Parametrized Inexact-ADMM based coordination games: A normalized Nash equilibrium approach

Abstract

Generalized Nash equilibrium problems are single-shot Nash equilibrium problems, whereby the decisions of all agents are coupled through a shared constraint. Such games are generally challenging to solve as they might give rise to a very large number of solutions. In this context, spanning many equilibria can be interesting to provide meaningful interpretations. In the literature, to compute equilibria, equilibrium problems are classically reformulated as optimization problems, potential games, relaxed and extended games. Applications of these reformulations to an economic dispatch problem under perfect and imperfect competition are provided. Unfortunately, these approaches only enable to describe a very limited part of the equilibrium set. To fill that gap, relying on normalized Nash equilibrium as solution concept, we provide a parametrized decomposition algorithm inspired by the Inexact-ADMM to span many more equilibrium points. Complexifying the setting, we consider an information structure in which the agents can withhold some local information from sensitive data, resulting in private coupling constraints. The convergence of the algorithm and deviations in the players' strategies at equilibrium are formally analyzed. In addition, the algorithm can be used to coordinate the agents on one specific equilibrium with desirable properties at the system level. The coordination game is formulated as a principal-agent problem, and a procedure is detailed to compute the equilibrium that minimizes a secondary cost function capturing systemlevel properties. Finally, the Inexact-ADMM is applied to a cellular resource allocation problem, exhibiting better convergence rate than vanilla ADMM, and to compute equilibria that achieve both system-level efficiency and maximum fairness.

Strategic Resource Pricing and Allocation in a 5G Network Slicing Stackelberg Game

“…To that purpose, we set x s s d s as SP s's own action, x s −s as SP s's estimate of the other SPs' actions, and x s col(x s s , x s −s ) as the concatenation of SP s's own action and estimate of the others' actions. Let Fs {x s s |x s s ≥ 0, ω T x s s = B s } be the strategy set of SP s. Following [38], [39], we decompose the pricing game G P per agent. Some slack variables (v ss ) s,s and (w ss ) s,s are introduced to guarantee the coincidence of the local copies.…”

“…Following [39], from (48a)-(48b), the primal update rule for SP s is obtained by solving a local optimization problem Dual update rule takes the form…”

TSO-DSOs Stable Cost Allocation for the Joint Procurement of Flexibility: A Cooperative Game Approach

“…Let C ⊆ N G . Relying on LP (30) structure, a direct application of [37], Prop. 6 enables us to prove that the set of solutions of LP (30) over C coincides with the set of solutions of the generalized Nash equilibrium problem GNEP(C), which has a generalized potential game structure, defined as:…”

“…It is quite obvious that C(G) = ∅ if and only if the optimum value of the linear program ( 37) is equal to v(N G ), in which case any optimal solution to (37) lies in C(G). Taking the linear program dual to (37), an equivalent condition for C(G) = ∅ can be obtained based on the concept of balanced sets. A collection B of nonempty subsets of N G is balanced if C∈B γ C v(C) ≤ v(N ) holds for every balanced collection B with weights (γ C ) C∈B .…”

Distributed and Constrained H2 Control Design via System Level Synthesis and Dual Consensus ADMM

“…It is therefore important to design intervention mechanisms, such as incentives and pricing, that drive the agents to a desirable equilibrium configuration. This inherently hierarchical structure appears in such diverse fields as wireless sensor networks [1], energy demand-response [2] and peer-to-peer trading [3], [4] in smart grids, traffic routing [5], and investment networks [6]. Recently, hierarchical decision making has attracted also the interest of the machine learning community in the context of hyperparameter optimization [7], deep equilibrium models [8], and meta-learning [9].…”