Odile Pourtallier

Mixed equilibrium (ME) for multiclass routing games

We consider a network shared by noncooperative two types of users, group users and individual users. Each user of the first type has a significant impact on the load of the network, whereas a user of the second type does not. Both group users as well as individual users choose their routes so as to minimize their costs. We further consider the case that the users may have side constraints. We study the concept of mixed equilibrium (mixing of Nash equilibrium and Wardrop equilibrium). We establish its existence and some conditions for its uniqueness. Then, we apply the mixed equilibrium to a parallel links network and to a case of load balancing.

Paradoxes in distributed decisions on optimal load balancing for networks of homogeneous computers

In completely symmetric systems that have homogeneous nodes (hosts, computers, or processors) with identical arrival processes, an optimal static load balancing scheme does not involve the forwarding of jobs among nodes. Using an appropriate analytic model of a distributed computer system, we examine the following three decision schemes for load balancing: completely distributed, intermediately distributed, and completely centralized. We show that there is no forwarding of jobs in the completely centralized and completely distributed schemes, but that in an intermediately distributed decision scheme, mutual forwarding of jobs among nodes is possible, leading to degradation in system performance for every decision maker. This result appears paradoxical, because by adding communication capacity to the system for the sharing of jobs between nodes, the overall system performance is degraded. We characterize conditions under which such paradoxical behavior occurs, and we give examples in which the degradation of performance may increase without bound. We show that the degradation reduces and finally disappears in the limit as the intermediately distributed decision scheme tends to a completely distributed one.

Approximations in Dynamic Zero-Sum Games II

We pursue in this paper our study of approximations of values and

On the Convergence to Nash Equilibrium in Problems of Distributed Computing

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Avoiding paradoxes in multi-agent competitive routing

-saddle-point policies in dynamic zero-sum games. After extending the general theorem for approximation, we study zero-sum stochastic games with countable state space and unbounded immediate reward. We focus on the expected average payoff criterion. We use some tools developed in [M. M. Tidball and E. Altman, SIAM J. Control Optim., 34 (1996), pp. 311--328] to obtain the convergence of the values as well as the convergence of the

Continuity of Optimal Values and Solutions for Control of Markov Chains with Constraints

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Approximating Nash Equilibria In Nonzero-Sum Games

saddle-point policies in various approximation problems. We consider several schemes of truncation of the state space (e.g., finite state approximation) and approximations of games with discount factor close to one for the game with expected average cost. We use the extension of the general theorem for approximation to study approximations in stochastic games with complete information. Finally, we consider the problem of approximating the sets of policies. We obtain some general results that we apply to a pursuit evasion differential game.

Avoiding paradoxes in routing games

This paper studies two problems that arise in distributed computing. We deal with these problems from a game theoretical approach. We are interested in the convergence to the Nash equilibrium of algorithms based on the best reply strategy in a special case of linear costs. We present three specific types of algorithm that converge to the equilibrium. In our first model, composed of two processors, the convergence is established through monotonicity of the sequence of updates generated by each of the three algorithms. In the second model, made up of N processors, the convergence is due to the contraction of the algorithms.

Optimal Selection of Observation Times in a Costly Information Game

Strange behavior may occur in networks due to the non-cooperative nature of decision making, when the latter are taken by individual agents. In particular, the well known Braess paradox illustrates that when upgrading a network by adding a link, the resulting equilibrium may exhibit larger delays for all users. We present here some guidelines to avoid the Braess paradox when upgrading a network. We furthermore present conditions for the delays to be monotone increasing in the total demand.

A Mixed Optimum in Symmetric Distributed Computer Systems

We consider in this paper constrained Markov decision processes. This type of control model has many applications in telecommunications and other fields [E. Altman and A. Shwartz, IEEE Trans. Automat. Control, 34 (1989), pp. 1089--1102, E. A. Feinberg and M. I. Reiman, Probab. Engrg. Inform. Sci., 8 (1994), pp. 463--489, A. Hordijk and F. Spieksma, Adv. in Appl. Probab., 21 (1989), pp. 409--431, A. Lazar, IEEE Trans. Automat. Control, 28 (1983), pp. 1001--1007, P. Nain and K. W. Ross, IEEE Trans. Automat. Control, 31 (1986), pp. 883--888, K. W. Ross and B. Chen, IEEE Trans. Automat. Control, 33 (1988), pp. 261--267]. We address the issue of the convergence of the value and optimal policies of the problem with discounted costs, to the ones for the problem with expected average cost. We consider the general multichain ergodic structure. We present two stability results in this paper. We establish the continuity of optimal values and solutions of as well as some type of robustness of some suboptimal solutions in the discount factor. Our proof relies on same general theory on continuity of values and solutions in convex optimization that relies on well-known notions of