Calibrated Smoothness in Auction Efficiency

• Calibrated smoothness programs in auctions are a refined framework that blends classical smoothness analysis with revenue guarantees and behavioral bidding restrictions.
• They leverage calibrated parameter settings under no-underbidding and no-overbidding constraints to achieve improved Price of Anarchy bounds in various auction formats.
• This approach has significant implications for auction design, enhancing both theoretical welfare efficiency and practical revenue performance in multi-item markets.

Calibrated smoothness programs in auctions constitute a methodological refinement for bounding the efficiency loss (Price of Anarchy, PoA) in equilibrium of complex market mechanisms. This framework blends the classical smoothness analysis with revenue guarantees and incorporates auction-specific behavioral restrictions such as no-underbidding and no-overbidding. It yields tight inefficiency guarantees across a broad spectrum of auction formats, notably in simultaneous item auctions (both first- and second-price), capturing intricate agent incentives and heterogeneity. Calibrated smoothness programs are now central to rigorous PoA analysis in both theoretical and applied auction research.

1. The Smoothness Paradigm and its Extension

The classical smoothness framework, originated by Roughgarden and Syrgkanis-Tardos, formalizes a sufficient condition for robust PoA bounds in games and auctions. Given a mechanism (such as a multi-item auction), $(\lambda,\mu)$-smoothness asserts that for every profile of valuations $\mathbf v$ and bids $\mathbf b$, there exist feasible deviations $b_i^*(\mathbf v)$ such that:

$$\sum_i u_i(b_i^*(\mathbf v),\,\mathbf b_{-i};v_i) \ge \lambda\,\mathrm{OPT}(\mathbf v) - \mu\,SW(\mathbf b,\mathbf v)$$

where $u_i$ is the utility of bidder $i$, $\mathrm{OPT}(\mathbf v)$ the optimal welfare for $\mathbf v$, and $SW$ the realized welfare. The PoA bound follows: any coarse correlated equilibrium (CCE) or Bayes-Nash equilibrium (BNE) under independent types satisfies $SW\ge\lambda/(1+\mu)\,\mathrm{OPT}$ (Syrgkanis, 2012).

Calibrated smoothness advances this framework by combining it with a parameterized revenue guarantee and optimizing (calibrating) the trade-offs between these ingredients, structured to the specific auction setting.

2. Revenue Guarantees and No-Underbidding

A $(\gamma,\delta)$-revenue guarantee supplements smoothness with a lower bound of the form:

$$\mathrm{Revenue}(\mathbf b) \ge \gamma\,\mathrm{OPT}(\mathbf v) - \delta\,SW(\mathbf b,\mathbf v)$$

For simultaneous second-price auctions (S2PA), imposing “no-underbidding” (i.e., bidders never bid below their marginal value for an item relative to their possible allocations) yields revenue bounds that cannot be achieved with no-overbidding alone (Feldman et al., 2020). Two main flavors are:

• Item No-Underbidding (iNUB): Each bidder's item bid is at least the marginal value for that item, relative to possible allocative changes.
• Set No-Underbidding (sNUB): For every bidder, the sum of her bids for an optimally winnable set exceeds her marginal value for that set.

Under sNUB, S2PA is $(1,1)$-revenue guaranteed for general monotone valuations, yielding PoA $\geq 1/2$ for CCE/BNE with arbitrary correlation structures (Feldman et al., 2020). For $\alpha$-submodular valuations and iNUB, the parameters become $(\alpha,\alpha)$, and PoA increases to $\alpha/(1+\alpha)$.

3. Combined Calibrated Smoothness Programs

Calibrated smoothness synthesizes the deviations and revenue lower bounds for stronger PoA guarantees by "calibrating" parameters. If a mechanism is both $(\lambda,\mu)$-smooth and $(\gamma,\delta)$-revenue guaranteed (possibly under equilibrium restrictions such as sNUB and no-overbidding), every CCE (or BNE under product priors) satisfies:

$$SW(\mathbf b) \ge \frac{\lambda+\gamma}{1+\mu+\delta}\,\mathrm{OPT}$$

For S2PA with submodular (or XOS) valuations, sNUB and no-overbidding together enable a tight PoA of $2/3$. Calibrated smoothness thus allows one to go beyond the individual $\lambda/(1+\mu)$ or $\gamma/(1+\delta)$ guarantees, and optimize over auction parameters and behavioral restrictions to derive the maximal attainable welfare efficiency bound.

