Designing Exploration Contracts

TL;DR

The paper investigates algorithmic contract design for sequential exploration, where a principal delegates the Pandora's Box‑style search to an agent and commits to non-negative transfers for observed prizes. It provides polynomial-time methods to compute optimal contracts in several natural scenarios, including optimal linear contracts and general contracts under $a_{ij}=0$, as well as specialized cases for binary boxes and IID boxes with a single principal‑positive prize. Central to the approach is the fair-cap index framework, which reduces the agent’s problem to the Pandora's Box policy while enabling the principal to shape exploration order and stopping decisions through transfers. The results highlight when simple linear contracts suffice and when more structured general contracts are necessary, offering tractable algorithms and insights for principled incentive design in exploration tasks with hidden actions and costly information gathering.

Abstract

We study a natural application of contract design in the context of sequential exploration problems. In our principal-agent setting, a search task is delegated to an agent. The agent performs a sequential exploration of $n$ boxes, suffers the exploration cost for each inspected box, and selects the content (called the prize) of one inspected box as outcome. Agent and principal obtain an individual value based on the selected prize. To influence the search, the principal a-priori designs a contract with a non-negative payment to the agent for each potential prize. The goal of the principal is to maximize her expected reward, i.e., value minus payment. Interestingly, this natural contract scenario shares close relations with the Pandora's Box problem. We show how to compute optimal contracts for the principal in several scenarios. A popular and important subclass is that of linear contracts, and we show how to compute optimal linear contracts in polynomial time. For general contracts, we obtain optimal contracts under the standard assumption that the agent suffers cost but obtains value only from the transfers by the principal. More generally, for general contracts with non-zero agent values for outcomes we show how to compute an optimal contract in two cases: (1) when each box has only one prize with non-zero value for principal and agent, (2) for i.i.d. boxes with a single prize with positive value for the principal.

Figures (1)

• Figure 1: Two cases for an intersection between $v_{ij}$ and ${\varphi_{i}}\xspace$. Left: The slope of $v_{ij}$ was less than slope of ${\varphi_{i}}\xspace$ before the intersection. By definition, it contributed to the weighted average slope that defines the slope of ${\varphi_{i}}\xspace$. After the intersection, it does not contribute anymore, and the weighted average only increases. Right: The slope of $v_{ij}$ was greater than slope of ${\varphi_{i}}\xspace$ before the intersection. Symmetrical arguments.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Theorem 3
• Definition 1
• Lemma 2
• proof
• Lemma 3
• proof