Evolutionary Equilibrium in Bayesian Routing Games: Specialization and Niche Formation

Berenbrink, Petra

doi:10.1007/978-3-540-75520-3_5

Cited by 3 publications

(8 citation statements)

References 22 publications

(21 reference statements)

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“…Proposition 1 is then an instance of the so-called folk-theorems of Evolutionary Game Theory [?]. For completeness, the proof goes as follows: From Corollary 1, clearly any Nash equilibria must also vanish the right-hand side of Equation (5).…”

Section: General Games and Replicator-like Dynamicsmentioning

confidence: 99%

“…Theorem 5 Consider a Lyapunov game with a multiaffine Lyapunov function F , with respect to (5). This includes ordinal, and hence potential games from above discussion.…”

Section: Replicator-like Dynamics For Multiaffine Lyapunov Gamesmentioning

confidence: 99%

“…The game whose costs are defined by u i is clearly isomorphic to the original game. Notice that when γ is affine, this is just introducing a(n other) rescaling in (5).…”

Section: General Games and Replicator-like Dynamicsmentioning

confidence: 99%

“…Now, by linearity of expectation, we have that F (q i , Q −i ) = ℓ q i,ℓ F (e ℓ , Q −i ), and hence ∂F ∂q i,ℓ (Q) = F (e ℓ , Q −i ). Now, for dynamics (5), left-hand side of (6) rewrites to…”

Section: Lyapunov Games Ordinal and Potential Gamesmentioning

confidence: 99%

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Learning Equilibria in Games by Stochastic Distributed Algorithms

Bournez

Cohen²

2012

Computer and Information Sciences III

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We consider a class of fully stochastic and fully distributed algorithms, that we prove to learn equilibria in games. Indeed, we consider a family of stochastic distributed dynamics that we prove to converge weakly (in the sense of weak convergence for probabilistic processes) towards their mean-field limit, i.e an ordinary differential equation (ODE) in the general case. We focus then on a class of stochastic dynamics where this ODE turns out to be related to multipopulation replicator dynamics. Using facts known about convergence of this ODE, we discuss the convergence of the initial stochastic dynamics: For general games, there might be non-convergence, but when convergence of the ODE holds, considered stochastic algorithms converge towards Nash equilibria. For games admitting Lyapunov functions, that we call Lyapunov games, the stochastic dynamics converge. We prove that any ordinal potential game, and hence any potential game is a Lyapunov game, with a multiaffine Lyapunov function. For Lyapunov games with a multiaffine Lyapunov function, we prove that this Lyapunov function is a super-martingale over the stochastic dynamics. This leads a way to provide bounds on their time of convergence by martingale arguments. This applies in particular for many classes of games that have been considered in literature, including several load balancing game scenarios and congestion games.

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Section: General Games and Replicator-like Dynamicsmentioning

confidence: 99%

“…Theorem 5 Consider a Lyapunov game with a multiaffine Lyapunov function F , with respect to (5). This includes ordinal, and hence potential games from above discussion.…”

Section: Replicator-like Dynamics For Multiaffine Lyapunov Gamesmentioning

confidence: 99%

“…The game whose costs are defined by u i is clearly isomorphic to the original game. Notice that when γ is affine, this is just introducing a(n other) rescaling in (5).…”

Section: General Games and Replicator-like Dynamicsmentioning

confidence: 99%

Section: Lyapunov Games Ordinal and Potential Gamesmentioning

confidence: 99%

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Learning Equilibria in Games by Stochastic Distributed Algorithms

Bournez

Cohen²

2012

Computer and Information Sciences III

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“…It is proved to terminate in expected O(log log n + m 4 ) rounds for uniform tasks, and uniform machines. This has been extended to weighted tasks and uniform machines in [3]. The expected time of convergence to an ǫ-Nash equilibrium is in O(nmW 3 ǫ −2 ) where W denotes the maximum weight of any task.…”

Section: Related Workmentioning

confidence: 99%

A Dynamic Approach for Load Balancing

Barth¹,

Bournez²,

Boussaton³

et al. 2009

Proceedings of the 4th International ICST Conference on Performance Evaluation Methodologies and Tools

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We study how to reach a Nash equilibrium in a load balancing scenario where each task is managed by a selfish agent and attempts to migrate to a machine which will minimize its cost. The cost of a machine is a function of the load on it. The load on a machine is the sum of the weights of the jobs running on it. We prove that Nash equilibria can be learned on that games with incomplete information, using some Lyapunov techniques.

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Evolutionary equilibrium in Bayesian routing games: Specialization and niche formation

Berenbrink¹

2010

Theoretical Computer Science

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In this paper we consider Nash Equilibria for the selfish routing model proposed in [12], where a set of n users with tasks of different size try to access m parallel links with different speeds. In this model, a player can use a mixed strategy (where he uses different links with a positive probability); then he is indifferent between the different link choices. This means that the player may well deviate to a different strategy over time. We propose the concept of evolutionary stable strategies (ESS) as a criterion for stable Nash Equilibria, i.e. Equilibria where no player is likely to deviate from his strategy. An ESS is a steady state that can be reached by a user community via evolutionary processes in which more successful strategies spread over time. The concept has been used widely in biology and economics to analyze the dynamics of strategic interactions. We establish that the ESS is uniquely determined for a symmetric Bayesian parallel links game (when it exists). Thus evolutionary stability places strong constraints on the assignment of tasks to links.

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Evolutionary Equilibrium in Bayesian Routing Games: Specialization and Niche Formation

Cited by 3 publications

References 22 publications

Learning Equilibria in Games by Stochastic Distributed Algorithms

Learning Equilibria in Games by Stochastic Distributed Algorithms

A Dynamic Approach for Load Balancing

Evolutionary equilibrium in Bayesian routing games: Specialization and niche formation

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