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Pricing AI Model Accuracy

Nikhil Kumar

TL;DR

The paper analyzes a duopoly where firms compete on FP and FN errors under heterogeneous consumer sensitivities, showing that optimal competition drives specialization: firms profit more by strengthening their advantaged error dimension than by closing the gap on the inferior one. By decomposing errors into FP and FN, and introducing investment and dataset-size considerations, it shows how prices and market shares respond under different correlation structures of consumer sensitivities (positive, negative, and mixed). A key result is that greater differentiation, achieved by investing in a firm’s dominant error dimension, raises equilibrium prices and firm revenues but lowers consumer welfare; however, the total welfare can rise due to efficiency gains and market dynamics. The model integrates fixed-endowment and scalable investment costs, linking AI model market dynamics to pricing, investment incentives, and welfare implications in a stylized competitive setting.

Abstract

This paper examines the market for AI models in which firms compete to provide accurate model predictions and consumers exhibit heterogeneous preferences for model accuracy. We develop a consumer-firm duopoly model to analyze how competition affects firms' incentives to improve model accuracy. Each firm aims to minimize its model's error, but this choice can often be suboptimal. Counterintuitively, we find that in a competitive market, firms that improve overall accuracy do not necessarily improve their profits. Rather, each firm's optimal decision is to invest further on the error dimension where it has a competitive advantage. By decomposing model errors into false positive and false negative rates, firms can reduce errors in each dimension through investments. Firms are strictly better off investing on their superior dimension and strictly worse off with investments on their inferior dimension. Profitable investments adversely affect consumers but increase overall welfare.

Pricing AI Model Accuracy

TL;DR

The paper analyzes a duopoly where firms compete on FP and FN errors under heterogeneous consumer sensitivities, showing that optimal competition drives specialization: firms profit more by strengthening their advantaged error dimension than by closing the gap on the inferior one. By decomposing errors into FP and FN, and introducing investment and dataset-size considerations, it shows how prices and market shares respond under different correlation structures of consumer sensitivities (positive, negative, and mixed). A key result is that greater differentiation, achieved by investing in a firm’s dominant error dimension, raises equilibrium prices and firm revenues but lowers consumer welfare; however, the total welfare can rise due to efficiency gains and market dynamics. The model integrates fixed-endowment and scalable investment costs, linking AI model market dynamics to pricing, investment incentives, and welfare implications in a stylized competitive setting.

Abstract

This paper examines the market for AI models in which firms compete to provide accurate model predictions and consumers exhibit heterogeneous preferences for model accuracy. We develop a consumer-firm duopoly model to analyze how competition affects firms' incentives to improve model accuracy. Each firm aims to minimize its model's error, but this choice can often be suboptimal. Counterintuitively, we find that in a competitive market, firms that improve overall accuracy do not necessarily improve their profits. Rather, each firm's optimal decision is to invest further on the error dimension where it has a competitive advantage. By decomposing model errors into false positive and false negative rates, firms can reduce errors in each dimension through investments. Firms are strictly better off investing on their superior dimension and strictly worse off with investments on their inferior dimension. Profitable investments adversely affect consumers but increase overall welfare.

Paper Structure

This paper contains 23 sections, 7 theorems, 32 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. The Model
  4. Perfectly Positively Correlated Cost Sensitivities
  5. Perfectly Negatively Correlated Cost Sensitivities
  6. Split Domination
  7. Strict Domination
  8. Combination of Positively and Negatively Correlated Sensitivities
  9. Firm Flexibility in FP and FN Rates
  10. Investments for Inferior Firm in Strict Domination
  11. Investments Under Split Domination
  12. Investment Costs
  13. Welfare Analysis
  14. Summary and Conclusions
  15. Derivations
  16. ...and 8 more sections

Key Result

Proposition 1

Assume $|FN_2 - FN_1| < |FP_2 - FP_1|$, and firm 1 strictly dominates firm 2 on both dimensions. An investment by firm 2 which lowers $FN_2$ to $FN_2$† results in strictly positive revenue iff the investment achieves FN domination: i.e. $FN_2$†$< FN_1$.

Figures (5)

  • Figure 1: Price equilibria under split domination with varying fixed FP and FN rate values.
  • Figure 2: Price equilibria under strict domination with varying fixed FP and FN rate values.
  • Figure 3: Effect on equilibrium prices when firm 2 reduces $FP_2$ from 0.5 to 0.2: $(p_1^*, p_2^*) = (0.15, 0) \Rightarrow (0.167, 0.033)$.
  • Figure 4: Investment distances to reach strictly positive profits.
  • Figure 5: Price and revenue of both firms across varying proportions of positively correlated consumers $\zeta$

Theorems & Definitions (13)

  • Claim 1: Price Equilibrium with $\alpha = \beta$
  • Claim 2: Price Equilibrium with $\alpha = 1- \beta$ and Split Domination
  • Claim 3: Monopoly Under Strict Domination
  • Claim 4: Relationship Between Prices and Consumer Heterogeneity
  • Proposition 1: Inferior Firm Investments on FN Rates
  • Theorem 1: Inferior Firm Investments on FP Rates
  • Proposition 2: Optimal Investment Dimension
  • Lemma 1: Investments in Dominated Dimension
  • Lemma 2: Investments in Dominating Dimension
  • Theorem 2: Optimal Investment Choices
  • ...and 3 more