Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Utility Design of Greedy Algorithms in Resource Allocation Games

Rohit Konda, Rahul Chandan, David Grimsman, Jason R. Marden

TL;DR

This paper investigates transient efficiency in resource-allocation games by fixing round-robin best-response dynamics and designing agent utilities to optimize short-term welfare. It proves that, for submodular welfare, optimal one-round designs can achieve performance near the asymptotic optimum, with a closed-form bound governed by curvature $\mathrm{C}$, and that running more than one round yields no general improvement. A detailed analysis reveals a Pareto frontier between transient efficiency and price of anarchy, including exact results for set-covering games that illustrate severe transient losses when asymptotic optimality is prioritized. An Illustrative wireless sensor coverage example demonstrates the practical impact: one-round optimal designs can outperform greedy and asymptotically optimal designs in the near term, with fast saturation of gains over a few rounds. Overall, the work highlights the importance of jointly considering local utility design, transient guarantees, and asymptotic performance in distributed, dynamic settings.

Abstract

Designing distributed algorithms for multi-agent problems is vital for many emerging application domains, and game-theoretic approaches are emerging as a useful paradigm to design such algorithms. However, much of the emphasis of the game-theoretic approach is on the study of equilibrium behavior, whereas transient behavior is often less explored. Therefore, in this paper we study the transient efficiency guarantees of best response processes in the context of resource-allocation games, which are used to model a variety of engineering applications. Specifically, the main focus of the paper is on designing utility functions of agents to induce optimal short-term system-level behavior under a best-response process. Interestingly, the resulting transient performance guarantees are relatively close to the optimal asymptotic performance guarantees. Furthermore, we characterize a trade-off that results when optimizing for both asymptotic and transient efficiency through various utility designs.

Optimal Utility Design of Greedy Algorithms in Resource Allocation Games

TL;DR

This paper investigates transient efficiency in resource-allocation games by fixing round-robin best-response dynamics and designing agent utilities to optimize short-term welfare. It proves that, for submodular welfare, optimal one-round designs can achieve performance near the asymptotic optimum, with a closed-form bound governed by curvature , and that running more than one round yields no general improvement. A detailed analysis reveals a Pareto frontier between transient efficiency and price of anarchy, including exact results for set-covering games that illustrate severe transient losses when asymptotic optimality is prioritized. An Illustrative wireless sensor coverage example demonstrates the practical impact: one-round optimal designs can outperform greedy and asymptotically optimal designs in the near term, with fast saturation of gains over a few rounds. Overall, the work highlights the importance of jointly considering local utility design, transient guarantees, and asymptotic performance in distributed, dynamic settings.

Abstract

Designing distributed algorithms for multi-agent problems is vital for many emerging application domains, and game-theoretic approaches are emerging as a useful paradigm to design such algorithms. However, much of the emphasis of the game-theoretic approach is on the study of equilibrium behavior, whereas transient behavior is often less explored. Therefore, in this paper we study the transient efficiency guarantees of best response processes in the context of resource-allocation games, which are used to model a variety of engineering applications. Specifically, the main focus of the paper is on designing utility functions of agents to induce optimal short-term system-level behavior under a best-response process. Interestingly, the resulting transient performance guarantees are relatively close to the optimal asymptotic performance guarantees. Furthermore, we characterize a trade-off that results when optimizing for both asymptotic and transient efficiency through various utility designs.
Paper Structure (15 sections, 13 theorems, 88 equations, 6 figures, 1 algorithm)

This paper contains 15 sections, 13 theorems, 88 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model
  3. One Round Guarantees
  4. Multiple Round Guarantees
  5. Tradeoffs of Short/Long Term Performance
  6. Illustrative Example
  7. Conclusion
  8. Linear Program Formulation of One Round Walk
  9. Proof of Theorem \ref{['thm:optLP']}
  10. Proof of Theorem \ref{['thm:oneroundC']}
  11. Proof of Proposition \ref{['prop:effMCC']}
  12. Proof of Theorem \ref{['thm:kroundC']}
  13. Proof of Theorem \ref{['thm:submodtrade']}
  14. Proof of Theorem \ref{['thm:poapobtradeoff']}
  15. Supermodular welfare rules

Key Result

Theorem 1

Suppose that $\mathcal{W}$ is spanned from a set of welfare rules $\{w^1, \dots, w^m\}$ where each $w^\ell$ is submodular. Then the optimal efficiency guarantees achievable with a one-round best response process is given by where $\beta^{\ell} \in \mathbb{R}_{\geq 0}$ is the solution to the following program. where we take $u^{\ell}(1) = 1$ and $y, z \in \mathbb{N}$. Furthermore, a utility desig

Figures (6)

  • Figure 1: If a given multi-agent scenario with $n$ agents is modeled as a game, the construction of distributed algorithms can be decoupled into two domains: the design of local objectives (utilities) and the design of the learning dynamics. In this paper, we fix the dynamics to the classical round-robin best response and study the effects of the utility design on the resulting efficiency bounds on the trajectory. Moreover, we characterize the efficiency guarantee as the number iterations $\kappa$ that the Algorithm is run for increases.
  • Figure 2: In the top figure, we visually depict the efficiency guarantees of Theorem \ref{['thm:oneroundC']} and Proposition \ref{['prop:effMCC']} with respect to the optimal asymptotic guarantees. Additionally, the fractional gains in the performance when moving from the greedy solution to the optimal one-round and the asymptotic solutions are depicted in the bottom figure.
  • Figure 3: We depict the Pareto-optimal frontier of the one-round efficiency ($\mathrm{Eff}(\mathcal{G}_{w_{\mathrm{sc}}, u}; 1)$) versus the asymptotic efficiency guarantees ($\mathrm{PoA}(\mathcal{G}_{w_{\mathrm{sc}}, u})$) that are possible with regards to the class of set-covering games. We note that the severe drop off in transient efficiency that results from optimizing the asymptotic efficiency.
  • Figure 4: We plot the average rate of event detection in a randomly generated set of wireless sensor coverage problems with respect to three utility designs: the one-round optimal, the common interest, and the asymptotically optimal utility design. We see that in the short term, the one-round optimal design performs better in the worst case than the greedy and the asymptotically optimal utility designs.
  • Figure 5: The worst case game construction achieving the one-round walk guarantee dictated by Lemma \ref{['lem:setpob']}. In this game, all the agents can either stack on the first resource set or spread out.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Definition 1: Basis
  • Definition 2: Submodularity
  • Theorem 1
  • Definition 3: Curvature
  • Theorem 2
  • Proposition 1
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • ...and 13 more