Overcoming limitations of game-theoretic distributed control

Abstract

Abstract-Recently, game theory has been proposed as a tool for cooperative control. Specifically, the interactions of a multiagent distributed system are modeled as a non-cooperative game where agents are self-interested. In this work, we prove that this approach of non-cooperative control has limitations with respect to engineering multi-agent systems. In particular, we prove that it is not possible to design budget balanced agent utilities that also guarantee that the optimal control is a Nash equilibrium. However, it is important to realize that game-theoretic designs are not restricted to the framework of non-cooperative games. In particular, we demonstrate that these limitations can be overcome by conditioning each player's utility on additional information, i.e., a state. This utility design fits into the framework of a particular form of stochastic games termed state-based games and is applicable in many application domains.

State based potential games

“…Irrespective of the state of the mission space, incremental marginal costs ensure (i) the interaction framework is a potential game and (ii) all pure Nash equilibria are guaranteed to be at least 50% efficient provided that the objective functions in (8) are submodular (Marden & Wierman, 2009). 9 However, the optimal allocation is not necessarily a Nash equilibrium as the following example highlights.…”

“…In general, it is impossible to design budget-balanced utility functions which ensures that the optimal allocation is a pure Nash equilibrium irrespective of the mission space (Marden & Wierman, 2009). …”

“…10 We will now present an approach for dynamically adjusting the order of the vehicles to provide stronger efficiency guarantees. This approach is adapted from the approach presented in Marden and Wierman (2009) and we will explicitly highlight the differences below. We focus predominantly on the vehicle target assignment problem for explanation; however, the approach and forthcoming guarantees immediately extend to any resource allocation problems where the objective functions W r (·) are submodular.…”

“…Incremental marginal costs represent a promising approach to utility design, or protocol design, for distributed engineering systems (Marden & Wierman, 2009). A protocol defined using incremental marginal costs assigns payoff shares to the agents in accordance with their marginal contribution relative to a given ordering x = {x i } i∈N where each agent i ∈ N is assigned a unique order x i ∈ {1, .…”

“…It turns out that it is theoretically impossible to model such systems as a potential game where all resulting Nash equilibria optimize the system level objective while at the same time satisfying the given constraint (Li & Marden, 2010). Alternative limitations focus on the derivation of budget-balanced agent objective functions in networked cost sharing problems (Chen, Roughgarden, & Valiant, 2008;Marden & Wierman, 2009) and the derivation of local agent objective functions for multiagent coordination in distributed engineering systems (Li & Marden, 2011). These inherent limitations provide the analytical justification for moving beyond potential games to games of a broader structure.…”