The Inference Bottleneck: A Formal Model of Vertical Foreclosure in AI Markets

Abstract: As generative AI commercializes, competitive advantage is shifting from model training toward inference, distribution, and routing. This paper develops a formal game-theoretic model of vertical foreclosure in inference markets, as the formal-model companion to Besanson and Celani (2026). The model isolates two foreclosure mechanisms operating without predatory pricing: quality-of-service (QoS) discrimination against downstream rivals via latency, throughput, context limits, or feature access; and routing bias in assistant-layer interfaces. An extension motivated by Anthropic's April 2026 release of Claude Opus 4.7 alongside the restricted-access Claude Mythos Preview introduces a third mechanism, tier-based access discrimination, parameterized by a tier gap (tau) and partner-exclusivity (kappa). The main result gives an explicit local equilibrium characterization of the QoS gap. Under logit demand and symmetric rivals, the gap is strictly increasing in inference-quality importance (alpha) and downstream margins, and strictly decreasing in API price and rival entry elasticity. Discrimination vanishes at a joint boundary rather than at a simple threshold in alpha alone. A stylized calibration to four providers using April 2026 data treats parameter values as inputs to a comparative risk mapping, not structural estimates. The mapping suggests Google and OpenAI face conditions most conducive to foreclosure; Microsoft's realized routing bias has been voluntarily constrained by a March 2026 multi-model pivot; Anthropic shows low consumer-channel risk and elevated risk in enterprise coding-agent segments. The policy section proposes Neutral Inference, a four-pillar conduct framework: QoS parity, routing transparency, FRAND-style non-discrimination, and tier transparency with release-pathway discipline. Illustrative welfare calculations suggest net gains in the tens of billions annually.

• The paper introduces a formal model that quantifies how vertically integrated AI providers may use QoS manipulation, routing bias, and tier-based access to foreclose rivals.
• The methodology employs a logit demand system to derive closed-form expressions for quality gaps and assess welfare losses, linking theory to practical market observables.
• It highlights dynamic regulatory implications by mapping discriminatory mechanisms to actionable audit protocols and policy interventions, such as the Neutral Inference framework.

The Inference Bottleneck: Formalizing Vertical Foreclosure in AI Markets

Overview

"The Inference Bottleneck: A Formal Model of Vertical Foreclosure in AI Markets" (2604.17431) articulates a rigorous economic framework for understanding anticompetitive risks in commercial AI inference markets. The paper adapts and extends classical vertical foreclosure theory to the context of "cognitive infrastructure": foundational AI systems provided as APIs upon which downstream application developers depend, frequently controlled by vertically integrated incumbents such as Google, OpenAI, Microsoft, or Anthropic. The model isolates and formally characterizes incentive conditions for non-price foreclosure, through both conventional and novel mechanisms specific to generative AI markets. These mechanisms are calibrated and mapped to policy-observable constructs, enabling both ex ante regulatory designation and empirical audit.

Model Structure and Foreclosure Mechanisms

The core setup models a two-layer market: a dominant upstream inference provider (e.g., foundational model vendor) sets API access terms and also operates downstream applications, in direct competition with API-dependent rivals. The provider can manipulate not only price but multiple quality-of-service (QoS) dimensions—latency, context size, feature access, etc.—that are not readily contractible nor transparent to regulators or rivals.

Three distinct foreclosure mechanisms are formulated:

1. Quality-of-Service Discrimination: The incumbent supplies inferior inference quality—via increased latency, lower throughput, limited context, or restricted features—to downstream rivals compared to its own applications.
2. Routing Bias: Where the provider controls an assistant or orchestrator (e.g., an AI agent layer), it can systematically favor its own or select partners' applications through non-transparent routing.
3. Tier-Based Access Discrimination: Motivated by recent real-world deployments (e.g., Anthropic’s Project Glasswing), the provider gates access to advanced model classes (frontier models) to privileged customers or partners, measured by a "tier gap" ($\tau$) and "partner exclusivity" fraction ($\kappa$), creating persistent, bimodal distributions in downstream capabilities.

Formal Equilibrium Characterization and Comparative Statics

A logit demand system with symmetric downstream rivals is analyzed. Application quality, $Q_i$, aggregates inference-layer QoS and application-layer effort with parameter $\alpha$ denoting the relative importance of inference quality. The equilibrium characterization yields an explicit closed-form expression for the QoS gap ($q_U^* - q_i^*$) between the vertically integrated app ($U$) and its rivals:

$$\Delta q^* = \frac{1}{\gamma}\left\{ \alpha m_U s_U \left[(1-s_U) + s\right] - \frac{p \rho \eta}{q_i^*} \right\}$$

Where:

• $m_U$: downstream margin of the incumbent,
• $s_U$: incumbent's downstream share,
• $s$: typical rival share,
• $\kappa$0: API access price,
• $\kappa$1: active rival share,
• $\kappa$2: elasticity of rival entry with respect to QoS,
• $\kappa$3: infrastructure cost curvature.

Key comparative statics:

• The QoS gap is increasing in the importance of inference quality ($\kappa$4) and downstream margin.
• The gap is decreasing in the API access price and rival entry elasticity, as higher API revenue and more elastic rival exit constrain discrimination incentives.
• Discriminatory equilibrium occurs only when a joint condition is satisfied—discrimination is not solely a function of inference’s importance, but of a multidimensional boundary involving all relevant margins.

