Proportional Response as Iterated Cobb-Douglas

We show that the proportional response algorithm for computing an economic equilibrium in a Fisher market model can be interpreted as iteratively approximating the economy by one with Cobb-Douglas utilities, for which a closed-form equilibrium can be obtained. We also extend the method to allow elasticities of substitution at most one

Tit-for-Tat Dynamics and Market Volatility

We study the tit-for-tat dynamic in production markets, where each player can make a good given as input various amounts of goods in the system. In the tit-for-tat dynamic, each player allocates its good to its neighbors in fractions proportional to how much they contributed in its production in the last round. Tit-for-tat does not use money and was studied before in pure exchange settings. We study the phase transitions of this dynamic when the valuations are symmetric (i.e. each good has the same worth to everyone) by characterizing which players grow or vanish over time. We also study how the fractions of their investments evolve in the long term, showing that in the limit the players invest only on players with optimal production capacity

Tracing Equilibrium in Dynamic Markets via Distributed Adaptation

Competitive equilibrium is a central concept in economics with numerous applications beyond markets, such as scheduling, fair allocation of goods, or bandwidth distribution in networks. Computation of competitive equilibria has received a significant amount of interest in algorithmic game theory, mainly for the prominent case of Fisher markets. Natural and decentralized processes like tatonnement and proportional response dynamics (PRD) converge quickly towards equilibrium in large classes of Fisher markets. Almost all of the literature assumes that the market is a static environment and that the parameters of agents and goods do not change over time. In contrast, many large real-world markets are subject to frequent and dynamic changes. In this paper, we provide the first provable performance guarantees of discrete-time tatonnement and PRD in markets that are subject to perturbation over time. We analyze the prominent class of Fisher markets with CES utilities and quantify the impact of changes in supplies of goods, budgets of agents, and utility functions of agents on the convergence of tatonnement to market equilibrium. Since the equilibrium becomes a dynamic object and will rarely be reached, our analysis provides bounds expressing the distance to equilibrium that will be maintained via tatonnement and PRD updates. Our results indicate that in many cases, tatonnement and PRD follow the equilibrium rather closely and quickly recover conditions of approximate market clearing. Our approach can be generalized to analyzing a general class of Lyapunov dynamical systems with changing system parameters, which might be of independent interest

Network slicing for guaranteed rate services: admission control and resource allocation games

Technologies that enable network slicing are expected to be a key component of next generation mobile networks. Their promise lies in enabling tenants (such as mobile operators and/or services) to reap the cost and performance benefits of sharing resources while retaining the ability to customize their own allocations. When employing dynamic sharing mechanisms, tenants may exhibit strategic behavior, optimizing their choices in response to those of other tenants. This paper analyzes dynamic sharing in network slicing when tenants support inelastic users with minimum rate requirements. We propose a NEtwork Slicing (NES) framework combining: 1) admission control; 2) resource allocation; and 3) user dropping. We model the network slicing system with admitted users as a NES game; this is a new class of game where the inelastic nature of the traffic may lead to dropping users whose requirements cannot be met. We show that, as long as admission control guarantees that slices can satisfy the rate requirements of all their users, this game possesses a Nash equilibrium. Admission control policies (a conservative and an aggressive one) are considered, along with a resource allocation scheme and a user dropping algorithm, geared at maintaining the system in Nash equilibria. We analyze our NES framework's performance in equilibrium, showing that it achieves the same or better utility than static resource partitioning, and bound the difference between NES and the socially optimal performance. Simulation results confirm the effectiveness of the proposed approach.The work of University of Texas at Austin was supported in part by a gift from Cisco. The work of University Carlos III of Madrid was supported by the H2020 5G-MoNArch project (Grant Agreement No. 761445) and the 5GCity project of the Spanish Ministry of Economy and Competitiveness (TEC2016-76795-C6-3-R). The work of NEC Europe Ltd. was supported by the H2020 5G-Transformer project (Grant agreement no. 761536)

Resource allocation for network slicing in mobile networks

This paper provides a survey of resource allocation for network slicing. We focus on two classes of existing solutions: (i) reservation-based approaches, which allocate resources on a reservation basis, and (ii) share-based approaches, which allocate resources based on static overall shares associated to individual slices. We identify the requirements that a slice-based resource allocation mechanism should satisfy, and evaluate the performance of both approaches against these requirements. Our analysis reveals that reservation-based approaches provide a better level of isolation as well as stricter guarantees, by enabling tenants to explicitly reserve resources, but one must pay a price in terms of efficiency unless reservations can be updated very dynamically; in particular, efficiency falls below 50\% when reservations are performed over long timescales. We provide further comparisons in terms of customizability, complexity, privacy and cost predictability, and discuss which approach might be more suitable depending on the network slices' characteristics. We also describe the additional mechanisms required to implement the desired resource allocations while meeting the latency and reliability requirements of the different slice types, and outline some issues for future work.The work of Albert Banchs was supported in part by the H2020 5G-TOURS European project under Grant 856950, and in part by the Spanish State Research Agency (TRUE5G project) under Grant PID2019-108713RB-C52/AEI/10.13039/501100011033. The work of Gustavo de Veciana was supported by NSF Grant CNS-1910112