Solving Problems of Unknown Difficulty

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.

Figures (12)

• Figure 1: Markov chain representation of the individual state of an arm. Each arm starts in an "undiscovered" state $U$, which transitions to a discovered state $D$ immediately with cost $c$; effort on a discovered state increases the historical effort associated with the arm, and an arm with historical effort $K$ transitions to a solved state at rate $\lambda \nu(K)$, where $\nu(K)$ is the posterior belief that the arm is valid given $K$ historical effort defined in \ref{['eqn:interim_belief']}.
• Figure 2: Non-monotone effect of time pressure on the threshold $K^*$. Example plotted for $\lambda = 1$, $c = 0.2$, $\nu_0 = 0.75$.
• Figure 3: The belief over whether approach 1 is valid at state $(K_1, K_2)$ as $K_2$ varies. Example plotted for $\lambda_H = 1$, $\lambda_E = 2$, $\delta_0 = 0.5$, $\nu_0 =0.5$, and $K_1$ fixed to 1.
• Figure 4: The original action path, and the interchanges which generate the first-order condition that determines $K^*_n$. Each color denotes effort on a distinct approach. Blue denotes effort on the $(n+1)$st approach, and $t_{n+1}$ denotes the time that the $(n+1)$st approach is brainstormed.
• Figure 5: The beliefs over $\omega_1$, $\theta$, and $\omega_2$ on the optimal path, for an example where $\nu_0 = 0.75$, $\delta_0 = 0.5$, $\lambda_E = 2$, $\lambda_H = 1$, $r=1$, and $c = 0.1$. Up until $t = 1.06$, the optimal policy only works on approach 1; for $t \in [1.06,2.12)$, the policy works on approach 2; for $t \in [2.12, 2.25)$, the policy splits effort on 1 and 2, and after $t \ge 2.25$ the policy works on approach 3. Note that in the intermediate region of time from $[1.06, 2.12)$, the policy works only on approach 2, but the belief over whether approach 1 is valid drifts upwards.

Theorems & Definitions (35)

• Proposition 1
• Corollary 1
• Proposition 2
• Theorem 1
• Theorem 1b
• Lemma 1
• Lemma 2
• Corollary 2
• Proposition 3
• Corollary 3