Solving Problems of Unknown Difficulty

Nicholas Wu

Papers from arXiv.org

Abstract: This paper studies how uncertainty about problem difficulty shapes problem-solving strategies. I develop a dynamic model where an agent solves a problem by brainstorming approaches of unknown quality and allocating a fixed effort budget among them. Success arrives from spending effort pursuing good approaches, at a rate determined by the unknown problem difficulty. The agent balances costly exploration (expanding the set of approaches) with exploitation (pursuing existing approaches). Failures could signal either a bad idea or a hard problem, and this uncertainty generates novel dynamics: optimal search alternates between trying new approaches and revisiting previously abandoned ones. I then examine a principal-agent environment, where moral hazard arises on the intensive margin: how the agent explores. Dynamic commitment creates a frontloading force, which can be counteracted by the presence of learning. The framework reflects scientific discovery, product development, and other creative work, providing insights into innovation and organizational design.

Date: 2026-03, Revised 2026-11
New Economics Papers: this item is included in nep-mic
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