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Optimal Utility Design with Arbitrary Information Networks

Vartika Singh, Will Wesley, Philip N. Brown

TL;DR

The paper addresses utility design in multi-agent coordination under general information networks, proposing LP-based methods to (i) exactly characterize the Price of Anarchy (PoA) for a fixed utility design and network, and (ii) optimize the PoA to derive the PoA-optimal local utilities. It extends Marden's full-information results to arbitrary information graphs by introducing a class-based network representation and deriving primal and dual LPs that yield the PoA and the optimal mechanism $f^{opt}$. A two-class refinement for networks with blind agents enables tractable analysis and reveals that marginal-contribution utilities are optimal in set-covering scenarios with blindness. Numerical results demonstrate robustness of the optimal mechanism to unanticipated communication failures, suggesting that full-information-optimized utilities can perform near-optimally even when some agents become isolated or blind, with practical implications for distributed resource allocation.

Abstract

We consider multi-agent systems with general information networks where an agent may only observe a subset of other agents. A system designer assigns local utility functions to the agents guiding their actions towards an outcome which determines the value of a given system objective. The aim is to design these local utility functions such that the Price of Anarchy (PoA), which equals the ratio of system objective at worst possible outcome to that at the optimal, is maximized. Towards this, we first develop a linear program (LP) that characterizes the PoA for any utility design and any information network. This leads to another LP that optimizes the PoA and derives the optimal utility design. Our work substantially generalizes existing approaches to the utility design problem. We also numerically show the robustness of proposed framework against unanticipated communication failures.

Optimal Utility Design with Arbitrary Information Networks

TL;DR

The paper addresses utility design in multi-agent coordination under general information networks, proposing LP-based methods to (i) exactly characterize the Price of Anarchy (PoA) for a fixed utility design and network, and (ii) optimize the PoA to derive the PoA-optimal local utilities. It extends Marden's full-information results to arbitrary information graphs by introducing a class-based network representation and deriving primal and dual LPs that yield the PoA and the optimal mechanism . A two-class refinement for networks with blind agents enables tractable analysis and reveals that marginal-contribution utilities are optimal in set-covering scenarios with blindness. Numerical results demonstrate robustness of the optimal mechanism to unanticipated communication failures, suggesting that full-information-optimized utilities can perform near-optimally even when some agents become isolated or blind, with practical implications for distributed resource allocation.

Abstract

We consider multi-agent systems with general information networks where an agent may only observe a subset of other agents. A system designer assigns local utility functions to the agents guiding their actions towards an outcome which determines the value of a given system objective. The aim is to design these local utility functions such that the Price of Anarchy (PoA), which equals the ratio of system objective at worst possible outcome to that at the optimal, is maximized. Towards this, we first develop a linear program (LP) that characterizes the PoA for any utility design and any information network. This leads to another LP that optimizes the PoA and derives the optimal utility design. Our work substantially generalizes existing approaches to the utility design problem. We also numerically show the robustness of proposed framework against unanticipated communication failures.

Paper Structure

This paper contains 13 sections, 6 theorems, 35 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Local Utility Functions and Resulting Class of Games
  4. Information Graph and Classes
  5. Price of Anarchy for Network $({\@fontswitch\mathcal{C}},{\@fontswitch\mathcal{N}})$
  6. Linear Program for Price of Anarchy
  7. Dual Linear Program to characterize PoA
  8. Optimal Price of Anarchy
  9. A Network with Blind Agents
  10. Numerical Results
  11. Set Covering Games with Blind Agents
  12. Robustness of Optimal Mechanism
  13. Conclusions

Key Result

Theorem 1

For a given network $({\@fontswitch\mathcal{C}},{\@fontswitch\mathcal{N}})$, basis function $w$ and utility generating mechanism $f=\{f_j\}_{j\in[k]}$,

Figures (3)

  • Figure 1: A general network with information available to various agents given by ${\@fontswitch\mathcal{N}}_1 = {\@fontswitch\mathcal{N}}_5 = \{1,2,3,4,5\}$, ${\@fontswitch\mathcal{N}}_2 ={\@fontswitch\mathcal{N}}_3= \{1,2,3,5\}$ and ${\@fontswitch\mathcal{N}}_4 =\{1,4\}$. Agent 1 and 5 observe the same set of agents, but 1 can be observed by 4 whereas 5 cannot be, thus actions of 1 affect 4 differently than actions of 5, hence, 1 and 5 are not similar. On the other hand 2 and 3 are similar. The agents can be partitioned as ${\@fontswitch\mathcal{C}}_1 =\{1\}$, ${\@fontswitch\mathcal{C}}_2 = \{2,3\}$, ${\@fontswitch\mathcal{C}}_3 = \{4\}$ and ${\@fontswitch\mathcal{C}}_4 =\{5\}$ based on similarity. Then ${\@fontswitch\mathcal{N}}_{{\@fontswitch\mathcal{C}}_1}= {\@fontswitch\mathcal{N}}_{{\@fontswitch\mathcal{C}}_4}={\@fontswitch\mathcal{C}}_1\cup{\@fontswitch\mathcal{C}}_2\cup {\@fontswitch\mathcal{C}}_3\cup {\@fontswitch\mathcal{C}}_4$, ${\@fontswitch\mathcal{N}}_{{\@fontswitch\mathcal{C}}_2} = {\@fontswitch\mathcal{C}}_1 \cup {\@fontswitch\mathcal{C}}_2\cup {\@fontswitch\mathcal{C}}_4$ and ${\@fontswitch\mathcal{N}}_{{\@fontswitch\mathcal{C}}_3} = {\@fontswitch\mathcal{C}}_1\cup {\@fontswitch\mathcal{C}}_3$, and C.1-C.3 are satisfied.
  • Figure 2: PoA at marginal contribution utility $f^{mc}$ and optimal PoA for set covering games of section \ref{['sec_set_cover']} as a function of number of blind agents in the network. The PoA at $f^{mc}$ matches with the optimal PoA if there is even one blind agent in the network, implying optimality of marginal contribution
  • Figure 3: PoA when agents keep using optimal utility mechanism corresponding to full information case $f^*$ even when there is a communication failure, compared to the optimal PoA for sub-modular system objective. The mechanism $f^*$ is near optimal, thus robust against communication failures.

Theorems & Definitions (8)

  • Definition 1: Nash Equilibrium Nash
  • Theorem 1: Primal LP to characterize PoA
  • Remark 1
  • Theorem 2: Dual LP to characterize PoA
  • Theorem 3: Optimal PoA for general network
  • Theorem 4
  • Lemma 1
  • Lemma 2