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Asuman Ozdaglar

Publications and source records attributed to Asuman Ozdaglar.

At least 127 records · Page 7Linked to original sources
arXiv2011

Structure of Extreme Correlated Equilibria: a Zero-Sum Example and its Implications

We exhibit the rich structure of the set of correlated equilibria by analyzing the simplest of polynomial games: the mixed extension of matching pennies. We show that while the correlated equilibrium set is convex and compact, the structure of its extreme points can be quite complicated. In finite games the ratio of extreme correlated to extreme Nash equilibria can be greater than exponential in the size of the strategy spaces. In polynomial games there can exist extreme correlated equilibria which are not finitely supported; we construct a large family of examples using techniques from ergodic theory. We show that in general the set of correlated equilibrium distributions of a polynomial game cannot be described by conditions on finitely many moments (means, covariances, etc.), in marked contrast to the set of Nash equilibria which is always expressible in terms of finitely many moments.

cs.GT↗
arXiv2010

Generalized Stable Matching in Bipartite Networks

In this paper we study the generalized version of weighted matching in bipartite networks. Consider a weighted matching in a bipartite network in which the nodes derive value from the split of the matching edge assigned to them if they are matched. The value a node derives from the split depends both on the split as well as the partner the node is matched to. We assume that the value of a split to the node is continuous and strictly increasing in the part of the split assigned to the node. A stable weighted matching is a matching and splits on the edges in the matching such that no two adjacent nodes in the network can split the edge between them so that both of them can derive a higher value than in the matching. We extend the weighted matching problem to this general case and study the existence of a stable weighted matching. We also present an algorithm that converges to a stable weighted matching. The algorithm generalizes the Hungarian algorithm for bipartite matching. Faster algorithms can be made when there is more structure on the value functions.

math.OC↗
arXiv2010

Existence of Stable Exclusive Bilateral Exchanges in Networks

In this paper we show that when individuals in a bipartite network exclusively choose partners and exchange valued goods with their partners, then there exists a set of exchanges that are pair-wise stable. Pair-wise stability implies that no individual breaks her partnership and no two neighbors in the network can form a new partnership while breaking other partnerships if any so that at least one of them improves her payoff and the other one does at least as good. We consider a general class of continuous, strictly convex and strongly monotone preferences over bundles of goods for individuals. Thus, this work extends the general equilibrium framework from markets to networks with exclusive exchanges. We present the complete existence proof using the existence of a generalized stable matching in \cite{Generalized-Stable-Matching}. The existence proof can be extended to problems in social games as in \cite{Matching-Equilibrium} and \cite{Social-Games}.

cs.GT↗
arXiv2010

A partial proof of Nash's Theorem via exchangeable equilibria

This document consists of two parts: the second part was submitted earlier as a new proof of Nash's theorem, and the first part is a note explaining a problem found in that proof. We are indebted to Sergiu Hart and Eran Shmaya for their careful study which led to their simultaneous discovery of this error. So far the error has not been fixed, but many of the results and techniques of the paper remain valid, so we will continue to make it available online. Abstract for the original paper: We give a novel proof of the existence of Nash equilibria in all finite games without using fixed point theorems or path following arguments. Our approach relies on a new notion intermediate between Nash and correlated equilibria called exchangeable equilibria, which are correlated equilibria with certain symmetry and factorization properties. We prove these exist by a duality argument, using Hart and Schmeidler's proof of correlated equilibrium existence as a first step. In an appropriate limit exchangeable equilibria converge to the convex hull of Nash equilibria, proving that these exist as well. Exchangeable equilibria are defined in terms of symmetries of the game, so this method automatically proves the stronger statement that a symmetric game has a symmetric Nash equilibrium. The case without symmetries follows by a symmetrization argument.

