[Paper Review] LQG Information Design

Miyashita, Masaki, Ui, Takashi|arXiv (Cornell University)|Dec 15, 2023

Game Theory and Applications18 references4 citations

TL;DR

This paper studies information design in linear-quadratic-Gaussian (LQG) games, showing that Gaussian information structures are optimal among all possible information structures due to the quadratic nature of payoff and objective functions. The optimal structure is found via semidefinite programming, with closed-form solutions derived for symmetric games and public information structures in asymmetric games, particularly when the covariance matrix of the state vector is not full rank.

ABSTRACT

This paper addresses information design in a workhorse model of network games, where agents have linear best responses, the information designer maximizes a quadratic objective, and the payoff-relevant state follows a multivariate Gaussian distribution. We formulate the problem as a semidefinite program and establish strong duality to characterize the optimal information structure. A necessary and sufficient condition for optimality is given by a simple linear relationship between the induced equilibrium strategy profile and the state. Leveraging this characterization, we show that the state is fully revealed in an aggregative form for welfare maximization, while individual agents may remain only partially informed. When agent roles are interchangeable, the optimal information structure inherits the same degree of symmetry, which facilitates computation. In such cases, we show that the optimal amount of information revealed to each agent is closely linked to the network's chromatic number.

Motivation & Objective

• To identify optimal information structures in linear-quadratic-Gaussian (LQG) games where payoffs are quadratic and states are Gaussian.
• To determine whether Gaussian information structures are optimal among all feasible information structures.
• To characterize optimal public information structures in asymmetric LQG games, especially when the state variance matrix is rank-deficient.
• To provide closed-form solutions for optimal information design using semidefinite programming and spectral decomposition.
• To extend prior results on Bayesian persuasion with quadratic objectives to general LQG environments beyond binary or full-rank settings.

Proposed method

• Formulate the LQG information design problem as maximizing a quadratic objective function over the covariance matrix of actions and states under Bayes-Nash equilibrium.
• Show that the expected objective value is affine in the covariance matrix, enabling reformulation as a linear function over covariance matrices.
• Prove that the set of feasible covariance matrices under all information structures coincides with that under Gaussian information structures, due to the positive semidefiniteness and construction of multivariate normal distributions.
• Reduce the problem to a semidefinite program (SDP) over positive semidefinite matrices subject to linear constraints on the covariance structure.
• Use spectral decomposition of the matrix $ D^ op V_Q D $ to construct the optimal public signal $ t_0 = U_m^ op (D^ op D)^{-1} D^ op heta $, where $ U_m $ contains eigenvectors corresponding to positive eigenvalues.
• Leverage properties of multivariate normal distributions to verify that the constructed signal achieves the theoretical upper bound on the objective function.

Experimental results

Research questions

• RQ1 Is a Gaussian information structure optimal among all possible information structures in LQG games with quadratic payoffs and Gaussian states?
• RQ2 Under what conditions is no information disclosure or full information disclosure optimal in LQG games?
• RQ3 Can the optimal public information structure be expressed in closed form when the state variance matrix is not full rank?
• RQ4 How does the structure of the matrix $ D^ op V_Q D $ determine the optimality of partial information disclosure?
• RQ5 To what extent can the results of Tamura (2018) on single-agent Bayesian persuasion be generalized to multi-agent LQG games with rank-deficient state variances?

Key findings

• Among all information structures, Gaussian information structures are optimal for LQG information design, enabling the designer to restrict attention to Gaussian signals without loss of optimality.
• The optimal information structure can be computed via semidefinite programming, which generalizes linear programming to positive semidefinite constraints.
• When $ D^ op V_Q D $ is negative semidefinite, no information disclosure is optimal; when positive semidefinite, full information disclosure is optimal.
• For intermediate cases where $ D^ op V_Q D $ has both positive and negative eigenvalues, partial information disclosure is optimal, with the optimal signal constructed from eigenvectors of the positive eigenvalue subspace.
• The maximum value of the objective function equals the sum of the positive eigenvalues of $ D^ op V_Q D $, and this bound is achieved by the constructed public signal $ t_0 = U_m^ op (D^ op D)^{-1} D^ op heta $.
• The solution generalizes Tamura (2018) by handling rank-deficient state variance matrices, providing a complete characterization even when $ \mathrm{var}(\theta) $ is not full rank.