Subsidizing Sequential Search

We study markets where firms compete for consumer attention by subsidizing costly product inspection. These subsidies do not change product quality, but they alter the order in which consumers search by lowering inspection costs. We establish a subsidy–sorting principle: in any equilibrium, higher–quality firms provide weakly larger subsidies, leading consumers to search in descending subsidy order. A unique equilibrium survives forward–induction reasoning in the spirit of the Intuitive Criterion: low–quality firms are never inspected, intermediate–quality firms separate with strictly increasing subsidies, and high–quality firms pool at the full subsidy. This equilibrium maximizes information revelation among all possible outcomes and ensures efficient inspection. We then extend the analysis to AI–mediated platforms that can create and price “inspection tokens.” The platform’s optimal linear pricing leads to excessive inspection relative to the social optimum. While this distortion does not reduce consumer welfare, it reallocates surplus from sellers to the platform and consumers.

Introduction. Markets run on attention. Buyers want to find a product that fits their needs, but evaluating options takes time and effort. This challenge is familiar in online markets: a consumer types a query, browses links, and exerts effort before finding what she wants. Firms compete for the consumer’s attention not only through price and product features but also by lowering the cost of evaluation, most visibly through sponsored placements and promoted listings. This phenomenon arises also beyond the web. For instance, graduate programs cover travel expenses so prospective students can visit campus and meet faculty; car dealers hand out cash for test drives; galleries and non–profits host receptions to draw people. Across these settings, in both online and off-line markets, the consumer faces search frictions and the other side competes to offset these search frictions. We formalize this problem as a new kind of “directed search”: a consumer chooses which options to evaluate, and each evaluation carries a cost. Firms compete for her attention by subsidizing some or all of her search cost. These subsidies play a dual role. They are instrumental, because they reduce the cost of being considered and enlarge the set of products the consumer is willing to examine. They are informative, because only firms confident in high match value find it worthwhile to subsidize. In this way, subsidies both broaden what is searched and determine the order in which search unfolds. We formalize these ideas in Section 2. We study a sequential–search model with vertical match uncertainty. A consumer seeks a product from one of n firms. Firm j has a privately known type tj P r0, 1s, which denotes the probability that its product is a match for the consumer’s needs. Firm types are drawn i.i.d. from a common prior. To learn whether a firm’s product matches, the consumer must “inspect” it at cost c ą 0, which is a commonly known search or inspection cost. Each firm j, after observing its type, simultaneously announces a subsidy sj P r0, cs paid to the consumer upon inspection. The consumer observes the subsidy profile s “ ps1, . . . , snq, updates beliefs about match probabilities, and chooses an inspection order that maximizes expected utility, trading off lower net costs pc ́ sjq against inferred quality. We analyze the Perfect Bayesian equilibria of this game and show how subsidized inspection reallocates attention in equilibrium. Before we describe the analysis of this setting, we relate it to two well-known studies. One way to view our analysis is as a Pandora’s Box problem in the spirit of Weitzman (1979) in which the “boxes” can play an active role by subsidizing search. In this spirit, the model captures decentralized search in which privately-informed firms try to attract consumers by reducing their inspection costs. A second way to view our analysis is as a platform in the spirit of Athey and Ellison (2011) in which firms go beyond paying for prominence, and instead can subsidize search. After describing our analysis, we highlight how this approach to search on platforms is relevant to the emerging trend of agentic search, where consumers delegate discovery and evaluation to AI assistants.1 Three questions guide our investigation of this directed-search framework. First, how do subsidies signal quality? Second, in this signaling game, do forward induction refinements in the spirit of Cho and Kreps (1987) refine equilibria? Third, how do market fundamentals, competition, search costs, and subsidy prices, affect consumer and social welfare? How do changes in market fundamentals, more competition, lower search costs, higher subsidy prices, affect consumer and social welfare?

