Efficiency of Proportional Mechanisms in Online Auto-Bidding Advertising

Abstract: The rise of automated bidding strategies in online advertising presents new challenges in designing and analyzing efficient auction mechanisms. In this paper, we focus on proportional mechanisms within the context of auto-bidding and study the efficiency of pure Nash equilibria, specifically the price of anarchy (PoA), under the liquid welfare objective. We first establish a tight PoA bound of 2 for the standard proportional mechanism. Next, we introduce a modified version with an alternative payment scheme that achieves a PoA bound of $1 + \frac{O(1)}{n-1}$ where $n \geq 2$ denotes the number of bidding agents. This improvement surpasses the existing PoA barrier of 2 and approaches full efficiency as the number of agents increases. Our methodology leverages duality and the Karush-Kuhn-Tucker (KKT) conditions from linear and convex programming. Despite its conceptual simplicity, our approach proves powerful and may offer broader applications for establishing PoA bounds.

• The paper establishes the Price of Anarchy of 2 for standard proportional allocation in auto-bidding ad auctions with heterogeneous agent objectives and constraints.
• It employs a primal-dual LP framework and KKT conditions to analyze pure Nash equilibria and measure liquid welfare efficiency.
• A modified mechanism with decoupled bid-payment rules achieves a PoA of 1+ε, indicating asymptotic full efficiency as the market scales.

Efficiency of Proportional Mechanisms in Online Auto-Bidding Advertising

Problem Setting and Motivation

The proliferation of automated bidding (auto-bidding) in online advertising, driven by the increasing complexity and scale of digital markets, necessitates a rigorous understanding of the efficiency of underlying auction mechanisms. In auto-bidding environments, advertisers often specify high-level objectives, delegating bid optimization under budget and performance constraints to automated agents. This transition exacerbates interaction complexity, prompting the need for robust mechanism design to ensure market efficiency.

The study considers a canonical divisible item ad auction model with $n$ agents and $m$ items, where each agent $i$ possesses a private, concave, continuous, non-decreasing valuation function $v_i$ and a budget $W_i$. Agents may be utility-maximizing, valuation-maximizing, or hybrid, distinguished by their objectives and constraints, notably budget and return-on-spend (RoS). The auctioneer's objective is the maximization of liquid welfare: the sum of each agent's valuation, capped by their respective budget. This metric aligns tightly with revenue benchmarks and reflects the welfare attainable in budget-constrained settings.

Proportional Mechanisms and Pure Nash Efficiency

The principal mechanism in focus is the proportional allocation (Kelly) mechanism, in which each item $j$ is divided among agents proportionally to their bids $b_{ij}$. Payments are determined directly by the submitted bids in the standard regime. The paper rigorously explores the equilibria of such mechanisms, restricting attention to pure Nash equilibria, and analyzes their efficiency via the canonical Price of Anarchy (PoA) measured with respect to liquid welfare.

Prior literature establishes a tight PoA bound of 2 in various auction settings involving budget or RoS-constrained agents, with both utility and valuation maximization, and this bound is known to be intractable for standard mechanisms under modest assumptions. The main result here confirms this bound for the standard proportional mechanism under the broadest setting—agents with both types of objectives and both budget and RoS constraints—thus closing the gap for this class.

The approach employs an advanced primal-dual analytical technique, formulating the welfare maximization as a configuration LP, analyzing agent best-response structures via KKT conditions, and constructing feasible dual solutions parameterized by equilibrium variables. This framework enables precise upper bounding of the liquid welfare at equilibria relative to the optimal.

Key Result:

• The Price of Anarchy of the standard proportional allocation mechanism for pure Nash equilibria under liquid welfare is tightly 2, even with heterogeneous agent objectives and constraints.

Surpassing the Efficiency Barrier: Mechanism Variant

The crux of the theoretical contribution is the construction of a modified proportional mechanism with an alternative payment rule that overcomes the previously established PoA barrier of 2. The new payment function, parameterized by a function $g(u)$ and an offset $h(u)$, is analytically specified so that payments may, in some cases, exceed the submitted bid—uncoupling bid and payment while respecting budget/utility constraints.

By setting $g(u) = u^{1 + (n-1)\epsilon}$ for a small $m$0, the variant mechanism achieves:

Strong Claim:

• The Price of Anarchy of pure Nash equilibria for this mechanism is at most $m$1 (where $m$2), implying asymptotic full efficiency as $m$3.

Notably, this circumvents previously proven lower bounds that apply to mechanisms with direct bid-payment linkage, leveraging the relaxation that bids can be decoupled from required payments as long as agent feasibility is preserved. The LP duality/KKT analytical toolkit is again employed to verify feasibility and tightness of the dual, extending to the complex case of mixed-objective hybrid agents.

The work also introduces a modification ensuring that payments never exceed bids, enabled by scaling and shifting the admissible bid region, conditional on the existence of a public upper bound on individual bids. Despite this safeguard, the mechanism preserves the tight PoA and is thus operationally appealing.

Implications and Theoretical Significance

The findings have significant implications for both theoretical and applied mechanism design in auto-bidding environments:

• They provide the first general mechanism achieving PoA strictly less than 2 under mild and natural assumptions, answering an open question in the field.
• The results challenge established impossibility frontiers, underscoring the possibility of improved efficiency given careful relaxation of payment constraints.
• Practically, mechanisms with asymptotically optimal efficiency present a blueprint for market designers and platform architects as competition intensifies and agent heterogeneity grows.

The analytical techniques—primal-dual LP constructions, KKT-based equilibrium characterization, and careful convexity arguments—are broadly applicable and may facilitate further advances in other resource allocation markets and blockchain-based protocols.

Future Research Directions

Several natural extensions emerge:

• Tightening of efficiency guarantees for mixed and correlated equilibrium concepts, extending beyond pure Nash.
• Exploration of robustness under dynamic or bandit-feedback learning dynamics as in repeated ad auctions.
• Empirical evaluation and possible adaptation of the proposed payment rules in production-scale ad exchanges, with concrete implications for real-world welfare and revenue.
• Application to other divisible-goods market settings, especially where budget constraints and payment-bid decoupling are tractable.
• Integration with machine learning-based forecasting or value estimation, which may further improve market efficiency.

Conclusion

The paper rigorously characterizes the efficiency of proportional mechanisms in the auto-bidding advertising context, establishing a tight upper bound of 2 for the classical scheme and surpassing this with a novel mechanism approaching full efficiency as the market scales. The theoretical advances in modeling, analysis, and mechanism construction enrich both the foundations and applications of mechanism design in digital economics, offering actionable insights for theory and practice alike (2604.12799).