Improved Bayes Risk Can Yield Reduced Social Welfare Under Competition

TL;DR

The paper investigates how competition among multiple model-providers alters the traditional scaling intuition that larger models and better representations always improve predictive accuracy. It develops a formal classification-competition framework where providers optimize market share and users choose providers based on predictive losses, revealing that equilibrium social welfare can be non-monotonic or even decrease as representation quality improves. Theoretical results in a stylized binary setting yield closed-form characterizations of equilibrium social loss, showing sharp transitions between heterogeneous and homogeneous predictions as Bayes risk changes, with extensions to multiclass and unequal-market scenarios. Empirical analyses with linear predictors and CIFAR-10 experiments corroborate non-monotonic welfare under competition across different representation qualities and provider counts. Overall, the work highlights that gains in single-provider performance do not straightforwardly translate to societal welfare improvements in competitive data markets, motivating evaluation of scaling in realistic competitive settings.

Abstract

As the scale of machine learning models increases, trends such as scaling laws anticipate consistent downstream improvements in predictive accuracy. However, these trends take the perspective of a single model-provider in isolation, while in reality providers often compete with each other for users. In this work, we demonstrate that competition can fundamentally alter the behavior of these scaling trends, even causing overall predictive accuracy across users to be non-monotonic or decreasing with scale. We define a model of competition for classification tasks, and use data representations as a lens for studying the impact of increases in scale. We find many settings where improving data representation quality (as measured by Bayes risk) decreases the overall predictive accuracy across users (i.e., social welfare) for a marketplace of competing model-providers. Our examples range from closed-form formulas in simple settings to simulations with pretrained representations on CIFAR-10. At a conceptual level, our work suggests that favorable scaling trends for individual model-providers need not translate to downstream improvements in social welfare in marketplaces with multiple model providers.

Figures (5)

• Figure 1: Comparison of equilibrium loss on two data distributions, one with high Bayes risk (left) and one with lower Bayes risk (right). Each plot shows the linear predictors chosen at equilibrium under competition between three model-providers (solid lines), along with two approximately Bayes-optimal predictors (dashed lines). The equilibrium social loss is lower in the left plot than the right plot, even though the Bayes risk is much higher. The intuition is that approximate Bayes optima disagree on more data points in the left plot than in the right plot; thus, users have a greater likelihood of at least one predictor offering them a correct prediction, which increases the overall predictive accuracy for users (i.e., the social welfare).
• Figure 2: Equilibrium social loss (y-axis) versus data representation quality (x-axis) given $m$ model-providers, for different function classes $\mathcal{F}$ (rows) and when representations are varied along different aspects (columns). Top row: $\mathcal{F} = \mathcal{F}^{\text{binary}}_{\text{all}}$, with closed-form formula from Proposition \ref{['prop:idealized']}. Bottom row: linear functions, computed via simulation (Section \ref{['sec:linear']}). We vary representations with respect to per-representation Bayes risk (a,d), noise level (b,e), and dimension (c,f). The dashed line indicates the Bayes risk (omitted if it is too high to fit on the axis). The Bayes risk is monotone, but the equilibrium social loss is non-monotone.
• Figure 3: Equilibrium social loss (y-axis) versus data representation quality (x-axis) given two model-providers with market reputations $[1 - w_{\text{min}}, w_{\text{min}}]$ when representations are varied along different aspects (columns). The equilibrium social loss is computed via the closed-form formula from Proposition \ref{['prop:2model-providers']}. We vary representations with respect to per-representation Bayes risk (a), noise level (b), and dimension (c). The dashed line indicates the Bayes risk. The Bayes risk is monotone for all 3 axes of varying representations; on the other hand, the equilibrium social loss is non-monotone in the per-representation Bayes risk and monotone in noise level and dimension.
• Figure 4: Equilibrium social loss (left) and Bayes risk (right) on a binary classification task on CIFAR-10 (Section \ref{['subsec:image']}). Representations are generated from different networks pre-trained on ImageNet. The points show the equilibrium social loss when $m$ model-providers compete with each other (left) and the Bayes risk of a single model-provider in isolation (right). While Bayes risk is decreasing in this representation ordering, the equilibrium social loss is non-decreasing in this ordering. The equilibrium social loss is thus non-monotonic in representation quality as measured by Bayes risk. Error bars are 1 standard error.
• Figure 5: Equilibrium social loss (left) and Bayes risk (right) on a 10-class classification task on CIFAR-10 (Section \ref{['subsec:image10']}). Representations are generated from different networks pre-trained on ImageNet. The points show the equilibrium social loss when $m$ model-providers compete with each other (left) and the Bayes risk of a single model-provider in isolation (right). While Bayes risk is decreasing in this representation ordering, the equilibrium social loss is non-decreasing in this ordering. The equilibrium social loss is thus non-monotonic in representation quality as measured by Bayes risk. Error bars are 1 standard error.

Theorems & Definitions (16)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Lemma 1
• proof
• proof : Proof of Proposition \ref{['prop:existence']}
• proof : Proof of Proposition \ref{['prop:idealized']}
• proof : Proof of Proposition \ref{['prop:idealizedmulticlass']}
• Lemma 2