Related work. Relation to the literature. The classical search literature treats inspection as a fixed burden borne by consumers. Early work frames search as an optimal stopping problem (Stigler, 1961; McCall, 1970; Diamond, 1971). Weitzman (1979) provides the canonical index rule when inspection costs are heterogeneous. Subsequent work extends this framework to directed and simultaneous search under match-quality uncertainty (Chade and Smith, 2006).

We build on this tradition but depart by making search costs contractible: firms can subsidize evaluation, turning a stopping problem into an equilibrium allocation of attention. A growing strand studies strategic information provision and disclosure in search environments. Firms strategically disclose more when they mainly compete to attract new customers and cannot condition on buyer beliefs; they disclose less when competition is about persuading already-arrived buyers or when firms can condition on beliefs (Board and Lu, 2018; Au and Whitmeyer, 2023). Recent work recasting Weitzman’s Pandora’s box in competitive and intermediary settings finds that competition often improves informativeness, yet principals who sell signals can implement efficient stopping while extracting surplus (Ding et al., 2025; Mekonnen et al., 2025). These contributions show that strategic information provision reshapes inspection order, and thus welfare. Our analysis complements this literature but shifts the instrument from beliefs to search frictions: rather than solely manipulating posteriors, firms in our model manipulate the cost of learning. Subsidies lower frictions directly and thereby reorder the search path, benefiting the consumer. A second strand examines how firms compete for consumer attention through advertising (Anderson and Renault, 2006; Edelman et al., 2007; Athey and Ellison, 2011; Chen and He, 2011). In sponsored search, firms bid for ad positions, with higher bidders receiving more prominent placement. The generalized second-price auction creates efficient rankings where firms with higher valuations for clicks secure better positions (Edelman et al., 2007). Athey and Ellison (2011) show this extends to organic search when firms can influence their ranking through quality investments. We demonstrate that subsidizing inspection achieves similar efficient sorting without auctions or slot constraints. The mechanism is simpler and more direct. Yet we uncover a crucial difference: subsidy caps create pooling at the top, generating excessive search that hurts social welfare while preserving consumer surplus. This distinguishes our setting from position auctions where firms can always separate through higher bids.

Method. 2 Environment A consumer seeks a product that matches her needs. The benefit that she accrues from finding such a product is u ą 0; otherwise, she obtains her outside option, H, which delivers zero utility. She searches for the product on a platform, inspecting the products of different firms. Each inspection costs c ą 0 and reveals whether the inspected firm’s product meets her needs.5 The consumer searches sequentially and terminates either when she finds a match or she opts out.6 Each firm j P J “ t1, . . . , nu has a type tj P T ” r0, 1s, drawn independently from a common distribution F with full support.7 The interpretation is that tj is the firm’s match probability upon inspection. Hence, we refer to a higher type firm as better and a lower type firm as worse. A firm earns v “ 1 if its product matches the consumer, and 0 otherwise. Before inspections occur, a firm may offer a subsidy sj P r0, cs that is paid if the consumer inspects the firm’s product. The subsidy reduces the consumer’s effective inspection cost from c to c ́ sj. Each unit of subsidy costs the firm p P R` per unit. In Section 6, we consider a profit-maximizing choice of p for a platform, but for the baseline model, p is a primitive.

Timing. Each firm privately observes tj and simultaneously chooses a subsidy sj P r0, cs. After observing the vector of subsidies s “ psjqjPJ , the consumer selects a sequential inspection rule. Upon inspecting firm j, the consumer learns whether the product satisfies her need (i.e., whether a match occurs).

