The Cost of Denied Observation in Multiagent Submodular Optimization

Abstract

A popular formalism for multiagent control applies tools from game theory, casting a multiagent decision problem as a cooperation-style game in which individual agents make local choices to optimize their own local utility functions in response to the observable choices made by other agents.When the system-level objective is submodular maximization, it is known that if every agent can observe the action choice of all other agents, then all Nash equilibria of a large class of resulting games are within a factor of 2 of optimal; that is, the price of anarchy is 1/2. However, little is known if agents cannot observe the action choices of other relevant agents. To study this, we extend the standard game-theoretic model to one in which a subset of agents either become blind (unable to observe others' choices) or isolated (blind, and also invisible to other agents), and we prove exact expressions for the price of anarchy as a function of the number of compromised agents. When k agents are compromised (in any combination of blind or isolated), we show that the price of anarchy for a large class of utility functions is exactly 1/(2 + k). We then show that if agents use marginal-cost utility functions and at least 1 of the compromised agents is blind (rather than isolated), the price of anarchy improves to 1/(1 + k). We also provide simulation results demonstrating the effects of these observation denials in a dynamic setting.

The Price of Anarchy is Fragile in Single-Selection Coverage Games

“…Previous results [19] show that a game G = (N, R, A, K, v) with |K| compromised agents have PoA(G) ≥ 1 1+|K| , in this paper we improve upon this bound that is parameterized by the distance that the game in question is from a game with an optimal Nash equilibrium. This effectively demonstrates that worst-case instances for this type of game are fragile in a strong sense; that is, arbitrarily-small perturbations to their specifications can render their equilibria optimal.…”

“…It is known that for any game G as defined in this paper, it holds that if K = ∅, PoA(G) ≥ 1/(1 + |K|) [19], and that if K = ∅, PoA(G) ≥ 1/2 [18]. A companion "optimistic" metric is the price of stability, defined as the best-case ratio between the objective value of a Nash equilibrium and an optimal action profile, or…”

“…Building on this, the recent work in [19] identifies a class of tight worst-case guarantees for communication-denied multiagent submodular maximization problems. There, the authors consider a scenario in which some agents cannot observe the actions or detect the contributions of other agents, perhaps due to failed communication or sensing equipment.…”

“…Although it is shown that this bound is tight, the family of worst-case problem instances is finelytuned; arbitrarily small perturbations to resource values can easily destabilize the low-quality Nash equilibria. In essence, the specific worst-case examples presented in [19] are fragile.…”

“…In this paper, we show that this fragility is generic, and is in fact a consequence of being worst-case problem instances. For the class of single-selection coverage games (which are used to generate the worst-case examples in [19]), we investigate the robustness of the worst-case guarantees offered by [19] both analytically and empirically. First, we show the following general principle analytically: if an instance of this game has a very poor price of anarchy, then the game is very close (in a sense which we make precise in Section II) to some game with an optimal Nash equilibrium.…”