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Recent developments in domains such as non-local games, quantum interactive proofs, and quantum generative adversarial networks have renewed interest in quantum game theory and, specifically, quantum zero-sum games. Central to classical game theory is the efficient algorithmic computation of Nash equilibria, which represent optimal strategies for both players. In 2008, Jain and Watrous proposed the first classical algorithm for computing equilibria in quantum zero-sum games using the Matrix Multiplicative Weight Updates (MMWU) method to achieve a convergence rate of $\mathcal{O}\left(d/{ϵ}^2\right)$iterations to $ϵ$-Nash equilibria in the $4^d$-dimensional spectraplex. In this work, we propose a hierarchy of quantum optimization algorithms that generalize MMWU via an extra-gradient mechanism. Notably, within this proposed hierarchy, we introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as $\mathcal{O}\left(d/ϵ\right)$iterations to $ϵ$-Nash equilibria. This quadratic speed-up relative to Jain and Watrous' original algorithm sets a new benchmark for computing $ϵ$-Nash equilibria in quantum zero-sum games. 

Competition between traditional platforms is known to improve user utility by aligning the platform's actions with user preferences. But to what extent is alignment exhibited in data-driven marketplaces? To study this question from a theoretical perspective, we introduce a duopoly market where platform actions are bandit algorithms and the two platforms compete for user participation. A salient feature of this market is that the quality of recommendations depends on both the bandit algorithm and the amount of data provided by interactions from users. This interdependency between the algorithm performance and the actions of users complicates the structure of market equilibria and their quality in terms of user utility. Our main finding is that competition in this market does not perfectly align market outcomes with user utility. Interestingly, market outcomes exhibit misalignment not only when the platforms have separate data repositories, but also when the platforms have a shared data repository. Nonetheless, the data sharing assumptions impact what mechanism drives misalignment and also affect the specific form of misalignment (e.g. the quality of the best-case and worst-case market outcomes). More broadly, our work illustrates that competition in digital marketplaces has subtle consequences for user utility that merit further investigation. 

Traditional systems for allocating finite cluster resources among competing jobs have either aimed at providing fairness, relied on users to specify their resource requirements, or have estimated these requirements via surrogate metrics (e.g. CPU utilization). These approaches do not account for a job’s real world performance (e.g. P95 latency). Existing performance-aware systems use offline profiled data and/or are designed for specific allocation objectives. In this work, we argue that resource allocation systems should directly account for real-world performance and the varied allocation objectives of users. In this pursuit, we build Cilantro. At the core of Cilantro is an online learning mechanism which forms feedback loops with the jobs to estimate the resource to performance mappings and load shifts. This relieves users from the onerous task of job profiling and collects reliable real-time feedback. This is then used to achieve a variety of user-specified scheduling objectives. Cilantro handles the uncertainty in the learned models by adapting the underlying policy to work with confidence bounds. We demonstrate this in two settings. First, in a multi-tenant 1000 CPU cluster with 20 independent jobs, three of Cilantro’s policies outperform 9 other baselines on three different performance-aware scheduling objectives, improving user utilities by up to 1.2 − 3.7x. Second, in a microservices setting, where 160 CPUs must be distributed between 19 inter-dependent microservices, Cilantro outperforms 3 other baselines, reducing the end-to-end P99 latency to x0.57 the next best baseline.