Nested search

I introduce and study a nested search problem modeled as a tree structure that generalizes Weitzman (1979) in two ways: (1) search progresses incrementally, reflecting real-life scenarios where agents gradually acquire information about the prizes; and (2) the realization of prizes can be correlated, capturing similarities among them. I derive the optimal policy, which takes the form of an index solution. I apply this result to study monopolistic competition in a market with two stages of product inspection. My application illustrates that regulations on drip pricing lower equilibrium price and raise consumer surplus.

💡 Research Summary

The paper introduces a novel “nested search” model that extends the classic Pandora’s box problem of Weitzman (1979) in two important directions. First, it captures incremental information acquisition: an agent moves from the root of a tree toward a terminal node, paying a cost at each edge to observe a random signal. This mirrors real‑world processes such as multi‑stage R&D, housing searches, degree selection, or online product browsing, where information is revealed step‑by‑step rather than all at once. Second, the model allows the realizations of the final prizes to be correlated. Correlation is encoded in the tree structure: the longer two terminal nodes share a common path, the more similar their underlying signals, and consequently the more correlated their final payoffs. The key technical assumption (Assumption 1) is a conditional independence property that guarantees all dependence is captured by the shared ancestry in the tree.

Within this framework the author defines an index for every non‑root node. The index is a random variable that depends only on the signals observed along the node’s ancestors and on the known distribution and cost structure of the subtree rooted at that node. The proposed “index policy” always inspects the node with the highest current index, provided that this index exceeds the best fully‑inspected prize available. The central theoretical contribution is Theorem 1, which proves that this index policy is optimal. The proof proceeds in three lemmas: (1) an upper bound on the expected payoff of any policy, (2) tightness of the bound for the index policy, and (3) maximality of the bound under the index rule. The construction is constructive and yields an explicit expression for the expected payoff, extending the classic Gittins‑index intuition to a setting with both multi‑stage inspection and correlation.

The second part of the paper applies the nested‑search solution to a market with monopolistic competition and two stages of product inspection. An infinite number of firms each offer a single product. A consumer may pay a first‑stage cost to learn a noisy signal about his match value for a firm’s product and, optionally, a second‑stage cost to learn the exact match value. The analysis distinguishes three timing regimes: (i) prices revealed at the first stage, (ii) prices revealed at the second stage, and (iii) a benchmark with only one inspection stage. Proposition 1 and Proposition 2 characterize equilibrium prices under (i) and (ii). Theorem 2 shows that when prices are disclosed early (first stage), equilibrium prices are always lower than in the benchmark. Two mechanisms drive this result: (a) the option to stop after a partially informative signal makes it more attractive to sample additional firms, intensifying competition (Lemma 6); and (b) consumers who receive an unfavorable early signal tend to quit immediately, further raising competitive pressure (Lemma 7). Consequently, consumer surplus is higher than in the benchmark (Corollary 1).

When prices are disclosed only at the second stage, the equilibrium price can be higher or lower than the benchmark, because the competition‑enhancing “stop‑early” effect is offset by the firm’s ability to raise the price after observing that the consumer has already received a favorable first‑stage signal (Proposition 3). Consumer surplus in this regime is ambiguous.

The paper then endogenizes the timing of price revelation and studies the impact of “drip‑pricing” regulation, which requires that any price disclosed at stage 1 be weakly lower than the price disclosed at stage 2. In an unregulated market, early price disclosure is merely cheap talk; any equilibrium can be replicated by a setting where firms reveal prices only at stage 2 (Proposition 4). Under the regulation, firms can credibly commit to the stage‑1 price, and the equilibrium price coincides with the outcome when price disclosure is exogenously fixed to stage 1 (Proposition 5). Because equilibrium prices are always lower when disclosed early (Proposition 3), the regulation improves consumer welfare (Corollary 2).

The literature review situates the work among three strands: (a) extensions of Pandora’s box (including multi‑stage inspection and correlated prizes), (b) branching bandit problems where index optimality has been known but without constructive proofs, and (c) the economics of drip pricing and price competition with search costs. The paper’s contributions are threefold: (1) a general, tractable model of incremental, correlated search; (2) a constructive proof of index optimality that yields explicit payoff formulas; and (3) an application showing how the timing of price disclosure and drip‑pricing regulation affect equilibrium prices and consumer surplus. The results suggest that policy makers aiming to protect consumers should consider mandating early price disclosure, as it both intensifies competition and reduces the final price paid by consumers. Future research could explore finite numbers of firms, dynamic entry and exit, or richer forms of correlation beyond the tree representation.