Search for: All records

We study the problem of online resource allocation, where customers arrive sequentially, and the seller must irrevocably allocate resources to each incoming customer while also facing a prespecified procurement cost function over the total allocation. The objective is to maximize the reward obtained from fulfilling the customers’ requests sans the cumulative procurement cost. We analyze the competitive ratio of a primal-dual algorithm in this setting and develop an optimization framework for designing a surrogate function for the procurement cost to be used by the algorithm to improve the competitive ratio of the primal-dual algorithm. We use the optimal surrogate function for polynomial procurement cost functions to improve on previous bounds. For general procurement cost functions, our design method uses quasiconvex optimization to find optimal design parameters. We then implement the design techniques and show the improved performance of the algorithm in numerical examples. Finally, we extend the analysis by devising a posted pricing mechanism in which the algorithm does not require the customers’ preferences to be revealed. Funding: M. Fazel’s work was supported in part by the National Science Foundation [Awards 2023166, 2007036, and 1740551]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoo.2021.0012 . 

Numerous online services are data-driven: the behavior of users affects the system’s parameters, and the system’s parameters affect the users’ experience of the service, which in turn affects the way users may interact with the system. For example, people may choose to use a service only for tasks that already works well, or they may choose to switch to a different service. These adaptations influence the ability of a system to learn about a population of users and tasks in order to improve its performance broadly. In this work, we analyze a class of such dynamics—where users allocate their participation amongst services to reduce the individual risk they experience, and services update their model parameters to reduce the service’s risk on their current user population. We refer to these dynamics as risk-reducing, which cover a broad class of common model updates including gradient descent and multiplicative weights. For this general class of dynamics, we show that asymptotically stable equilibria are always segmented, with sub-populations allocated to a single learner. Under mild assumptions, the utilitarian social optimum is a stable equilibrium. In contrast to previous work, which shows that repeated risk minimization can result in representation disparity and high overall loss with a single learner (Hashimoto et al., 2018; Miller et al., 2021), we find that repeated myopic updates with multiple learners lead to better outcomes. We illustrate the phenomena via a simulated example initialized from real data. 

Numerous online services are data-driven: the behavior of users affects the system’s parameters, and the system’s parameters affect the users’ experience of the service, which in turn affects the way users may interact with the system. For example, people may choose to use a service only for tasks that already works well, or they may choose to switch to a different service. These adaptations influence the ability of a system to learn about a population of users and tasks in order to improve its performance broadly. In this work, we analyze a class of such dynamics—where users allocate their participation amongst services to reduce the individual risk they experience, and services update their model parameters to reduce the service’s risk on their current user population. We refer to these dynamics as risk-reducing, which cover a broad class of common model updates including gradient descent and multiplicative weights. For this general class of dynamics, we show that asymptotically stable equilibria are always segmented, with sub-populations allocated to a single learner. Under mild assumptions, the utilitarian social optimum is a stable equilibrium. In contrast to previous work, which shows that repeated risk minimization can result in representation disparity and high overall loss with a single learner (Hashimoto et al., 2018; Miller et al., 2021), we find that repeated myopic updates with multiple learners lead to better outcomes. We illustrate the phenomena via a simulated example initialized from real data. 

We study the sample complexity of identifying an approximate equilibrium for two-player zero-sum n × 2 matrix games. That is, in a sequence of repeated game plays, how many rounds must the two players play before reaching an approximate equilibrium (e.g., Nash)? We derive instance-dependent bounds that define an ordering over game matrices that captures the intuition that the dynamics of some games converge faster than others. Specifically, we consider a stochastic observation model such that when the two players choose actions i and j, respectively, they both observe each other’s played actions and a stochastic observation Xij such that E [Xij ] = Aij . To our knowledge, our work is the first case of instance-dependent lower bounds on the number of rounds the players must play before reaching an approximate equilibrium in the sense that the number of rounds depends on the specific properties of the game matrix A as well as the desired accuracy. We also prove a converse statement: there exist player strategies that achieve this lower bound.