🤖 machine learning

Model Monotonicity in Autobidding Auctions: When Do Better Predictions Lead to Better Outcomes?

This paper introduces a framework to analyze when improvements in machine learning prediction models lead to better platform-level outcomes in autobidding auctions, revealing that while revenue monotonicity holds for tCPA bidders in first-price auctions without budgets, it can be broken by second-price formats or budget constraints.

Original authors: Ashwinkumar Badanidiyuru

Published 2026-06-01

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ashwinkumar Badanidiyuru

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are running a massive, high-speed digital auction house. Every second, millions of people are browsing the web, and advertisers want to show them ads. To decide who gets to show an ad, the platform uses three main ingredients:

The Crystal Ball (The Model): A machine learning system that guesses how likely a user is to click or buy something.
The Auction Ring: The rules of the game (like "highest bidder wins" or "second-highest bidder pays").
The Autobidders: Robots working for advertisers. They automatically place bids based on the Crystal Ball's predictions and the advertiser's goals (like "spend no more than $10 per sale").

The Big Question:
The paper asks a simple but tricky question: If we make the Crystal Ball smarter and more accurate, does the auction house always make more money or do a better job?

Intuitively, you'd think "Yes, of course! Better predictions mean better decisions." But the authors, Ashwinkumar Badanidiyuru, discovered that this is often not true. Sometimes, making the Crystal Ball more detailed actually hurts the platform's results.

Here is the breakdown of their findings using everyday analogies:

1. The "Refinement" Analogy: Sorting the Fruit Basket

The authors define a "better" model as a refined one.

The Coarse Model: Imagine a fruit basket where you just say, "This whole basket is 50% apples." You treat every apple the same.
The Refined Model: Now, you look closer. You separate the basket into "Red Delicious," "Granny Smith," and "Golden Delicious." You know exactly which specific apples are sweet and which are sour.

In the paper's math, moving from the "Coarse" basket to the "Refined" basket is what counts as an "improvement." The question is: Does this extra detail always help?

2. The Surprising Answer: It Depends on the Rules

The paper finds that the answer depends entirely on how the auction is run and what the advertisers are trying to do.

The "Safe" Scenario: The First-Price Auction (The "Pay What You Bid" Game)

The Setup: Advertisers want a specific cost per sale (tCPA). They use a "First-Price" auction (if you bid $5, you pay $5).
The Result: If advertisers do not have a spending limit (budget), then YES, a better model always leads to more revenue and better outcomes.
The Analogy: Think of this like a group of friends splitting a bill. If everyone is honest about what they can afford and there's no cap on the total bill, getting a more accurate receipt (better model) just ensures everyone pays exactly what they owe. No one loses out. The math behind this is called Jensen's Inequality, which basically says that averaging a "bumpy" curve (like bidding strategies) always gives you a lower result than looking at the bumps individually.

The "Dangerous" Scenarios: Where Better Models Backfire

The paper shows that in almost every other situation, a better model can actually lower revenue or efficiency. Here are the traps:

A. The "Second-Price" Trap (VCG/SPA)

The Setup: In these auctions, the winner pays the second-highest bid, not their own.
The Problem: When you get a better model, you might realize that a specific user is only valuable to one specific advertiser. In the "Coarse" model, that user looked like a generic target, so many advertisers bid aggressively, driving the price up. In the "Refined" model, everyone realizes only one person wants that user, so competition drops, and the price plummets.
The Analogy: Imagine a rare collectible.
- Coarse View: "This is a cool vintage toy." Five people bid $100. Winner pays $100.
- Refined View: "This is a vintage toy that only appeals to people who love 1990s cartoons." Only one person wants it. They bid $10. The winner pays $10.
- Result: The model got smarter, but the auction house made less money.

