Collaborative Decision-Making and the k-Strong Price of Anarchy in Common Interest Games

TL;DR

This work studies the benefits and costs of collaborative communication in large-scale multi-agent systems by modeling them as common-interest games with coalitions up to size $k$ and analyzing the resulting $k$-strong Nash equilibria. It introduces the $k$-strong price of anarchy (SPoA) and a coalitionally smooth framework to bound how far these equilibria are from the optimum, plus tractable linear programs for resource allocation settings. The paper also analyzes coalitional dynamics (round-robin and asynchronous best-response) and provides transient performance guarantees, showing finite convergence and welfare bounds that depend on coalition size. Furthermore, it develops a utility-design approach, via generalized coalitional smoothness, to improve system performance by shaping agents’ objectives $U$ while preserving tractable analysis. Overall, the results offer practical tools for designing collaborative multi-agent systems with tunable communication structures and predictable efficiency guarantees, applicable to resource allocation and beyond.

Abstract

The control of large-scale, multi-agent systems often entails distributing decision-making across the system components. However, with advances in communication and computation technologies, we can consider new collaborative decision-making paradigms that exist somewhere between centralized and distributed control. In this work, we seek to understand the benefits and costs of increased collaborative communication in multi-agent systems. We specifically study this in the context of common interest games in which groups of up to k agents can coordinate their actions in maximizing the common objective function. The equilibria that emerge in these systems are the k-strong Nash equilibria of the common interest game; studying the properties of these states can provide relevant insights into the efficacy of inter-agent collaboration. Our contributions come threefold: 1) provide bounds on how well k-strong Nash equilibria approximate the optimal system welfare, formalized by the k-strong price of anarchy, 2) study the run-time and transient performance of collaborative agent-based dynamics, and 3) consider the task of redesigning objectives for groups of agents which improve system performance. We study these three facets generally as well as in the context of resource allocation problems, in which we provide tractable linear programs that give tight bounds on the k-strong price of anarchy.

Figures (6)

• Figure 1: Illustration of the $k$-strong Nash equilibrium local optimality guarantee for a three-agent common-interest game where $k \in \{1,2,3\}$. In each case, if the dark cube is a $k$-strong Nash equilibrium, then it is optimal over the highlighted region with respect to the shared objective function $W$. As $k$ (the size of collaborative groups) increases, the local optimality is strengthened by holding overall $k$-lateral deviations.
• Figure 2: Strong Price of Anarchy in resource covering games with $n=20$ players and coalitions up to size $k$ (horizontal axis). As the size of groups that are allowed to collaborate grows, so too does the approximation ratio (i.e., strong price of anarchy) of a $k$-strong Nash equilibrium. The efficiency of an equilibrium can be further improved by designing the utility functions agents are set to maximize. The solid green line is the $k$-strong price of anarchy when agents maximize the system objective (generated by \ref{['thm:spoa']}). The dashed red line is an upper bound on the $k$-strong price of anarchy while using an optimal utility design (generated by \ref{['prop:util_upper_bound']}).
• Figure 3: Tight $k$-strong price of anarchy bounds for resource allocation games with various welfare functions. We illustrate four settings of local welfare function (top, left to right), and for each, we use \ref{['thm:spoa']} to generate tight bounds on the $k$-strong price of anarchy for all $1 \leq k \leq n$. The bottom figures show these bounds and illustrate how increased inter-agent collaboration increases our efficiency guarantees on equilibrium system welfare.
• Figure 4: Numerical example of the coalitional asynchronous best response dynamics. In \ref{['fig:num_sim:group_rev']}, the system welfare is plotted over the number of group action revisions, and in \ref{['fig:num_sim:welf_check']} it is plotted over the number of welfare evaluations. From this data, we can observe that group revisions offer superior system transient and long-term performance but require more welfare evaluations to compute group actions.
• Figure 5: Bounds on the $k$-strong Price of Anarchy using the optimal utility function in the class of resource allocation games with welfare function $w$. Upper bound on $\mathrm{SPoA}_k^\star(\mathcal{G}_n,w)$ generated by \ref{['prop:util_upper_bound']} and lower bound and utility rule that attains it generated by \ref{['thm:gsmoothLP']}. Compared with the $k$-strong price of anarchy when agents optimize the system welfare (lighter line), we demonstrate the possible and guaranteed gain in equilibrium performance attainable by designing group decision-making for collaborative multi-agent systems.

Theorems & Definitions (20)

• Definition 1
• Proposition II.1
• Definition 2
• Proposition III.1
• proof
• Proposition III.2
• Theorem III.3
• Proposition IV.1
• proof
• Theorem IV.2