2.1 Strategies Each per–unit price p P Rinduces a Bayesian game Gppq between the consumer and the set of firms J “ t1, . . . , nu. A subsidy policy is a measurable function σ : T Ñ r0, cs, mapping a type t to a subsidy s “ σptq. A subsidy policy profile pσjqjPJ is a collection of subsidy policy stating each firm j policy. Given a type realization t, a subsidy policy profile pσjqjPJ induces a realized subsidy profile s “ psjqjPJ with sj “ σjptjq. The consumer observes s and then implements an inspection rule. An inspection rule r P R is a planned order of inspections with an outside-option stop: inspections proceed sequentially until a match is found or the outside option is reached. Formally, let R be the set of permutations of ̄J “ J Y tHu. For r P R, write i ąr j to mean “i is scheduled before j,” i.e., rpiq ą rpjq; by convention, if H ąr j then j is never inspected. The consumer’s strategy is an inspection policy, a stochastic kernel ι : R ˆ r0, csn Ñ r0, 1s, where ιpr | sq denotes the probability that inspection rule r is used at subsidy profile s. The consumer’s beliefs are given by a measurable mapping μ : r0, csn Ñ ∆pTqn that assigns to each subsidy profile s a posterior over types; our formulation assumes both on- and off-path that the consumer’s beliefs about a firm j do not depend on the subsidies offered by other firms.8 An assessment is a triple pσjqjPJ , ι, μ ̆ , where σj is firm j’s subsidy policy, ι is the consumer’s inspection policy, and μ is the consumer’s belief system. Throughout, “increasing” means weakly increasing, i.e. x1 ě x ùñ fpx1q ě fpxq. We use “strictly increasing” when the inequality is strict. Analogous definitions apply for decreasing functions and monotone functions.

2.2 Equilibrium Concept Our solution concept is Perfect Bayesian Equilibrium (PBE). An assessment ppσjqjPJ , ι, μq is a PBE if (i) each firm’s strategy maximizes expected profit, (ii) the consumer’s strategy maximizes expected utility given beliefs, and (iii) beliefs are updated by Bayes’ rule whenever possible. We restrict attention to symmetric equilibria. In a symmetric Perfect Bayesian Equilibrium, two restrictions apply. First, exchangeability: the consumer treats firms offering identical subsidies identically, both on and off the equilibrium path.9 This implies that the belief system can be characterized by a measurable mapping μ : r0, cs Ñ ∆pTq to which we refer as the consumer’s belief function.10 Second, all firms adopt the same subsidy policy σ. Thus a symmetric assessment can be written as pσ, ι, μq, where σ is the common subsidy policy, ι the inspection policy, and μ the consumer’s belief function.

Discussion. 5 Welfare and Comparative Statics We now compute welfare under the reasonable equilibrium, decomposing it into consumer surplus, producer surplus, and total efficiency. We then establish comparative statics with respect to the primitives pp, c, nq. The next lemma summarizes key properties of the lower and upper cutoff, the attention function, and the separating subsidy policy.

Lemma 5. The following properties hold:

(i) Lower cutoff. The lower cutoff is given by tpp, cq “ pc{p1 ` pq. It is strictly increasing in both c and p, and independent of n.

(ii) Attention function. For any fixed t0, the separating attention function qsep n pt; t0q is strictly decreasing in n for all t P pt0, 1q. Moreover, for each n and t, qsep n pt; ̈q is decreasing. Therefore, qsep n p ̈; tq is decreasing in c and p.

(iii) Separating policy. For all t ě t1 0, σsep n pt; t1 0q ą σsep n pt; t0q with t1 0 ą t0. In addition, σsep n`1pt; t0q ą σsep n pt; t0q for all t ą t0. We have that σsep n pt; tq is increasing in c and decreasing in p.

(iv) Upper cutoff. The upper cutoff ̄t is strictly increasing in both p and n.

A higher inspection cost c or a higher price of subsidies p raises the entry cutoff t, excluding more low–quality firms. As the number of firms n grows, competition for attention intensifies and the inspection probability of each individual firm falls. To compensate, firms respond with higher subsidies, making the separating schedule steeper. Finally, the upper cutoff ̄t increases with p and p, since more expensive subsidies compress the set of types willing to separate before pooling at the cap.

Notation. For convenience, let q ̊ptq denote the attention that a type t receives in the reasonable equilibrium: it equals qseppt; tq for t ă ̄t, and qpoolp ̄tq for t ě ̄t.

