Separable and low-rank continuous games

Stein, N.; Ozdaglar, Asuman; Parrilo, Pablo A.

doi:10.1007/s00182-008-0129-2

Cited by 36 publications

(47 citation statements)

References 21 publications

(33 reference statements)

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“…This feature was central in our model for the motor-muscle system as it was shown to be the source of a bias in favor of the less active MN. An appropriate modeling of the cerebellar game may be as a continuous game (i.e., the strategy set is continuous), in which each player (i.e., each CF cell) “chooses” where to send its axon initially (Stein et al, 2008). …”

Section: Discussionmentioning

confidence: 99%

Losing the battle but winning the war: game theoretic analysis of the competition between motoneurons innervating a skeletal muscle

Nowik¹,

Zamir²,

Segev³

2012

Front. Comput. Neurosci.

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The fibers in a skeletal muscle are divided into groups called “muscle units” whereby each muscle unit is innervated by a single neuron. It was found that neurons with low activation thresholds have smaller muscle units than neurons with higher activation thresholds. This results in a fixed recruitment order of muscle units, from smallest to largest, called the “size principle.” It is thought that the size principle results from a competitive process—taking place after birth—between the neurons innervating the muscle. The underlying mechanism of the competition was not understood. Moreover, the results in the majority of experiments that manipulated the activity during the competition period seemed to contradict the size principle. Experiments at the isolated muscle fibers showed that the competition is governed by a Hebbian-like rule, whereby neurons with low activation thresholds have a competitive advantage at any single muscle fiber. Thus neurons with low activation thresholds are expected to have larger muscle units in contradiction to what is seen empirically. This state of affairs was termed “paradoxical.” In the present study we developed a new game theoretic framework to analyze such competitive biological processes. In this game, neurons are the players competing to innervate a maximal number of muscle fibers. We showed that in order to innervate more muscle fibers, it is advantageous to win (as the neurons with higher activation thresholds do) later competitions. This both explains the size principle and resolves the seemingly paradoxical experimental data. Our model establishes that the competition at each muscle fiber may indeed be Hebbian and that the size principle still emerges from these competitions as an overall property of the system. Thus, the less active neurons “lose the battle but win the war.” Our model provides experimentally testable predictions. The new game-theoretic approach may be applied to competitions in other biological systems.

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Section: Discussionmentioning

confidence: 99%

Losing the battle but winning the war: game theoretic analysis of the competition between motoneurons innervating a skeletal muscle

Nowik¹,

Zamir²,

Segev³

2012

Front. Comput. Neurosci.

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“…Recall the fact that both the NG-game cost J, (3), and the constraint g, (4), are separable in the second argument and the dual cost function D(µ), (7), can be decomposed as in (8). Thus we have the following decomposition result.…”

Section: B Osnr Model and Nash Game Formulationsmentioning

confidence: 99%

“…) Since J i and g l are continuously differentiable and convex functions, by Corollary 1, the dual cost function D(µ) can be decomposed as in (8). Using (26) we see that ∀i ∈ M r(i) , L i is given here as…”

Section: Proofmentioning

confidence: 99%

“…where g k = [g k,l ], (31), are separable in the second argument and the dual cost function D(µ k ) can be decomposed as in 46th IEEE CDC, New Orleans, USA, Dec. [12][13][14]2007 ThB05.6 (8) or (24), which we restate here:…”

Section: Hierarchical Decomposition Over Stagesmentioning

confidence: 99%

“…As an alternative to the traditional system-wide network optimization approach, the game-theoretic approach has been used for network flow optimization, [3], [4], as well as for network power allocation (power control), [1], [2], and channel optical signal-to-noise ratio (OSNR) optimization in optical networks, [5], [6]. Insightful theoretical results have been obtained for computation of equilibria in classes of games with structures, such as two-player polynomial games, [7], separable games, [8], or potential games, [9], [10], or large population stochastic dynamic games, [11]. Recently it was shown that Nash equilibria in games with coupled utilities and constraints can be computed based on an extension of duality to a game theoretical framework, [12].…”

Section: Introductionmentioning

confidence: 99%

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Games with coupled propagated constraints in optical networks: The multi-link case

Pan

Pavel

2007

2007 46th IEEE Conference on Decision and Control

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This work extends previous results on games with coupled constraints in optical links to general multi-link topologies. Nash equilibria of such games can be computed based on recent extension of duality to a game theoretical framework. Unlike capacity constraints in flow control, coupled constraints in optical networks are propagated along links. This introduces additional complications for analysis. Specifically, convexity of the propagated constraints is no longer automatically ensured. We show that convexity is satisfied for the special case of multi-links with a single sink. The general case of multi-links with arbitrary sources and sinks is dealt with by a partitioned (modified) game with stages. We exploit the single sink structure of each stage and the ladder-nested form of the game.

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Strategic Form Games and Nash Equilibrium

Ozdaglar¹

2013

Encyclopedia of Systems and Control

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This chapter introduces strategic form games, which provide a framework for the analysis of strategic interactions in multi-agent environments. We present the main solution concept in strategic form games, Nash equilibrium, and provide tools for its systematic study. We present fundamental results for existence and uniqueness of Nash equilibria and discuss their efficiency properties. We conclude with current research directions in this area. IntroductionMany problems in communication, decision, and technological networks as well as in social and economic situations depend on human choices, which are made in anticipation of the behavior of the others in the system. Examples include how to map your drive over a road network, how to use the communication medium, and how to choose strategies for resource use and more conventional economic, financial, and social decisions such as which products to buy, which technologies to invest in, or who to trust. The defining feature of all of these interactions is the dependence of an agent's objective (payoff, utility, or survival) on others' actions. Game theory focuses on formal analysis of such strategic interactions. Here, we will review strategic form games, which focus on static game-theoretic interactions and present the relevant solution concept. Strategic Form GamesA strategic form game is a model for a static game in which all players act simultaneously without knowledge of other players' actions. Definition 1 (Strategic Form Game)A strategic form game is a triplet hI; .S i / i2I ; .u i / i2I i where:1. I is a finite set of players, I D f1; : : : ; I g. 2. S i is a nonempty set of available actions for player i . 3. u i W S ! R is the utility (payoff) function of player i where S D Q i2I S i .

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Separable and low-rank continuous games

Cited by 36 publications

References 21 publications

Losing the battle but winning the war: game theoretic analysis of the competition between motoneurons innervating a skeletal muscle

Losing the battle but winning the war: game theoretic analysis of the competition between motoneurons innervating a skeletal muscle

Games with coupled propagated constraints in optical networks: The multi-link case

Strategic Form Games and Nash Equilibrium

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