B. The "Budget" Trap

The Setup: Advertisers have a hard limit on how much they can spend (e.g., "I only have $1,000 today").
The Problem: When the model gets better, it helps advertisers find the "best" users very quickly. They spend their $1,000 in the first hour of the day on the top-tier users. By the time the day is over, they have no money left to bid on the average users.
The Analogy: Imagine a buffet with a $20 limit.
- Coarse View: You don't know exactly which dishes are the best, so you spread your $20 out over the whole day, eating a little bit of everything.
- Refined View: You know exactly which three dishes are the absolute best. You eat them all immediately and spend your whole $20 in 10 minutes. The buffet owner (the platform) loses out on the sales from the rest of the day because you ran out of money too fast.

C. The "Value Maximizer" Trap

The Setup: Advertisers who just want to get the most value possible, regardless of the specific cost-per-acquisition target.
The Problem: Similar to the budget issue, better models allow these robots to "game" the system more effectively, often reducing the competitive pressure that drives up prices for the platform.

3. The One Exception: The "Centralized Chef" (LP Benchmark)

The paper mentions one scenario where better models always help: If a central computer (not a strategic auction) directly assigns ads to maximize total value while respecting budgets.

The Catch: This isn't a real auction. It requires the platform to know the advertisers' private secrets (their exact budgets and values) to do the math. In the real world, advertisers won't share those secrets. So, while this works in theory, it's not a practical auction format.

The Bottom Line

The paper's main takeaway is a warning to advertising platforms: Do not assume that a smarter AI model automatically means more money.

Better models are a double-edged sword. They can help advertisers target better, but that can reduce competition or cause them to run out of money too fast.
The rules matter. If you use a "First-Price" auction with advertisers who have no spending limits, you are safe. In almost every other case (Second-Price, Budgets, different bidder types), a better model can actually make things worse.
Test before you trust. The authors suggest that platforms shouldn't just look at the model's accuracy scores. They must run simulations or A/B tests to see if the actual business metrics (revenue, welfare) go up or down when the model gets smarter.

In short: More information doesn't always mean a better outcome; it depends entirely on the game you are playing.

Technical Summary: Model Monotonicity in Autobidding Auctions

1. Problem Definition

Online advertising platforms rely on a complex interplay between three components: machine learning models predicting user behavior (pCTR/pCVR), auction mechanisms for allocating impressions, and autobidding systems that act on behalf of advertisers. A prevailing industry intuition suggests that improving prediction model accuracy should inherently lead to better platform-level outcomes, such as increased revenue or welfare.

However, this paper challenges that intuition. It posits that model improvements can paradoxically degrade platform metrics due to the feedback loop between refined predictions and strategic autobidder behavior. The central research question is: Under what conditions do improvements in prediction models (defined as model refinement) guarantee improvements in Evaluation Criteria Metrics (ECM) like revenue, welfare, or liquid welfare?

2. Methodology and Framework

2.1. Model Refinement via Filtrations

The authors formalize "model improvement" not through standard pointwise metrics (e.g., AUC or log-loss) but through a structural relation inspired by filtrations in probability theory.

User Partition: A prediction model $M$ is defined by a partition $\mathcal{P}$ of the user universe $U$ into clusters. For each cluster, the model outputs the average true conversion probability.
Refinement Relation: A model $M_A$ refines $M_B$ (denoted $M_A \succeq M_B$ ) if every cluster in $M_A$ 's partition is contained within a cluster of $M_B$ 's partition. This implies $M_A$ makes finer distinctions between users while maintaining calibration (predictions equal true cluster averages).
Implication: This definition ensures that any concave Bayes risk associated with a proper scoring loss improves in expectation under refinement (via Jensen's inequality), providing a rigorous mathematical basis for "better" models.

2.2. Bidder Models and Auction Formats

The study analyzes the interaction of model refinement across specific combinations of:

Bidder Types:
- tCPA (Target CPA): Advertisers maximize conversion volume subject to an average cost-per-acquisition constraint.
- MAX-CPA: Advertisers maximize quasi-linear utility (value minus payment) with a maximum willingness-to-pay per conversion.
Auction Formats:
- First-Price Auction (FPA): Highest bidder wins and pays their bid.
- VCG/Second-Price Auction (SPA): Highest bidder wins and pays the second-highest bid (externality).
- LP Benchmark: A centralized, non-strategic fractional allocation mechanism used as a theoretical baseline.
Constraints: Settings are analyzed both with and without budget constraints.

