Failure of Equilibrium Selection Methods for Multiple-Principal, Multiple-Agent Problems with Non-Rivalrous Goods: An Analysis of Data Markets

The advent of machine learning tools has led to the rise of data markets. These data markets are characterized by multiple data purchasers interacting with a set of data sources. Data sources have more information about the quality of data than the data purchasers; additionally, data itself is a non-rivalrous good that can be shared with multiple parties at negligible marginal cost. In this paper, we study the multiple-principal, multiple-agent problem with non-rivalrous goods. Under the assumption that the principal's payoff is quasilinear in the payments given to agents, we show that there is a fundamental degeneracy in the market of non-rivalrous goods. Specifically, for a general class of payment contracts, there will be an infinite set of generalized Nash equilibria. This multiplicity of equilibria also affects common refinements of equilibrium definitions intended to uniquely select an equilibrium: both variational equilibria and normalized equilibria will be non-unique in general. This implies that most existing equilibrium concepts cannot provide predictions on the outcomes of data markets emerging today. The results support the idea that modifications to payment contracts themselves are unlikely to yield a unique equilibrium, and either changes to the models of study or new equilibrium concepts will be required to determine unique equilibria in settings with multiple principals and a non-rivalrous good

Beyond Monotone Variational Inequalities: Solution Methods and Iteration Complexities

In this paper, we discuss variational inequality (VI) problems without monotonicity from the perspective of convergence of projection-type algorithms. In particular, we identify existing conditions as well as present new conditions that are sufficient to guarantee convergence. The first half of the paper focuses on the case where a Minty solution exists (also known as Minty condition), which is a common assumption in the recent developments for non-monotone VI. The second half explores alternative sufficient conditions that are different from the existing ones such as monotonicity or Minty condition, using an algorithm-based approach. Through examples and convergence analysis, we show that these conditions are capable of characterizing different classes of VI problems where the algorithms are guaranteed to converge.Comment: 29 page

Non-convex power allocation games in MIMO cognitive radio networks

Consideramos un escenario de reparto del espectro, basado en la detección, en una red de radio cognitiva MIMO donde el objetivo general es maximizar el rendimiento total de cada usuario de radio cognitiva optimizando conjuntamente la operación de detección y la asignación de potencia en todos los canales, bajo una restricción de interferencia para los usuarios primarios. Los problemas de optimización resultantes conducen a un juego no convexo, que presenta un nuevo desafío a la hora de analizar los equilibrios de este juego. Con el fin de hacer frente a la no convexidad del juego, utilizamos un nuevo concepto relajado de equilibrio, el equilibrio cuasi-Nash (QNE). Se demuestran las condiciones suficientes para la existencia y la unicidad de un QNE. El trabajo también presenta un método de optimización de punto interior primal-dual que converge a un QNE. Los resultados de la simulación muestran que el juego propuesto puede lograr una considerable mejora del rendimiento con respecto a un juego determinista.TEC2010- 19545-C04-04 “COSIMA”,CONSOLIDER-INGENIO 2010 CSD2008-00010 “COMONSENS”“HYDROBIONETS” FP7 Grant no. 287613 FP7We consider a sensing-based spectrum sharing scenario in a MIMO cognitive radio network where the overall objective is to maximize the total throughput of each cognitive radio user by jointly optimizing both the detection operation and the power allocation over all the channels, under a interference constraint bound to primary users. The resulting optimization problems lead to a non-convex game, which presents a new challenge when analyzing the equilibria of this game. In order to deal with the non-convexity of the game, we use a new relaxed equilibria concept, namely, quasi-Nash equilibrium (QNE). We show the sufficient conditions for the existence and the uniqueness of a QNE. A primal-dual interior point optimization method that converges to a QNE is also discussed in this paper. Simulation results show that the proposed game can achieve a considerable performance improvement with respect to a deterministic game

Quasi-Nash Equilibria for Non-Convex Distributed Power Allocation Games in Cognitive Radios

In this paper, we consider a sensing-based spectrum sharing scenario in cognitive radio networks where the overall objective is to maximize the sum-rate of each cognitive radio user by optimizing jointly both the detection operation based on sensing and the power allocation, taking into account the influence of the sensing accuracy and the interference limitation to the primary users. The resulting optimization problem for each cognitive user is non-convex, thus leading to a non-convex game, which presents a new challenge when analyzing the equilibria of this game where each cognitive user represents a player. In order to deal with the non-convexity of the game, we use a new relaxed equilibria concept, namely, quasi-Nash equilibrium (QNE). A QNE is a solution of a variational inequality obtained under the first-order optimality conditions of the player's problems, while retaining the convex constraints in the variational inequality problem. In this work, we state the sufficient conditions for the existence of the QNE for the proposed game. Specifically, under the so-called linear independent constraint qualification, we prove that the achieved QNE coincides with the NE. Moreover, a distributed primal-dual interior point optimization algorithm that converges to a QNE of the proposed game is provided in the paper, which is shown from the simulations to yield a considerable performance improvement with respect to an alternating direction optimization algorithm and a deterministic game