5.1 Welfare Accounting Define the average inspection probability per firm as the expected attention a firm generates:

Qnpp, cq ” Etrq ̊ptqs “ ż ̄t qsep n ptq dFptq ` qpool n p ̄tq r1 ́ Fp ̄tqs. (9) A type-t firm on the separating branch pays p σseppt; tq upon inspection; envelope arguments yield for t P rt, ̄tq p σseppt; tq qsep n pt; tq “ t qsep n pt; tq ́ ż t qsep n px; tq dx.

Pooled types t ě ̄t pay p c upon inspection, so their expected payment equals p c qpool n p ̄tq. Thus, on expectation a firm expected transfer costs is given by φF npp, cq :“ Etrp σ ̊ptq q ̊ptqs “ ż ̄t ́ t qsep n pt; tq ́ ż t qsep n px; tq dx ̄ dFptq ` p c qpool n p ̄tq r1 ́ Fp ̄tqs.

No match occurs precisely when all inspected firms fail to match which happens with probability ́ 1 ́ ş1 t x dFpxq ̄n .21 From this observation, the probability of a match is mnpp, cq “ 1 ́ 1 ́ ż 1 x dFpxq ̧n .

The consumer expected net inspection cost for a single firms equal Cnpp, cq :“ Etrpc ́ σ ̊ptqq q ̊ptqs “ c Qnpp, cq ́ 1 pφF npp, cq.

Welfare decomposition. Write consumer surplus and producer surplus as CSnpp, c; t0q :“ u mnpp, cq ́ n Cnpp, cq, (10) PSnpp, c; t0q :“ n Et “ pt ́ p σ ̊ptqq q ̊ptq ‰ “ n Etrt q ̊ptqs ́ n φF npp, cq, (11) Social welfare adds up consumer and producer surplus:

Wnpp, cq :“ CSnpp, c; t0q PSnpp, c; t0q “ mnpp, c; t0q n ` Etrpt ́ cq q ̊ptqs ̆ ́ n ˆ 1 ́ 1 ̇ φF npp, cq (12) In the special case where p “ 1 social welfare collapses and results into the expected match value minus the search costs incurred into achieving such match.

5.2 Comparative Statics We record the main monotonicity properties of the allocation and welfare components.

Theorem 3 (Comparative statics in the reasonable equilibrium). Then:

(i) Search intensity. Qnpp, cq is decreasing in p and in c, and increasing in n.

(ii) Match probability. mnpp, cq is decreasing in p and in c, and increasing in n.

(iii) Consumer surplus. CSnpp, cq is decreasing in p and in c, and increasing in n.

Conclusion. 7 Concluding Remarks This paper develops a theory of attention markets in which firms can directly subsidize costly search. In such environments, subsidies play a dual role. They serve as signals, conveying information about a firm’s expected match value. At the same time, they are instrumental, reducing the consumer’s effective inspection costs and thereby altering the sequence of search. In equilibrium, these two roles jointly determine which firms participate, which reveal their type, and which pool. In this way, subsidies endogenously structure the allocation of attention. We show that every equilibrium satisfies a subsidy–sorting principle: higher types weakly offer higher subsidies, and the consumer follows a descending–subsidy inspection order. Within this structure, the unique equilibrium outcome surviving refinement is the reasonable equilibrium. It takes a step–increasing–step form: no inspection below a participation cutoff, revelation on an intermediate region, and a pooling block at the cap. This equilibrium is efficient in the sense of being most informative: it maximizes the set of types that can induce inspection, subject to incentive constraints. Any information that is lost is either uninspected altogether or free to obtain. These results provide a benchmark for platform design. If a platform can mint and sell inspection tokens, its problem reduces to a linear–cost mechanism. Yet profit motives may distort attention. By thickening the top pool, a platform can steer consumers to inspect too much, raising revenue but lowering social efficiency. The framework also delivers empirical predictions. Subsidies should be weakly increasing in observable proxies for match quality. High–subsidy pools should exhibit congestion, with inspection probabilities falling as the pool thickens. Changes in token prices should shift the participation margin and alter match rates and transfers in predictable ways. Such patterns can be measured in platform data, offering a path for empirical validation. More broadly, the analysis highlights how inspection can be organized without auctions.