Hamiltonian Potential Functions for Differential Games

Dragone, Davide; Lambertini, Luca; Leitmann, G.; Палестини, Арсен

doi:10.3182/20090506-3-sf-4003.00002

Cited by 10 publications

(7 citation statements)

References 13 publications

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“…This is the first condition that Dragone [18] states, now confirmed by our analysis. Furthermore, we can establish that the equivalence between the dynamic game and the control problem is related to the existence of a conservative vector field such that the gradient of the potential function:…”

Section: Equivalence Between the Game And The Mocpsupporting

confidence: 82%

“…In [16], this scenario is considered and the conditions are presented from a Hamiltonian perspective. More recently, [18] shows rigorously the general procedure for the continuos time model also exploiting the Hamiltonian perspective. Our contribution follows a different approach starting with the proposal of a more general Euler equation (GEE) that is able to represent utility functions in terms of states and actions instead of the reduced form just in terms of states.…”

Section: Introductionmentioning

confidence: 99%

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A new framework for solving dynamic scheduling games

Zazo

Valcarcel

Sanchez-Fernandez

et al. 2015

2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Optimum scheduling is a key objective in many communications systems where different users have to share a common resource. Typically, centralized implementations are capable of guaranteeing certain fairness. In our approach, we follow a different path modeling the scheduling process as a dynamic infinite horizon discrete-time game. This formulation allows us to include any kind of dynamics and distributed implementations. Despite, these games are very difficult to solve, we are able to show that they are in fact dynamic potential games equivalent to a non-stationary multivariate optimum control problem. The dynamic control problem is solved via an augmented Bellman equation including time as an extra state.

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Section: Equivalence Between the Game And The Mocpsupporting

confidence: 82%

Section: Introductionmentioning

confidence: 99%

A new framework for solving dynamic scheduling games

Zazo

Valcarcel

Sanchez-Fernandez

et al. 2015

2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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show abstract

“…If, the social objective of the game coincides with the potential function then we see that the social optimum can be implemented in a noncooperative manner. Dragone et al [22] present some preliminary work towards the extension of potential games in a differential game setting and 15 study games that arise in advertising.…”

Section: Decentralizationmentioning

confidence: 99%

Prospects of Tools from Differential Games in the Study of Macroeconomics of Climate Change

Engwerda¹

2012

SSRN Journal

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In this note we sketch a dynamic framework within which the discussion on the macro economic effects of climate change take place. The problem setting is characterized by scientific uncertainties about the development of climate, potential large economic losses and human beings having their specific features. Some considerations about climate change, macroeconomics and their relationship are given. A characteristic feature of the problem setting is that there are multiple decision makers interacting in a dynamic world with large uncertainties. Problems of this type have been studied extensively by (dynamic) game theory. A rough literature review is provided and some items open for future research are indicated.

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“…However, in many cases, it is not possible to find such reduced form of the game (i.e., we cannot isolate the action) because the state-transition function is not invertible (e.g., when the state transition function is quadratic in the action variable). The more general case of DPG in nonreduced form was studied with the Pontryagin's maximum principle approach by [5] and [8] for discrete and continuous time models, respectively. However, in all these studies [3]- [8], the games have been analyzed without explicitly considering constraints for the state and action sets.…”

Section: Introductionmentioning

confidence: 99%

“…The more general case of DPG in nonreduced form was studied with the Pontryagin's maximum principle approach by [5] and [8] for discrete and continuous time models, respectively. However, in all these studies [3]- [8], the games have been analyzed without explicitly considering constraints for the state and action sets.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Potential Games With Constraints: Fundamentals and Applications in Communications

Zazo

Macua

Sanchez-Fernandez

et al. 2016

IEEE Trans. Signal Process.

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In a noncooperative dynamic game, multiple agents operating in a changing environment aim to optimize their utilities over an infinite time horizon. Time-varying environments allow to model more realistic scenarios (e.g., mobile devices equipped with batteries, wireless communications over a fading channel, etc.). However, solving a dynamic game is a difficult task that requires dealing with multiple coupled optimal control problems. We focus our analysis on a class of problems, named dynamic potential games, whose solution can be found through a single multivariate optimal control problem. Our analysis generalizes previous studies by considering that the set of environment's states and the set of players' actions are constrained, as it is required by most of the applications. We also show that the theoretical results are the natural extension of the analysis for static potential games. We apply the analysis and provide numerical methods to solve four key example problems, with different features each: i) energy demand control in a smartgrid network, ii) network flow optimization in which the relays have bounded link capacity and limited battery life, iii) uplink multiple access communication with users that have to optimize the use of their batteries, and iv) two optimal scheduling games with nonstationary channels.

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Hamiltonian Potential Functions for Differential Games

Cited by 10 publications

References 13 publications

A new framework for solving dynamic scheduling games

A new framework for solving dynamic scheduling games

Prospects of Tools from Differential Games in the Study of Macroeconomics of Climate Change

Dynamic Potential Games With Constraints: Fundamentals and Applications in Communications

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