Rediscovery

TL;DR

The paper studies a forward-looking rediscovery problem where a searcher knows that high-quality discoveries exist but not their locations, and shows that robust optimization under a Lipschitz payoff landscape yields a simple, optimal policy. The main result is the existence of a directional, threshold, index-based policy that ignores past discoveries and is dynamically consistent, operationalized by a left-to-right search with a history-dependent threshold $\phi(l)$. The analysis combines a two-period intuition with a general proof, revealing how a shrinking search window and increasing stopping thresholds guide efficient exploration under worst-case uncertainty. The findings illuminate how knowledge of achievable targets shapes exploration, offering practical implications for innovation policy and platform design in settings where rediscovery drives iterative progress.

Abstract

We model search in settings where decision makers know what can be found but not where to find it. A searcher faces a set of choices arranged by an observable attribute. Each period, she either selects a choice and pays a cost to learn about its quality, or she concludes search to take her best discovery to date. She knows that similar choices have similar qualities and uses this to guide her search. We identify robustly optimal search policies with a simple structure. Search is directional, recall is never invoked, there is a threshold stopping rule, and the policy at each history depends only on a simple index.

Figures (8)

• Figure 1: The red line represents the search window. The solid black line instead represents $\overline{q}_{h_t}$, the upper envelope of qualities given the current history.
• Figure 2: Visualizing $\sigma_{L \to R}$ at some ordered search history $h$, where the search window is given by the solid red line.
• Figure 3:
• Figure 4: The agent in the first figure explores $x$ and faces bifurcation risk. Given what she knows, it is possible that target locations exist either exclusively to the left (e.g., if $q=q_l$) or right of her initial search (e.g., if $q = q_r$), and she guesses the wrong side to search next. The agent in the second figure does not face bifurcation risk. Due to the Lipschitz constraint, any feasible $q$ must lie on or below the dashed black line. Therefore, target locations must exist exclusively near the right end of the search space.
• Figure 5: The agent explores $x$ and in the first figure she discovers a low quality $q(x)$. Lipschitz continuity then narrows down the possible target locations on the right end of the interval. Instead, in the second figure the agent discovers a high quality, which does not help significantly in narrowing down the location of the target.

Theorems & Definitions (19)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Theorem 1
• Definition 6
• Lemma 1
• Proposition 1
• Theorem 2