cs.GT↗
arXiv2010

Flows and Decompositions of Games: Harmonic and Potential Games

In this paper we introduce a novel flow representation for finite games in strategic form. This representation allows us to develop a canonical direct sum decomposition of an arbitrary game into three components, which we refer to as the potential, harmonic and nonstrategic components. We analyze natural classes of games that are induced by this decomposition, and in particular, focus on games with no harmonic component and games with no potential component. We show that the first class corresponds to the well-known potential games. We refer to the second class of games as harmonic games, and study the structural and equilibrium properties of this new class of games. Intuitively, the potential component of a game captures interactions that can equivalently be represented as a common interest game, while the harmonic part represents the conflicts between the interests of the players. We make this intuition precise, by studying the properties of these two classes, and show that indeed they have quite distinct and remarkable characteristics. For instance, while finite potential games always have pure Nash equilibria, harmonic games generically never do. Moreover, we show that the nonstrategic component does not affect the equilibria of a game, but plays a fundamental role in their efficiency properties, thus decoupling the location of equilibria and their payoff-related properties. Exploiting the properties of the decomposition framework, we obtain explicit expressions for the projections of games onto the subspaces of potential and harmonic games. This enables an extension of the properties of potential and harmonic games to "nearby" games. We exemplify this point by showing that the set of approximate equilibria of an arbitrary game can be characterized through the equilibria of its projection onto the set of potential games.

cs.GT↗
arXiv2010

Correlated Equilibria in Continuous Games: Characterization and Computation

We present several new characterizations of correlated equilibria in games with continuous utility functions. These have the advantage of being more computationally and analytically tractable than the standard definition in terms of departure functions. We use these characterizations to construct effective algorithms for approximating a single correlated equilibrium or the entire set of correlated equilibria of a game with polynomial utility functions.

cs.GT↗
arXiv2010

Access-Network Association Policies for Media Streaming in Heterogeneous Environments

We study the design of media streaming applications in the presence of multiple heterogeneous wireless access methods with different throughputs and costs. Our objective is to analytically characterize the trade-off between the usage cost and the Quality of user Experience (QoE), which is represented by the probability of interruption in media playback and the initial waiting time. We model each access network as a server that provides packets to the user according to a Poisson process with a certain rate and cost. Blocks are coded using random linear codes to alleviate the duplicate packet reception problem. Users must take decisions on how many packets to buffer before playout, and which networks to access during playout. We design, analyze and compare several control policies with a threshold structure. We formulate the problem of finding the optimal control policy as an MDP with a probabilistic constraint. We present the HJB equation for this problem by expanding the state space, and exploit it as a verification method for optimality of the proposed control law.

math.OC↗
arXiv2010

Distributed Multi-Agent Optimization with State-Dependent Communication

We study distributed algorithms for solving global optimization problems in which the objective function is the sum of local objective functions of agents and the constraint set is given by the intersection of local constraint sets of agents. We assume that each agent knows only his own local objective function and constraint set, and exchanges information with the other agents over a randomly varying network topology to update his information state. We assume a state-dependent communication model over this topology: communication is Markovian with respect to the states of the agents and the probability with which the links are available depends on the states of the agents. In this paper, we study a projected multi-agent subgradient algorithm under state-dependent communication. The algorithm involves each agent performing a local averaging to combine his estimate with the other agents' estimates, taking a subgradient step along his local objective function, and projecting the estimates on his local constraint set. The state-dependence of the communication introduces significant challenges and couples the study of information exchange with the analysis of subgradient steps and projection errors. We first show that the multi-agent subgradient algorithm when used with a constant stepsize may result in the agent estimates to diverge with probability one. Under some assumptions on the stepsize sequence, we provide convergence rate bounds on a "disagreement metric" between the agent estimates. Our bounds are time-nonhomogeneous in the sense that they depend on the initial starting time. Despite this, we show that agent estimates reach an almost sure consensus and converge to the same optimal solution of the global optimization problem with probability one under different assumptions on the local constraint sets and the stepsize sequence.

math.OC↗
arXiv2010

Optimal Reverse Carpooling Over Wireless Networks - A Distributed Optimization Approach

