Distributed control and game design: From strategic agents to programmable machines

TL;DR

This dissertation develops a comprehensive framework for distributed control and game design in large-scale systems. Part I models many agents as aggregative games with costs depending on population averages, deriving variational reformulations, bounds on Nash–Wardrop distance, and decentralized algorithms that converge to equilibria with provable efficiency. Part II shifts to programmable machines, introducing a utility-design approach that maximizes worst-case equilibrium performance via linear-programming formulations, and applying to submodular, covering, and maximum-coverage problems with explicit PoA guarantees. The work yields distributed, scalable methods with certified performance, enabling applications in demand-response, sensor placement, distributed caching, and resource allocation. Overall, it provides a rigorous bridge from strategic agent coordination to programmable multiagent systems, with provable guarantees on efficiency and convergence under realistic information-sharing constraints.

Abstract

Large scale systems are forecasted to greatly impact our future lives thanks to their wide ranging applications including cooperative robotics, mobility on demand, resource allocation, supply chain management. While technological developments have paved the way for the realization of such futuristic systems, we have a limited grasp on how to coordinate the individual components to achieve the desired global objective. This thesis deals with the analysis and coordination of large scale systems without the need of a centralized authority. In the first part of this thesis, we consider non-cooperative decision making problems where each agent's objective is a function of the aggregate behavior of the population. First, we compare the performance of an equilibrium allocation with that of an optimal allocation and propose conditions under which all equilibrium allocations are efficient. Towards this goal, we prove a novel result bounding the distance between the strategies at a Nash and Wardrop equilibrium that might be of independent interest. Second, we show how to derive scalable algorithms that guide agents towards an equilibrium allocation. In the second part of this thesis, we consider large-scale cooperative problems, where a number of agents need to be allocated to a set of resources with the goal of jointly maximizing a given submodular or supermodular set function. Since this class of problems is computationally intractable, we aim at deriving tractable algorithms for attaining approximate solutions. We approach the problem from a game-theoretic perspective and ask the following: how should we design agents' utilities so that any equilibrium configuration is almost optimal? To answer this question we introduce a novel framework that allows to characterize and optimize the system performance as a function of the chosen utilities by means of a tractable linear program.

Figures (28)

• Figure 1:
• Figure 2: Two dimensional representation of nonexpansive operator (NE), co-coercive operator with constant $\eta$ ($\eta-$COC), and strongly monotone operator with constant $\alpha$ ($\alpha-$SMON). For each of these properties, the corresponding colored region represents the locus of points where $F(1,0)$ must lie, under the assumption that $\mathbbold{0}_{2}$ is a fixed point of $F$, i.e., that $F(\mathbbold{0}_{2})=\mathbbold{0}_{2}$. The regions can be easily derived from the corresponding definitions.
• Figure 3: On the left: illustration of the condition \ref{['eq:videf']} for a general variational inequality. The point $\bar{x}$ is a solution of VI$(\mathcal{X},F)$ since the scalar product of $F(\bar{x})$ with any other vector attached to $\bar{x}$ and pointing inside the set $\mathcal{X}$ is non-negative. With a similar reasoning, it is immediate to note that the point $\bar{y}$ is not a solution of VI$(\mathcal{X},F)$. On the right: the special case of variational inequality VI$(\mathcal{X},\nabla_x f)$ corresponding to the convex optimization program $\min_{x\in\mathcal{X}}f(x)$. Similarly to the case on the left, $\bar{x}$ is a solution of VI$(\mathcal{X},\nabla_x f)$ and thus a global minimizer of $f$ (see \ref{['prop:minimumprinc']}), while $\bar{y}$ is not.
• Figure 4: The set of Nash equilibria (NE), normalized NE and variational NE are all nested.
• Figure 6: Information flow for \ref{['alg:asp']}

Theorems & Definitions (131)

• Example 1: Guess 2/3 of the average
• Definition 1: Lipschitz, nonexpansive, contractive
• Definition 2: Monotone and strongly monotone facchinei2007finite
• Proposition 1: Convex functions have monotone gradients bauschke2011convex
• Definition 3: Co-coercive
• Definition 4: Variational inequality
• Proposition 2: Existence and uniqueness facchinei2007finite
• Proposition 3: Minimum principle bertsekas1989parallel
• Definition 5: Metric projection
• Proposition 4: facchinei2007finite