Resource-Aware Cost-Sharing Mechanisms with Priors

Abstract

In a decentralized system with machines, we study the selfish scheduling problem where each user strategically chooses which machine to use. Each machine incurs a cost, which is a function of the total load assigned to it, and some cost-sharing mechanism distributes this cost among the machine's users. The users choose a machine aiming to minimize their own share of the cost, so the cost-sharing mechanism induces a game among them. We approach this problem from the perspective of a designer who can select which cost-sharing mechanism to use, aiming to minimize the price of anarchy (PoA) of the induced games.Recent work introduced the class of resource-aware cost-sharing mechanisms, whose decisions can depend on the set of machines in the system, but are oblivious to the total number of users. These mechanisms can guarantee low PoA bounds for instances where the cost functions of the machines are all convex or concave, but can suffer from very high PoA for cost functions that deviate from these families.In this paper we show that if we enhance the class of resource-aware mechanisms with some prior information regarding the users, then they can achieve low PoA for a much more general family of cost functions. We first show that, as long as the mechanism knows just two of the participating users, then it can assign special roles to them and ensure a constant PoA. We then extend this idea to settings where the mechanism has access to the probability with which each user is present in the system. For all these instances, we provide a mechanism that achieves an expected PoA that is logarithmic in the expected number of users.CCS Concepts: • Theory of computation Quality of equilibria; Algorithmic game theory.

Improved Price of Anarchy via Predictions

“…For each edge of the graph, we allow its cost function to be an arbitrary non-decreasing function of the number of agents using it. This is in contrast to most of the prior literature which imposes some type of structure or parameterization on the allowable cost functions (e.g., concavity, convexity, or some type of boundedness) (e.g., [von Falkenhausen and Harks, 2013, Christodoulou et al, 2017, Gkatzelis et al, 2021). For this class of games, the best known price of anarchy upper bound via a resource-aware protocol is O(n), and prior work has shown that without information regarding the number of users, no stable cost-sharing protocol (i.e., a protocol that admits a pure Nash equilibrium) can achieve a price of anarchy better than O( √ n) [Christodoulou et al, 2017].…”

“…This online algorithm, generalizes an algorithm that Gkatzelis et al [2021] introduced in the context of parallel-link graphs. The main difficulty in generalizing this algorithm beyond parallellink graphs is the fact that in series-parallel graphs each path may need to use multiple edges to connect the source to the sink.…”

“…Following-up on this work, extended the constant PoA to include Bayesian Nash equilibria. Recently, Gkatzelis et al [2021] showed that with some information about the users the PoA of resource-aware protocols can be significantly improved for the class of scheduling games with bounded cost functions. In this work, two different types of information are considered: knowing two of the participating agents' IDs in advance, or knowing the probability with which each of the agents appears in the system.…”

“…The main difficulty is that the optimal assignment OPT(n), i.e., the way in which the players are allocated to paths in this assignment, can change a lot even with small changes in n. As a result, a mechanism that tries to enforce the optimal assignment is bound to be very sensitive to prediction errors. To overcome this obstacle, we follow the approach of Gkatzelis et al [2021] and, rather than aiming to enforce the optimal outcome, we instead aim to enforce the outcome of an online algorithm which may be suboptimal, but is much more "well-behaved". Specifically, the outcome of an online algorithm does not change radically as a function of the number n of arriving users: an online algorithm decides how to assign each arriving user irrevocably, without knowing how many other users would arrive in the future.…”