2.3. Evaluation Criteria Metrics (ECM)

The paper evaluates three primary metrics:

Revenue: Total payments collected by the platform.
Welfare: Total expected value of conversions ( $\sum v_i \cdot p_{i,u}$ ).
Liquid Welfare: Welfare capped by advertiser budgets, serving as the efficiency benchmark for budget-constrained settings.

3. Key Contributions

The paper provides a systematic characterization of ECM Monotonicity—the property where $M_A \succeq M_B \implies ECM(M_A) \geq ECM(M_B)$ . The authors establish that monotonicity is the exception rather than the rule, holding only in specific, narrow configurations.

3.1. Positive Results (Monotonicity Holds)

The paper proves that model refinement guarantees metric improvements only in the following three scenarios:

tCPA Bidders in FPA (No Budgets): Under uniform bidding with an optimal multiplier $\mu=1$ , revenue (and consequently welfare, given $v_i=t_i$ ) is monotone. This is proven via Jensen's inequality applied to the convex function of the maximum bid.
MAX-CPA Bidders in VCG (No Budgets): Welfare is monotone. Since VCG allocates to maximize welfare and truthful bidding is dominant, refinement improves the allocation quality. However, revenue is not guaranteed to be monotone.
LP-Based Allocation (With Budgets): A centralized linear programming benchmark guarantees welfare monotonicity for both bidder types. For tCPA bidders, surrogate revenue is also monotone. This result relies on a "lifting argument" where a feasible coarse allocation can be mapped to the refined partition without violating constraints.

3.2. Negative Results (Monotonicity Fails)

The paper demonstrates that monotonicity fails in all other studied settings, providing explicit numerical constructions for these counterexamples:

VCG/SPA with tCPA Bidders: Both revenue and welfare can decrease. Refinement can alter the ranking of bidders across sub-clusters, reducing competitive pressure and lowering the second-highest bid (externality).
FPA with Budget Constraints: Revenue can decrease. Budget constraints force bidders to pace (shade bids), and refinement can reveal segments where constrained bidders win more cheaply, reducing overall revenue.
VCG with MAX-CPA Bidders: While welfare is monotone, revenue is not, as externalities can decrease under refinement.
MAX-CPA in FPA: Both revenue and welfare can decrease when multipliers are allowed to vary with the information structure, breaking the Jensen's inequality argument.
Liquid Welfare: Even with budgets, liquid welfare is not monotone in strategic auctions due to "crowding out" effects.

4. Significance and Claims

The paper claims to be the first work to formally study the conditions under which prediction model improvements guarantee platform metric improvements in autobidding settings. Its significance lies in the following insights:

Rarity of Monotonicity: The assumption that "better models equal better outcomes" is theoretically unsound for most practical auction configurations. Platforms cannot rely on model refinement alone to drive business metrics.
Trade-offs: There is a fundamental tension between incentive compatibility and monotonicity.
- FPA with tCPA (No Budgets) offers monotonicity but fails with budgets.
- LP Allocation offers monotonicity with budgets but requires the platform to know private bidder parameters (targets/values), rendering it incentive-incompatible.
- No known mechanism achieves both monotonicity and incentive compatibility simultaneously under budget constraints.
Practical Implications: The findings suggest that advertising platforms should not assume model improvements will translate to metric gains. Instead, improvements must be validated through simulation or A/B testing. The interaction between ML models, auction formats, and autobidder strategies creates complex dynamics where component-level optimizations may not yield system-level gains.

The paper concludes that while refined models provide more information, the strategic response of autobidders in specific auction environments (particularly those involving budgets or second-price mechanisms) can neutralize or reverse the benefits of that information.