Solving Problems of Unknown Difficulty

TL;DR

This paper models how uncertainty about problem difficulty influences problem-solving strategies, highlighting the trade-offs between exploration and exploitation in creative and organizational contexts.

Contribution

It introduces a dynamic model of problem-solving under unknown difficulty, incorporating exploration-exploitation trade-offs and analyzing incentive issues in principal-agent settings.

Findings

Optimal search alternates between exploring new approaches and revisiting old ones.

Uncertainty about difficulty creates novel dynamics in problem-solving strategies.

Contracts can be designed to incentivize exploration despite learning challenges.

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 leads contracts to frontload incentives, 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.