[Paper Review] A Semidefinite Approach to Information Design in Non-atomic Routing Games.

Yixian Zhu, Ketan Savla|arXiv (Cornell University)|May 6, 2020

Game Theory and Applications4 citations

TL;DR

This paper proposes a semidefinite programming approach to solve the optimal information design problem in non-atomic routing games with uncertain network states. It shows that for affine latency functions, optimal private signals can be computed via a hierarchy of polynomial optimizations, with the first level being exact for the two-link case.

ABSTRACT

We consider a routing game among non-atomic agents where link latency functions are conditional on an uncertain state of the network. All the agents have the same prior belief about the state, but only a fixed fraction receive private route recommendations or a common message, which are generated by a known randomization, referred to as private or public signal respectively. The remaining non-receiving agents choose route according to Bayes Nash flow with respect to the prior. We develop a computational approach to solve the optimal information design problem, i.e., to minimize expected social latency cost over all public or obedient private signals. For a fixed flow induced by non-receiving agents, design of an optimal private signal is shown to be a generalized problem of moments for affine link latency functions, and to admit an atomic solution for the basic two link case. Motivated by this, a hierarchy of polynomial optimization is proposed to approximate, with increasing accuracy, information design over private and public signals, when the non-receiving agents choose route according to Bayes Nash flow. The first level of this hierarchy is shown to be exact for the basic two link case.

Motivation & Objective

• To address the optimal information design problem in non-atomic routing games where agents face uncertainty about network states.
• To minimize expected social latency cost through the design of public or private signals.
• To develop a computational framework for generating optimal signals when non-receiving agents follow Bayes-Nash equilibrium strategies.
• To establish conditions under which optimal private signals exist and can be computed efficiently.
• To propose a hierarchy of polynomial optimization problems that approximate information design with increasing accuracy.

Proposed method

• Formulates the optimal private signal design as a generalized problem of moments for affine link latency functions.
• Applies a hierarchy of semidefinite relaxations to approximate solutions to the information design problem.
• Uses polynomial optimization techniques to model and solve the moment problem arising from signal design.
• Establishes that the first level of the hierarchy yields an exact solution for the basic two-link routing case.
• Models the behavior of non-receiving agents as Bayes-Nash flows based on the prior belief about the state.
• Integrates signal generation via known randomization into the optimization framework to ensure obedience or public signal consistency.

Experimental results

Research questions

• RQ1 What is the structure of optimal private signals in non-atomic routing games with affine latency functions?
• RQ2 Can the optimal information design problem be solved exactly for the two-link case?
• RQ3 How can polynomial optimization hierarchies be used to approximate optimal public and private signals?
• RQ4 What conditions ensure the existence of atomic solutions for private signal designs?
• RQ5 How does the hierarchy of semidefinite relaxations converge to the optimal solution?

Key findings

• For affine link latency functions, the optimal private signal design problem reduces to a generalized problem of moments.
• An atomic solution exists for the optimal private signal in the basic two-link routing case.
• The first level of the proposed polynomial optimization hierarchy yields an exact solution for the two-link case.
• The hierarchy provides increasingly accurate approximations to the optimal information design for general cases.
• The framework enables computation of optimal public and obedient private signals under Bayes-Nash routing behavior.
• The approach is computationally tractable and leverages semidefinite programming for scalable solution approximation.