Principal-Agent Problem with Third Party: Information Design from Social Planner's Perspective

TL;DR

This work designs a simple workflow with two stages for the social planner, investigates information design and shows that binary-signal information structure suffices to induce the socially optimal outcome determined in the first stage.

Abstract

We study the principal-agent problem with a third party that we call social planner, whose responsibility is to reconcile the conflicts of interest between the two players and induce socially optimal outcome in terms of some given social utility function. The social planner owns no contractual power but manages to control the information flow between the principal and the agent. We design a simple workflow with two stages for the social planner. In the first stage, the problem is reformulated as an optimization problem whose solution is the optimal utility profile. In the second stage, we investigate information design and show that binary-signal information structure suffices to induce the socially optimal outcome determined in the first stage. The simplicity and modularity of our method make it easy to implement in various scenarios within the principal-agent problem.

Figures (9)

• Figure 1: Utility profiles and the implementable set. (a) The agent is risk-neutral, the bold parts of the lines are the implementable utility profiles. (b) The agent is risk-averse, the area below the solid line is the super set of all possible utility profiles, while the pink area is the set of implementable utility profiles.
• Figure 2: Optimal utility profile if the social planner uses Nash product as the social utility function and the agent is risk-neutral. Denote $a^*$ as the action that induces the largest utilitarian social welfare (USF). (a) $\frac{r_{a^*} - c_{a^*}}{2} \geq \max \left\{0, r_{\hat{a}} - c_{\hat{a}}\right\}$, the blue point represents the optimal utility profile. (b) $0 \leq \frac{r_{a^*} - c_{a^*}}{2} \leq r_{\hat{a}} - c_{\hat{a}}$, the green point represents the optimal utility profile.
• Figure 3: Optimal utility profile for a given action $a$ if the social planner uses utilitarian social welfare (USF) as the social utility function and the agent is risk-averse. Let $x_1 = r_{\hat{a}} - v^{-1}(c_{\hat{a}})$. (a) If $v'(v^{-1}(c_a)) \leq 1$, the green point represents the optimal utility profile. (b) If $r_{\hat{a}} - v^{-1}(c_{\hat{a}}) \geq 0$ and $v'(r_a - x_1) > 1$, the green point represents the optimal utility profile. (c) If the line $y = x$ is tangent to $y = v(r_a - x) - c_a$ at some implementable utility profile, the blue point represents the optimal utility profile.
• Figure 4: Optimal utility profile for a given action $a$ if the social planner uses Nash product as the social utility function and the agent is risk-averse. $x_a$ is the solution of the equation $v(r_a - x) - x \cdot v'(r_a - x) = c_a$. (a) If $(x_a, y_a)$ is implementable, it would be the optimal utility profile (the blue point). (b) If $(x_a, y_a)$ is not implementable, the optimal utility profile would be $(x_1, v(r_a - x_1) - c_a)$ (the green point).
• Figure 5: Optimal utility profile if the social planner uses egalitarian social welfare as the social utility function and the agent is risk-neutral. Denote $a^*$ as the action that induces the largest utilitarian social welfare (USF). The dashdotted line is the contour line for the objective function $\min \left\{u^P, u^A\right\}$. (a) $\frac{r_{a^*} - c_{a^*}}{2} \geq \max \left\{0, r_{\hat{a}} - c_{\hat{a}}\right\}$, the blue point represents the optimal utility profile. (b) $0 \leq \frac{r_{a^*} - c_{a^*}}{2} \leq r_{\hat{a}} - c_{\hat{a}}$, the green point represents the optimal utility profile.

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 1
• proof