Best Response Sequences and Tradeoffs in Submodular Resource Allocation Games

TL;DR

The paper addresses transient efficiency in distributed submodular resource allocation via $k$-round round-robin best-response dynamics, examining how local utility designs shape short-term trajectories. By parameterizing welfare with curvature $\mathrm{C} \in [0,1]$ and mapping welfare rules to agent utilities through $\Gamma$, the authors derive tight transient bounds: $\mathrm{Eff}^*(\mathcal{W}; k) \leq 1 - \mathrm{C}/2$ and $\mathrm{Eff}(\mathcal{W}, \Gamma_{\rm{id}}; k) = (1+\mathrm{C})^{-1}$ for all finite $k$, with $\mathrm{Eff}^*(\mathcal{W}; \infty) = 1 - \mathrm{C}/e$. They further show that increasing $k$ does not improve these guarantees and that there is a fundamental trade-off between transient and asymptotic efficiency, including a detailed Pareto frontier for set-covering games. The results provide design principles for distributed utility design and reveal that short-term performance can diverge significantly from long-run equilibria, as corroborated by illustrative simulations. Overall, the work broadens analysis beyond equilibrium to transient dynamics in distributed resource allocation.

Abstract

Deriving competitive, distributed solutions to multi-agent problems is crucial for many developing application domains; Game theory has emerged as a useful framework to design such algorithms. However, much of the attention within this framework is on the study of equilibrium behavior, whereas transient behavior is often ignored. Therefore, in this paper we study the transient efficiency guarantees of best response processes in the context of submodular resource allocation games, which find application in various engineering contexts. Specifically the main focus of this paper is on characterizing the optimal short-term system-level behavior under the best-response process. Interestingly, the resulting transient performance guarantees are relatively close to the optimal asymptotic performance guarantees. Furthermore, we characterize the trade-offs that result when optimizing for both asymptotic and transient efficiency through various utility designs.

Figures (8)

• 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 $k$ round-robin best response and study the effects of the utility design on the efficiency bounds for the resulting decision trajectory. Moreover, we characterize the guarantees as the number of rounds $k$ increases.
• Figure 2: Game example depicting resource values and agent selections.
• Figure 3: We depict the Pareto-optimal frontier of the one-round efficiency $\mathrm{Eff}(w_{\mathrm{sc}}, \tilde{w}; 1)$ versus the asymptotic efficiency guarantees $\mathrm{Eff}(w_{\mathrm{sc}}, \tilde{w}; \infty)$ 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 defense in a randomly generated set of weapon-target assignment 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: In this figure, rows represent players and columns represent resources. Red circles represent selections in $a^{\mathrm{ne}}$ and blue circles represent selections in $a^{\mathrm{opt}}$.

Theorems & Definitions (21)

• Example 1: Weapon-Target Assignment
• Example 2: Wireless Transmission over a Network
• Example 3: Set Covering continued
• Definition 1: Curvature
• Proposition 1
• Remark
• Theorem 1
• Remark
• Theorem 2
• Theorem 3