Nash Social Welfare Approximation for Strategic Agents

The fair division of resources is an important age-old problem that has led to a rich body of literature. At the center of this literature lies the question of whether there exist fair mechanisms despite strategic behavior of the agents. A fundamental objective function used for measuring fair outcomes is the Nash social welfare, defined as the geometric mean of the agent utilities. This objective function is maximized by widely known solution concepts such as Nash bargaining and the competitive equilibrium with equal incomes. In this work we focus on the question of (approximately) implementing the Nash social welfare. The starting point of our analysis is the Fisher market, a fundamental model of an economy, whose benchmark is precisely the (weighted) Nash social welfare. We begin by studying two extreme classes of valuations functions, namely perfect substitutes and perfect complements, and find that for perfect substitutes, the Fisher market mechanism has a constant approximation: at most 2 and at least e1e. However, for perfect complements, the Fisher market does not work well, its bound degrading linearly with the number of players. Strikingly, the Trading Post mechanism---an indirect market mechanism also known as the Shapley-Shubik game---has significantly better performance than the Fisher market on its own benchmark. Not only does Trading Post achieve an approximation of 2 for perfect substitutes, but this bound holds for all concave utilities and becomes arbitrarily close to optimal for Leontief utilities (perfect complements), where it reaches $(1+\epsilon)$ for every $\epsilon > 0$. Moreover, all the Nash equilibria of the Trading Post mechanism are pure for all concave utilities and satisfy an important notion of fairness known as proportionality

Strategyproof Scheduling with Predictions

In their seminal paper that initiated the field of algorithmic mechanism design, Nisan and Ronen [Noam Nisan and Amir Ronen, 1999] studied the problem of designing strategyproof mechanisms for scheduling jobs on unrelated machines aiming to minimize the makespan. They provided a strategyproof mechanism that achieves an n-approximation and they made the bold conjecture that this is the best approximation achievable by any deterministic strategyproof scheduling mechanism. After more than two decades and several efforts, n remains the best known approximation and very recent work by Christodoulou et al. [George Christodoulou et al., 2021] has been able to prove an ?(?n) approximation lower bound for all deterministic strategyproof mechanisms. This strong negative result, however, heavily depends on the fact that the performance of these mechanisms is evaluated using worst-case analysis. To overcome such overly pessimistic, and often uninformative, worst-case bounds, a surge of recent work has focused on the "learning-augmented framework", whose goal is to leverage machine-learned predictions to obtain improved approximations when these predictions are accurate (consistency), while also achieving near-optimal worst-case approximations even when the predictions are arbitrarily wrong (robustness). In this work, we study the classic strategic scheduling problem of Nisan and Ronen [Noam Nisan and Amir Ronen, 1999] using the learning-augmented framework and give a deterministic polynomial-time strategyproof mechanism that is 6-consistent and 2n-robust. We thus achieve the "best of both worlds": an O(1) consistency and an O(n) robustness that asymptotically matches the best-known approximation. We then extend this result to provide more general worst-case approximation guarantees as a function of the prediction error. Finally, we complement our positive results by showing that any 1-consistent deterministic strategyproof mechanism has unbounded robustness

Getting More by Knowing Less: Bayesian Incentive Compatible Mechanisms for Fair Division

We study fair resource allocation with strategic agents. It is well-known that, across multiple fundamental problems in this domain, truthfulness and fairness are incompatible. For example, when allocating indivisible goods, there is no truthful and deterministic mechanism that guarantees envy-freeness up to one item (EF1), even for two agents with additive valuations. Or, in cake-cutting, no truthful and deterministic mechanism always outputs a proportional allocation, even for two agents with piecewise-constant valuations. Our work stems from the observation that, in the context of fair division, truthfulness is used as a synonym for Dominant Strategy Incentive Compatibility (DSIC), requiring that an agent prefers reporting the truth, no matter what other agents report. In this paper, we instead focus on Bayesian Incentive Compatible (BIC) mechanisms, requiring that agents are better off reporting the truth in expectation over other agents' reports. We prove that, when agents know a bit less about each other, a lot more is possible: using BIC mechanisms we can overcome the aforementioned barriers that DSIC mechanisms face in both the fundamental problems of allocation of indivisible goods and cake-cutting. We prove that this is the case even for an arbitrary number of agents, as long as the agents' priors about each others' types satisfy a neutrality condition. En route to our results on BIC mechanisms, we also strengthen the state of the art in terms of negative results for DSIC mechanisms.Comment: 26 page

Resource-Aware Protocols for Network Cost-Sharing Games

We study the extent to which decentralized cost-sharing protocols can achieve good price of anarchy (PoA) bounds in network cost-sharing games with $n$ agents. We focus on the model of resource-aware protocols, where the designer has prior access to the network structure and can also increase the total cost of an edge(overcharging), and we study classes of games with concave or convex cost functions. We first consider concave cost functions and our main result is a cost-sharing protocol for symmetric games on directed acyclic graphs that achieves a PoA of $2+\varepsilon$ for some arbitrary small positive $\varepsilon$, which improves to $1+\varepsilon$ for games with at least two players. We also achieve a PoA of 1 for series-parallel graphs and show that no protocol can achieve a PoA better than $\Omega(\sqrt{n})$ for multicast games. We then also consider convex cost functions and prove analogous results for series-parallel networks and multicast games, as well as a lower bound of $\Omega(n)$ for the PoA on directed acyclic graphs without the use of overcharging