We focus on a particular form of network coding, reverse carpooling, in a wireless network where the potentially coded transmitted messages are to be decoded immediately upon reception. The network is fixed and known, and the system performance is measured in terms of the number of wireless broadcasts required to meet multiple unicast demands. Motivated by the structure of the coding scheme, we formulate the problem as a linear program by introducing a flow variable for each triple of connected nodes. This allows us to have a formulation polynomial in the number of nodes. Using dual decomposition and projected subgradient method, we present a decentralized algorithm to obtain optimal routing schemes in presence of coding opportunities. We show that the primal sub-problem can be expressed as a shortest path problem on an \emph{edge-graph}, and the proposed algorithm requires each node to exchange information only with its neighbors.

cs.NI↗
arXiv2009

Spread of Misinformation in Social Networks

We provide a model to investigate the tension between information aggregation and spread of misinformation in large societies (conceptualized as networks of agents communicating with each other). Each individual holds a belief represented by a scalar. Individuals meet pairwise and exchange information, which is modeled as both individuals adopting the average of their pre-meeting beliefs. When all individuals engage in this type of information exchange, the society will be able to effectively aggregate the initial information held by all individuals. There is also the possibility of misinformation, however, because some of the individuals are "forceful," meaning that they influence the beliefs of (some) of the other individuals they meet, but do not change their own opinion. The paper characterizes how the presence of forceful agents interferes with information aggregation. Under the assumption that even forceful agents obtain some information (however infrequent) from some others (and additional weak regularity conditions), we first show that beliefs in this class of societies converge to a consensus among all individuals. This consensus value is a random variable, however, and we characterize its behavior. Our main results quantify the extent of misinformation in the society by either providing bounds or exact results (in some special cases) on how far the consensus value can be from the benchmark without forceful agents (where there is efficient information aggregation). The worst outcomes obtain when there are several forceful agents and forceful agents themselves update their beliefs only on the basis of information they obtain from individuals most likely to have received their own information previously.

cs.IT↗
arXiv2009

On Distributed Averaging Algorithms and Quantization Effects

We consider distributed iterative algorithms for the averaging problem over time-varying topologies. Our focus is on the convergence time of such algorithms when complete (unquantized) information is available, and on the degradation of performance when only quantized information is available. We study a large and natural class of averaging algorithms, which includes the vast majority of algorithms proposed to date, and provide tight polynomial bounds on their convergence time. We also describe an algorithm within this class whose convergence time is the best among currently available averaging algorithms for time-varying topologies. We then propose and analyze distributed averaging algorithms under the additional constraint that agents can only store and communicate quantized information, so that they can only converge to the average of the initial values of the agents within some error. We establish bounds on the error and tight bounds on the convergence time, as a function of the number of quantization levels.

math.OC↗
arXiv2008

Constrained Consensus

We present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time-varying connectivity. Our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem, where the global objective function is a combination of local agent objective functions. Our main focus is on constrained problems where the estimate of each agent is restricted to lie in a different constraint set. To highlight the effects of constraints, we first consider a constrained consensus problem and present a distributed ``projected consensus algorithm'' in which agents combine their local averaging operation with projection on their individual constraint sets. This algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents. We establish convergence and convergence rate results for the projected consensus algorithm. We next study a constrained optimization problem for optimizing the sum of local objective functions of the agents subject to the intersection of their local constraint sets. We present a distributed ``projected subgradient algorithm'' which involves each agent performing a local averaging operation, taking a subgradient step to minimize its own objective function, and projecting on its constraint set. We show that, with an appropriately selected stepsize rule, the agent estimates generated by this algorithm converge to the same optimal solution for the cases when the weights are constant and equal, and when the weights are time-varying but all agents have the same constraint set.

math.OC↗
arXiv2008

Resource Allocation in Multiple Access Channels

We consider the problem of rate allocation in a Gaussian multiple-access channel, with the goal of maximizing a utility function over transmission rates. In contrast to the literature which focuses on linear utility functions, we study general concave utility functions. We present a gradient projection algorithm for this problem. Since the constraint set of the problem is described by exponentially many constraints, methods that use exact projections are computationally intractable. Therefore, we develop a new method that uses approximate projections. We use the polymatroid structure of the capacity region to show that the approximate projection can be implemented by a recursive algorithm in time polynomial in the number of users. We further propose another algorithm for implementing the approximate projections using rate-splitting and show improved bounds on its convergence time.

