Competitive Information Design for Pandora's Box

TL;DR

This paper extends Pandora's Box by introducing competitive information design where each box (sender) commits to an information-revelation policy. The agent’s optimal search/stopping strategy, given the signaling, follows a Weitzman-style reservation-value framework based on posterior-mean distributions, enabling a reformulation of signaling as distributions over posterior means subject to mean-preserving constraints. The authors provide a necessary-and-sufficient condition for the existence of a pure symmetric Nash equilibrium in the symmetric setting, and show how equilibrium structure depends on reservation-value matching, convexity of $G^{n-1}$, and a no-deviation constraint; they further establish that more informative (Blackwell-order) distributions strictly improve the agent’s payoff and that full information across boxes yields the highest welfare. These results illuminate how competition and inspection costs shape information disclosure and agent outcomes, with practical implications for platforms and markets employing strategic information design.

Abstract

We study a natural competitive-information-design strategic variant for the celebrated Pandora's Box problem (Weitzman, 1979), where each box is associated with a strategic information sender who can design what information about the box's prize value to be revealed to the agent when the agent inspects the box. This variant with strategic boxes is motivated by a wide range of real-world economic applications for Pandora's Box. Our contributions are three-fold: (1) given the boxes' information policies, we characterize the agent's optimal search and stopping strategy; (2) we fully characterize the pure symmetric equilibrium for the game of boxes' competitive information revelation in a symmetric environment; and (3) we reveal various insights regarding information competition and the resultant agent payoff at equilibrium, and additionally, we study informational properties of Pandora's Box by establishing an intrinsic connection between informativeness of any box's value distribution and the utility order of the search agent.

Figures (1)

• Figure 1: In both figures, the prior $H$ is the gray solid line, the distribution $G$ that satisfies conditions $(i)$--$(ii)$ in \ref{['thm_nash_iff']} is the deep gray solid line. The profitable deviation $F$ is then the black dashed line. See the detailed descriptions in \ref{['ex_no_Nash_a']} and \ref{['ex_no_Nash_b']}. (a): Equilibrium does not exist as it violates the the case $(a)$ in condition $(iii)$. (b): Equilibrium does not exist as it violates the case $(b)$ in condition $(iii)$.

Theorems & Definitions (57)

• Example 1.1: Open Houses in Housing Markets
• Example 1.2: Free Trials of Digital Services
• Definition 3.1: Reservation Value
• Theorem 3.1: wei-79
• Theorem 3.2
• Corollary 3.3
• Definition 3.2: Mean-preserving Spread
• Lemma 3.4: Blackwell-79
• Definition 4.1: Alternating $(n-1)$-linear MPS -- HKB-19
• Theorem 4.1