The following table summarizes the calibrated PoA lower bounds for S2PA under different valuation classes and equilibrium restrictions (Feldman et al., 2020):

Valuation Class	Restrictions	PoA Lower Bound
Unit-demand, Submodular ($\alpha=1$)	iNUB + NOB	$2/3$
XOS	sNUB + NOB	$2/3$
Subadditive	strong NOB + sNUB	$2/3^*$
Monotone	sNUB only	$1/2$

$^*$For Bayes-Nash with arbitrary priors under sNUB: $1/2$ applies.

4. Calibration Procedures and Theoretical Intuition

The critical step is the calibration of $(\lambda,\mu)$ and $(\gamma,\delta)$ to the strongest values supported by the structure of the auction and equilibrium constraints. For submodular valuations, smoothness alone provides $(1,1)$ (leading to $1/2$ PoA), but revenue guarantees under sNUB enable $(1,1)$ as well. Combined, this yields $2/3$.

No-underbidding (either iNUB or sNUB) is empirically observed and is justified because bidding below one’s marginal value is weakly dominated in S2PA. This behavioral restriction translates directly into improved, tight lower bounds on achievable auction revenue, which classical smoothness neglects. By summing (smoothness inequality + revenue guarantee), the calibrated approach systematically pushes PoA bounds above $\lambda/(1+\mu)$ thresholds characteristic of the pure smoothness framework.

5. Extension to First-Price Auctions and Heterogeneous Agent Types

The calibrated smoothness methodology generalizes to simultaneous first-price auctions (FPA), including settings with heterogeneous agent objectives such as in autobidding for online advertising (Colini-Baldeschi et al., 26 Jun 2025). Here, per-type $(\lambda_t,\mu_t)$-smoothness is established for value maximizers and utility maximizers, and a calibration vector $\delta$ is introduced to balance contributions from diverse agent types.

An extension theorem elevates single-item smoothness to the full FPA with XOS valuations, under the constraint:

$$\max_{t\in T}(\delta_t\mu_t) + \max_{t\in T}(\delta_t(1-\sigma_t)) \leq 1$$

The global PoA is determined by solving a mathematical program:

$$\alpha = \min \bigg\{\min_{t\in T} \lambda_t, \Big(\max_{t\in T} \frac{\mu_t}{\lambda_t} + \max_{t\in T} \frac{1-\sigma_t}{\lambda_t}\Big)^{-1}\bigg\}$$

For the mixed autobidding model, this yields the tight PoA of $2.18$ by explicit optimization over $(\mu_t, \lambda_t)_{t\in T}$ (Colini-Baldeschi et al., 26 Jun 2025). There is no need for additional combinatorial argumentation beyond verifying feasibility and reading off the program’s solution.

6. Implications, Significance, and Limitations

Calibrated smoothness programs:

• Unify and refine PoA analysis for multi-item auctions by integrating revenue, behavioral, and incentive constraints via explicit mathematical programs.
• Enable robust bounds for both S2PA and FPA in a variety of environments, handling generality in valuation classes (from additive to subadditive and XOS), equilibrium concepts (CCE/BNE), and agent heterogeneity (utility vs value maximizers, budget/ROI constraints).
• Provide exact or tight bounds in instances where classical smoothness or revenue arguments alone are insufficient.

A key insight is that the natural no-underbidding behavior empirically seen in S2PA brings tight efficiency guarantees matching or exceeding those achievable in purely theoretical or idealized assumptions.

A plausible implication is that future auction design and analysis can leverage calibrated smoothness not only for welfare efficiency but as a diagnostic for the impact of behavioral and market design constraints across rich agent populations. However, the approach fundamentally depends on the viability of establishing both strong smoothness and revenue-guarantee bounds under relevant behavioral restrictions; where these are weak, the PoA bounds revert to classical limits.

7. Related Methodologies and Ongoing Directions

The calibrated smoothness framework aligns with, and extends, the established smoothness theory for Bayesian auctions (Syrgkanis, 2012), adapting it to richer behavioral models, multidimensional types, and mechanisms with intricate strategic environments (e.g., autobidding ecosystems). The introduction of calibration vectors and tight mathematical programming of smoothness parameters is an active topic for extensions to yet more general utility functions and market designs. This methodology is central to the latest progress in tightly characterizing the equilibrium inefficiency of online market mechanisms, with practical relevance for online advertising, combinatorial procurement, and decentralized exchange markets (Feldman et al., 2020, Colini-Baldeschi et al., 26 Jun 2025).