Technical Summary: Subsidizing Sequential Search

1. Problem Statement

The paper investigates markets where firms compete for consumer attention by subsidizing the costly process of product inspection. Unlike traditional models where search costs are fixed burdens borne solely by consumers, this framework treats search costs as contractible. Firms can offer subsidies $s_j \in [0, c]$ to reduce the consumer's effective inspection cost from $c$ to $c - s_j$.

The core tension lies in the dual role of subsidies:

1. Instrumental: They lower the net cost of evaluation, expanding the set of products a consumer is willing to examine.
2. Informative: Because subsidies are costly to provide, they act as signals of quality. Only firms confident in a high match probability find it profitable to subsidize inspection.

The authors formalize this as a directed search game with vertical match uncertainty. A consumer seeks a product from $n$ firms. Firm types $t_j \in [0, 1]$ (representing the probability of a match) are private information. The consumer observes a profile of subsidies, updates beliefs, and chooses a sequential inspection order to maximize expected utility.

2. Methodology

The analysis employs a Perfect Bayesian Equilibrium (PBE) framework within a sequential search model.

• Game Structure: Firms simultaneously choose subsidy policies $\sigma(t)$ after observing their private types. The consumer observes the subsidy profile and implements an inspection rule (a permutation of firms with a stopping rule).
• Equilibrium Refinement: To address equilibrium multiplicity, the authors apply a forward-induction refinement (specifically the equilibrium-dominance restriction in the spirit of Cho and Kreps, 1987, and Banks and Sobel, 1987). This refinement eliminates equilibria supported by implausible off-path beliefs, requiring that off-path subsidies not be attributed to types for whom such a deviation is strictly dominated.
• Extension to Agentic Economies: The model is extended to a platform setting where a monopolist sells "inspection tokens" (computational units) to firms. The platform sets a price $p$ per token, which firms use to subsidize the consumer's search cost.

3. Key Contributions and Results

A. The Subsidy-Sorting Principle

The paper establishes a fundamental structural result for any PBE:

• Monotonicity: Higher-quality firms (higher types) offer weakly larger subsidies.
• Descending-Subsidy Index Rule (DSIR): Consumers inspect firms in descending order of their realized subsidies.
• Mechanism: Higher subsidies signal higher match probabilities and simultaneously lower the net inspection cost. Consequently, the consumer's optimal strategy is to prioritize firms with higher subsidies, stopping only when a match is found or the expected surplus of the next inspection becomes negative.

B. Equilibrium Uniqueness and the "Reasonable" Outcome

While multiple equilibria exist, the forward-induction refinement selects a unique equilibrium outcome, termed the Reasonable Equilibrium. This outcome is characterized by a Step-Increasing-Step (SIS) subsidy structure:

1. Low Types ($t < \underline{t}$): These firms offer zero subsidies and are never inspected. The cutoff $\underline{t} = \frac{pc}{1+p}$ is determined by the intersection of the firm's participation constraint (non-negative profit) and the consumer's inspection-worthiness constraint (non-negative expected surplus).
2. Intermediate Types ($\underline{t} \le t < \bar{t}$): These firms separate strictly. They offer strictly increasing subsidies that reveal their types, allowing the consumer to distinguish between them.
3. High Types ($t \ge \bar{t}$): These firms pool at the maximum subsidy cap ($s = c$). They fully reimburse the inspection cost.
   • Trade-off: Pooling at the cap makes inspection free for the consumer but obscures quality differences among the best firms. This leads to "congestion" where high-quality firms are inspected with lower probability due to random tie-breaking within the pool.

C. Welfare and Comparative Statics

Under the Reasonable Equilibrium:

• Consumer Surplus: Increases with the number of firms ($n$) and decreases with search costs ($c$) and subsidy prices ($p$).
• Social Surplus: While consumer surplus is maximized by the Reasonable Equilibrium among all PBEs, social surplus suffers from the inefficiency caused by the pooling of high types. The loss of information at the top leads to excessive search (inspecting too many options) relative to the first-best social optimum.
• Comparative Statics: Higher search costs or subsidy prices raise the entry cutoff, excluding more low-quality firms. Increased competition ($n$) intensifies the race for attention, leading to steeper separating schedules.

D. Platform Pricing and Agentic Search

The paper extends the model to a platform selling inspection tokens at price $p$.

• Platform Optimization: The platform maximizes revenue by setting a price $p^*$ that induces excessive search.
• Mechanism of Distortion: The optimal price thickens the top pooling region. By lowering the price, the platform induces more firms to enter the pooling block (where they pay the full subsidy $c$). While this increases the total volume of inspections (and thus token sales), it distorts the allocation of attention.
• Welfare Implications:
  • Consumer Welfare: Consumers benefit from the lower per-inspection cost and higher match probability, so consumer surplus is not reduced.
  • Social Welfare: Social surplus is reduced because the market searches more than is socially optimal (inspecting firms with negative virtual values).
  • Surplus Redistribution: The platform extracts rents from the opacity it creates, reallocating surplus from sellers to the platform and consumers, while creating a deadweight loss in total efficiency.

4. Significance and Claims

The paper claims to provide a new theoretical foundation for understanding directed search where search frictions are endogenous and contractible.

• Theoretical Contribution: It bridges the gap between the classical "Pandora's Box" search literature (Weitzman, 1979) and strategic information provision. It demonstrates that subsidies, rather than just price or ad placement, are a primary mechanism for sorting and attention allocation.
• Refinement Power: The authors show that forward-induction reasoning is crucial for selecting a unique, informative equilibrium in signaling games with search costs, ruling out equilibria where low-quality firms might mimic high-quality ones or where pooling occurs inefficiently.
• Agentic Economy Relevance: The paper positions itself as a microfoundation for the emerging "agentic economy" (AI assistants). It argues that as AI agents automate search, the mechanism of "subsidizing inspection" becomes explicit (e.g., via token pricing). The analysis warns that while platforms may optimize for revenue by encouraging excessive search, this creates a wedge between consumer welfare and social efficiency, suggesting a need for policy intervention or platform design that aligns incentives with social surplus.

The paper concludes that subsidies effectively decentralize attention allocation without the need for complex auctions, but the resulting equilibrium is inherently prone to inefficiencies (pooling and excessive search) when driven by profit-maximizing intermediaries.