-driven mechanisms cannot implement socially efficient allocations when agents have multidimensional types, even under favorable conditions.

To overcome this limitation, we propose data-driven mechanisms that leverage additional post-allocation information, modeled as an estimator of the payoff-relevant state. Our data-driven mechanisms extend the classic Vickrey-Clarke-Groves class. We show that they achieve exact implementation in posterior equilibrium when the state is either fully revealed or the utility is affine in an unbiased estimator. We also show that they achieve approximate implementation with a consistent estimator, converging to exact implementation as the estimator converges, and present bounds on the convergence rate.

We demonstrate applications to digital advertising auctions and large language model (LLM)-based mechanisms, where user engagement naturally reveals relevant information.

We analyze a nonlinear pricing model where the seller controls both product pricing (screening) and buyer information about their own values (persuasion).

We prove that the optimal mechanism always consists of finitely many signals and items, even with a continuum of buyer values. The seller optimally pools buyer values and reduces product variety to minimize informational rents.

We show that value pooling is optimal even for finite value distributions if their entropy exceeds a critical threshold. We also provide sufficient conditions under which the optimal menu restricts offering to a single item.

In digital advertising, the allocation of sponsored search, sponsored product, or display advertisements is mediated by auctions. The generation of bids in these auctions for attention is increasingly supported by auto-bidding algorithms and platform-provided data. We analyze the equilibrium properties of a sequence of increasingly sophisticated auto-bidding algorithms. First, we consider the equilibrium bidding behavior of an individual advertiser who controls the auto-bidding algorithm through the choice of their budget. Second, we examine the interaction when all bidders use budget-controlled bidding algorithms. Finally, we derive the bidding algorithm that maximizes the platform’s revenue while ensuring all advertisers continue to participate.

We consider a seller who offers services to a buyer with multi-unit demand. Prior to the realization of demand, the buyer receives a noisy signal of their future demand, and the seller can design contracts based on the reported value of this signal. Thus, the buyer can contract with the service provider for an unknown level of future consumption, such as in the market for cloud computing resources or software services. We characterize the optimal dynamic contract, extending the classic sequential screening framework to a nonlinear and multi-unit setting. The optimal mechanism gives discounts to buyers who report higher signals, but in exchange they must provide larger fixed payments. We then describe how the optimal mechanism can be implemented by two common forms of contracts observed in practice, the two-part tariff and the committed spend contract. Finally, we use extensions of our base model to shed light on policy-focused questions, such as analyzing how the optimal contract changes when the buyer faces commitment costs, or when there are liquid spot markets.