One-Shot Pricing for Hands-Off-the-Wheel Advertising Markets

Technical Summary: One-Shot Pricing for Hands-Off-the-Wheel Advertising Markets

1. Problem Statement

The paper addresses the allocation mechanism in "hands-off-the-wheel" (HOTW) or "autobidding" online advertising markets. In these markets, advertisers declare a budget ($b_i$) and a return-on-investment (ROI) target ($r_i$), while the exchange (e.g., a DSP or integrated publisher like Google or Meta) manages the procurement of clicks using machine learning models.

Currently, despite the exchange possessing all necessary information—predicted click values ($v_{ij}$), declared budgets, ROI targets, and forecasted supply ($s_j$)—the industry relies on millions of real-time, per-impression auctions (typically first-price or second-price). The authors argue that this architecture is a legacy of an era where exchanges lacked value information and needed auctions for price discovery. In the HOTW regime, where values are known and inputs are declared, per-impression auctions introduce unnecessary noise, convergence uncertainty, and operational latency without providing additional information. The core problem is to determine if a centralized, one-shot pricing mechanism can replace these sequential auctions while satisfying all advertiser constraints and maximizing exchange revenue.

2. Methodology and Model

The authors model the HOTW market as a Fisher market with a specific modification to account for ROI constraints.

• Market Structure:

  • Agents: $N$ advertisers and $M$ click types (homogeneous impressions).
  • Parameters: Advertiser $i$ has a value $v_{ij}$ for click type $j$ (predicted by ML), a budget $b_i$, and an ROI target $r_i$. Supply $s_j$ is known.
  • ROI Constraint: Modeled as a personalized outside option. An advertiser can allocate budget to an outside option yielding a normalized "bang-per-buck" of 1 (cost $r_i$, utility $r_i$). This implies an advertiser only purchases a click if $v_{ij}/\alpha_j \geq r_i$, where $\alpha_j$ is the price.
  • Exchange Objective: Maximize total revenue $\sum p_i$ subject to supply constraints, budget constraints ($p_i \leq b_i$), and ROI constraints ($p_i \leq \frac{1}{r_i} \sum v_{ij}x_{ij}$). The exchange is restricted to uniform, anonymous per-type prices ($\alpha_j$).

• Optimization Approach:
  Instead of solving a bilevel optimization problem (searching for prices $\alpha$ that induce optimal advertiser responses), the authors utilize the Eisenberg-Gale (EG) convex program.

  • They formulate an augmented EG program that maximizes budget-weighted log utility (Nash social welfare) with a penalty term for the outside option:
    $$\max_{x,z} \sum_{i \in N} \left[ b_i \log \left( \sum_{j \in M} u_{ij} x_{ij} + r_i z_i \right) - r_i z_i \right]$$
    subject to supply constraints, where $u_{ij} = v_{ij}/r_i$ is the normalized willingness-to-pay, and $z_i$ is the quantity of the outside option.
  • The primal solution $x^*$ yields the optimal allocation, and the KKT multipliers on the supply constraints yield the equilibrium prices $\alpha^*$.

• Algorithmic Implementation:
  The paper proposes a quasi-ascending auction algorithm (adapted from Devanur et al. [2008] (DPSV)) to compute the equilibrium. This algorithm iteratively raises prices on active click types until supply equals demand or advertisers switch to the outside option, effectively solving the convex program via a primal-dual approach.

3. Key Contributions and Results

A. Equivalence to Competitive Equilibrium
The paper establishes that the HOTW market is isomorphic to a quasi-Fisher market (Finster, Gao, and Lahaie [2025]). The competitive equilibrium of this market, computable via the Eisenberg-Gale program, provides market-clearing prices and allocations that satisfy all budget and ROI constraints simultaneously.

B. Revenue Optimality of Competitive Prices
A central and counter-intuitive result is that the competitive equilibrium prices $\alpha^*$ are revenue-optimal for the exchange among all uniform-price mechanisms.

• In standard monopoly pricing, a seller restricts quantity to raise prices above the competitive level.
• In this HOTW setting, demand exhibits a "corner-solution" property: once a price exceeds an advertiser's threshold ($u_{ij}$), the advertiser exits entirely to the outside option rather than gradually reducing demand.
• Consequently, there is no "marginal buyer" whose reduced demand can be offset by higher payments from remaining buyers. The exchange cannot increase revenue by restricting supply; the market-clearing price is the revenue ceiling.

C. Outcome Equivalence to Paced Auctions
The authors prove that the one-shot competitive equilibrium outcome is outcome-equivalent to the equilibrium of sequential first-price auctions with pacing (a mechanism currently used in practice). The pacing multipliers used in sequential auctions can be computed ex-ante by the exchange as the equilibrium prices $\alpha^*$. This implies that the complex machinery of real-time bidding can be replaced by a single convex program without altering the final allocation or revenue.

D. Arbitrage-Free Dynamics
The model is extended to time-varying supply and demand. The resulting prices are shown to be arbitrage-free: an advertiser's realized ROI is equalized across all periods in which they are active, preventing strategic timing advantages.

4. Significance and Claims

The paper claims that the current reliance on real-time auctions in HOTW markets is an operational inefficiency. By recognizing that the exchange acts as a monopolist facing a downward-sloping demand curve derived from known values and declared constraints, the exchange can solve for optimal prices centrally.

• Operational Simplicity: The approach replaces millions of dynamic, strategic bidding interactions with a single convex optimization problem (or a polynomial-time ascending auction algorithm), which is described as operationally simpler than dynamically evolving bidding strategies.
• Revenue Guarantee: Contrary to the intuition that auctions are necessary for revenue maximization in multi-unit settings, the paper demonstrates that the competitive equilibrium price is the revenue-maximizing uniform price.
• Pareto Efficiency: The resulting allocation maximizes the budget-weighted log-utility (Nash social welfare), ensuring Pareto efficiency for the advertisers while maximizing the exchange's revenue.

The authors conclude that the "one-shot" approach is not only theoretically sound but practically superior, offering a simpler and revenue-optimal alternative to the current auction-based infrastructure.