cs.IT↗
arXiv2008

Dynamic Rate Allocation in Fading Multiple-access Channels

We consider the problem of rate allocation in a fading Gaussian multiple-access channel (MAC) with fixed transmission powers. Our goal is to maximize a general concave utility function of transmission rates over the throughput capacity region. In contrast to earlier works in this context that propose solutions where a potentially complex optimization problem must be solved in every decision instant, we propose a low-complexity approximate rate allocation policy and analyze the effect of temporal channel variations on its utility performance. To the best of our knowledge, this is the first work that studies the tracking capabilities of an approximate rate allocation scheme under fading channel conditions. We build on an earlier work to present a new rate allocation policy for a fading MAC that implements a low-complexity approximate gradient projection iteration for each channel measurement, and explicitly characterize the effect of the speed of temporal channel variations on the tracking neighborhood of our policy. We further improve our results by proposing an alternative rate allocation policy for which tighter bounds on the size of the tracking neighborhood are derived. These proposed rate allocation policies are computationally efficient in our setting since they implement a single gradient projection iteration per channel measurement and each such iteration relies on approximate projections which has polynomial-complexity in the number of users.

cs.IT↗
arXiv2008

Rate and Power Allocation in Fading Multiple Access Channels

We consider the problem of rate and power allocation in a fading multiple-access channel. Our objective is to obtain rate and power allocation policies that maximize a utility function defined over average transmission rates. In contrast with the literature, which focuses on the linear case, we present results for general concave utility functions. We consider two cases. In the first case, we assume that power control is possible and channel statistics are known. In this case, we show that the optimal policies can be obtained greedily by maximizing a linear utility function at each channel state. In the second case, we assume that power control is not possible and channel statistics are not available. In this case, we define a greedy rate allocation policy and provide upper bounds on the performance difference between the optimal and the greedy policy. Our bounds highlight the dependence of the performance difference on the channel variations and the structure of the utility function.

cs.IT↗
arXiv2008

Information Theory vs. Queueing Theory for Resource Allocation in Multiple Access Channels

We consider the problem of rate allocation in a fading Gaussian multiple-access channel with fixed transmission powers. The goal is to maximize a general concave utility function of the expected achieved rates of the users. There are different approaches to this problem in the literature. From an information theoretic point of view, rates are allocated only by using the channel state information. The queueing theory approach utilizes the global queue-length information for rate allocation to guarantee throughput optimality as well as maximizing a utility function of the rates. In this work, we make a connection between these two approaches by showing that the information theoretic capacity region of a multiple-access channel and its stability region are equivalent. Moreover, our numerical results show that a simple greedy policy which does not use the queue-length information can outperform queue-length based policies in terms of convergence rate and fairness.

cs.IT↗
arXiv2008

On Resource Allocation in Fading Multiple Access Channels - An Efficient Approximate Projection Approach

We consider the problem of rate and power allocation in a multiple-access channel. Our objective is to obtain rate and power allocation policies that maximize a general concave utility function of average transmission rates on the information theoretic capacity region of the multiple-access channel. Our policies does not require queue-length information. We consider several different scenarios. First, we address the utility maximization problem in a nonfading channel to obtain the optimal operating rates, and present an iterative gradient projection algorithm that uses approximate projection. By exploiting the polymatroid structure of the capacity region, we show that the approximate projection can be implemented in time polynomial in the number of users. Second, we consider resource allocation in a fading channel. Optimal rate and power allocation policies are presented for the case that power control is possible and channel statistics are available. For the case that transmission power is fixed and channel statistics are unknown, we propose a greedy rate allocation policy and provide bounds on the performance difference of this policy and the optimal policy in terms of channel variations and structure of the utility function. We present numerical results that demonstrate superior convergence rate performance for the greedy policy compared to queue-length based policies. In order to reduce the computational complexity of the greedy policy, we present approximate rate allocation policies which track the greedy policy within a certain neighborhood that is characterized in terms of the speed of fading.

cs.IT↗