The welfare loss decomposes into (a) direct consumer surplus loss from reduced rival quality, (b) indirect loss from lower downstream rival investment due to QoS–effort complementarity, and (c) standard business-stealing deadweight loss.

Dynamic and Tier-Based Extensions

The model is extended along several notable axes:

• Dynamic Foreclosure: Incumbents may initially grant high-QoS API access to attract ecosystem development and later degrade it after rivals are locked-in, exploiting switching costs (mirroring "open early, closed late" strategies).
• Routing Bias: Formalization of assistant-layer self-preferencing, showing routing bias is monotonically increasing in the incumbent's downstream margin and the opacity to external observers.
• Tier-Based Access: By parameterizing capability distance ($\kappa$5) and exclusivity ($\kappa$6), the paper formalizes persistent and more durable forms of foreclosure based on access to unreleased or pilot model classes, with different welfare and audit implications compared to conventional QoS discrimination.

The paper distinguishes between temporary/eroding rents via QoS manipulation and durable rents via tier-gating, making the latter an essential regulatory consideration as providers gate frontier model access.

Calibration, Auditability, and Empirical Bridge

A stylized calibration is implemented using April 2026 market data for Google (Gemini), OpenAI (GPT), Microsoft (Copilot), and Anthropic (Claude), explicitly distinguishing observable, inferred, and judgment-based parameters. Comparative risk mapping is presented (not as point identification but as scenario analysis), revealing:

• Google and OpenAI possess parameter constellations most conducive to profitable foreclosure, with high predicted equilibrium QoS gaps.
• Microsoft structurally could exercise significant routing bias, but recent voluntary adoption of multi-model routing (e.g., with Claude integration) has attenuated realized discrimination, likely due to competitive and regulatory pressures.
• Anthropic presents a split profile: low risk in consumer channels (API revenue predominates, disincentivizing discrimination), but higher latent risk in verticals such as enterprise coding agents and clear, measurable tier-gated access for advanced models like Claude Mythos under the Project Glasswing initiative.

Model primitives (e.g., $\kappa$7, $\kappa$8, $\kappa$9, $Q_i$0, $Q_i$1) are systematically mapped to empirical observables, enabling practical regulatory triggers and audit protocols. The audit section specifies practical matched-pair and benchmark-based strategies to distinguish discriminatory patterns amidst unavoidable measurement noise.

Regulatory Implications and Policy Framework

The exposited framework culminates in a four-pillar "Neutral Inference" regulatory regime, operationalizing the model’s constructs:

1. QoS Parity: Prohibiting material QoS degradation for downstream rivals.
2. Routing Transparency: Mandating auditable records for assistant/orchestrator routing.
3. FRAND-Style Non-Discrimination: Requiring fair, reasonable, and non-discriminatory access and terms.
4. Tier Transparency: Imposing transparency and discipline on restricted-access (tier-gated) capability rollouts—requiring justification, partner set audits, and published release pathways, with explicit safety carve-outs.

The optimal policy tolerance band is characterized as interior: strict neutrality is suboptimal given investment incentives, but high tolerance permits excessive foreclosure. Illustrative welfare analysis, calibrated to segment-level AI adoption, suggests annual static welfare losses from current discriminatory practices are on the order of $Q_i$2–$Q_i$3 billion USD, with net welfare gains of $Q_i$4–$Q_i$5 billion plausible under neutrality obligations—though these figures are scenario bounds, not point estimates.

Implications and Directions for Future Research

The formal framework evidences that—as in prior digital infrastructure cases—vertical integration in AI cognitive infrastructure generates persistent incentives for strategically raising rivals’ costs through non-price means. Crucially, the nature of modern AI APIs, with their multi-dimensional and opaque quality parameters, renders both detection and remedy of foreclosure more challenging.

Practically, the empirical bridge provided enables regulators to move beyond abstract theory, equipping audit and enforcement with testable predictions and concrete designation criteria. The tier-gated mechanism, substantiated by real deployment patterns, requires explicit regulatory treatment distinct from standard product-discrimination frameworks.

Theoretically, the work highlights the necessity of multidimensional characterization for discrimination boundaries (no single sufficient statistic such as inference quality importance), as well as new dynamic and multi-tier concerns unique to the AI sector. This constitutes both a refinement of classical vertical foreclosure models and a direct guide for future extensions, notably: robust structural estimation, richer demand heterogeneity, oligopolistic upstream supply, and empirical studies to calibrate and validate the tier and QoS gap measurements.

Conclusion

This paper supplies a tractable, empirically-linked mechanism model for vertical foreclosure in AI inference markets, integrating new market realities (multi-dimensional QoS, assistant-layer routing, and tier-gated model classes) and bridging them to actionable regulatory observables and audit strategies. The work establishes that anticompetitive risks manifest through both traditional and AI-specific discriminatory mechanisms, and makes precise both the incentive conditions for such behavior and the potential scale of social welfare implications. The Neutral Inference framework provides a structured, audit-friendly policy response that preserves innovation incentives while mitigating exclusionary practices, pending further advances in audit methodologies and empirical validation.