### Abstract

A linear-quadratic-Gaussian (LQG) game is an incomplete information game with quadratic payoff functions and Gaussian information structures. It has many applications such as a Cournot game, a Bertrand game, a beauty contest game, and a network game among others. LQG information design is a problem to find an information structure from a given collection of feasible Gaussian information structures that maximizes a quadratic objective function when players follow a Bayes Nash equilibrium. This paper studies LQG information design by formulating it as semidefinite programming, which is a natural generalization of linear programming. Using the formulation, we provide sufficient conditions for optimality and suboptimality of no and full information disclosure. In the case of symmetric LQG games, we characterize the optimal symmetric information structure, and in the case of asymmetric LQG games, we characterize the optimal public information structure, each of which is in a closed-form expression.

### Introduction

An equilibrium outcome in an incomplete information game depends not only upon a payoff structure, which consists of payoff functions together with a probability distribution of a payoff state, but also upon an information structure, which maps a payoff state to possibly stochastic signals of players. Information design analyzes the influence of an information structure on equilibrium outcomes, and in particular, characterizes an optimal information structure that induces an equilibrium outcome maximizing the expected value of an objective function of an information designer, who is assumed to be able to choose and commit to the information structure.1 General approaches to information design are presented by Bergemann and Morris (2013, 2016a,b, 2019), Taneva (2019), and Mathevet et al. (2020). A rapidly growing body of literature have investigated the economic application of information design in areas such as matching markets (Ostrovsky and Schwarz, 2010), voting games (Alonso and Camara, 2016), congestion games (Das et al., 2017), auctions (Bergemann et al., 2017), contests (Zhang and Zhou, 2016), and stress testing (Inostroza and Pavan, 2018), among others.

WP018

2020.03

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