Stable Utility Design for Distributed Resource Allocation* Ragavendran Gopalakrishnan 1 and Sean D. Nixon 2 and Jason R. Marden 3 Abstract— The framework of resource allocation games is be- coming an increasingly popular modeling choice for distributed control and optimization. In recent years, this approach has evolved into the paradigm of game-theoretic control, which consists of first modeling the interaction between the distributed agents as a strategic form game, and then designing local utility functions for these agents such that the resulting game possesses a stable outcome (e.g., a pure Nash equilibrium) that is efficient (e.g., good “price of anarchy” properties). One then appeals to the large, existing literature on learning in games for distributed algorithms for agents that guarantee convergence to such an equilibrium. An important first problem is to obtain a characterization of stable utility designs, that is, those that guarantee equilibrium existence for a large class of games. Recent work has explored this question in the general, multi- selection context, that is, when agents are allowed to choose more than one resource at a time, showing that the only stable utility designs are the so-called “weighted Shapley values”. It remains an open problem to obtain a similar characterization in the single-selection context, which several practical problems such as vehicle target assignment, sensor coverage, etc. fall into. We survey recent work in the multi-selection scenario, and show that even though other utility designs become stable for specific single-selection applications, perhaps surprisingly, in a broader context, the limitation to “weighted Shapley value” utility design continues to prevail. I. INTRODUCTION Resource allocation is a fundamental problem that is at the core of several application domains ranging from socioeco- nomic systems to systems engineering. A persistent example in the computer science literature is that of routing data through a shared-link network, where the global objective is to minimize the average delay [1], [2]. Another example in multiagent systems is the problem of deploying sensors in a given mission space where the global objective is to maximize the area covered and/or the quality of coverage [3]. The central objective in all these problems is to allocate resources to optimize some global objective. Increasingly, these problems need to be solved in a distributed, decentral- ized manner, especially in large scale engineering systems. Game-theoretic control has emerged as a promising ap- proach for distributed resource allocation (see [4] and refer- ences therein). This approach is motivated by the fact that the underlying decision making architecture in economic *This research was supported by AFOSR grant #FA9550-12-1-0359, ONR grant #N00014-09-1-0751, and NSF grant #ECCS-1351866 1 Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, CO 80309, USA. raga@colorado.edu 2 Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05405, USA. Sean.Nixon@uvm.edu 1 Department of Electrical, Computer, and Energy Engi- neering, University of Colorado, Boulder, CO 80309, USA. jason.marden@colorado.edu systems and distributed engineering systems is identical. That is, local decisions based on local information result in emergent global behavior. Game theoretic control consists of two distinct steps. First, the interactions of autonomous agents are modeled within the framework of a strategic form game where the agents are modeled as independent decision making entities. This involves specifying decision makers (“players”), their respective choices (“action sets”), and a local utility function for each agent, so that the resulting game has an equilibrium (e.g., pure Nash equilibria), which takes on the role of a stable operating point. The second step is to specify a local “learning rule” for the agents according to which they process available information to make individual decisions that collectively steer the system towards an equilibrium of the game. The goal is to complete these two design steps, referred to as utility design and learning design respectively, in order to ensure that the emergent global behavior is desirable [4]. Our focus in this paper is on utility design for separable resource allocation problems, where “separable” means that the global objective function can be decomposed into local objective functions for each resource. A key constraint is that the design must be “local”, meaning that an agent’s utility can only depend on the resources selected, the objective at each resource, and other agents that selected the same re- sources. Therefore, designing local utility functions reduces to the problem of defining a “distribution rule” that specifies how the welfare garnered from each resource is distributed to the players who have chosen that particular resource. A fundamental research problem in utility design is to characterize the space of distribution rules that guarantee equilibrium existence in resource allocation games. Such a characterization would provide a structured search space while optimizing for distribution rule(s) whose equilibria have the best efficiency properties (how well the equilibrium outcome performs in relation to the globally optimal out- come) and/or are “easy” to converge to. First, [5], [6] showed the existence of a special, worst-case, welfare function for which, any budget-balanced distribution rule (one that com- pletely distributes the welfare garnered among the agents without surplus or deficit) that guarantees the existence of a pure Nash equilibrium in any resource allocation game must be equivalent to a “weighted Shapley value”. 1 Then, [7] generalized this, showing that the weighted Shapley value characterization holds for any welfare function, and this holds true even if the budget-balance constraint is dropped. 1 The (weighted) Shapley value, defined later in (4), is a game-theoretic solution concept of enormous importance in the economics literature.