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Register a new userEfficient algorithm for approximating Nash equilibrium of distributed aggregative games
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In this paper, we aim to design a distributed approximate algorithm for seeking Nash equilibria of an aggregative game. Due to the local set constraints of each player, projectionbased algorithms have been widely employed for solving such problems actually. Since it may be quite hard to get the exact projection in practice, we utilize inscribed polyhedrons to approximate local set constraints, which yields a related approximate game model. We first prove that the Nash equilibrium of the approximate game is the $epsilon$-Nash equilibrium of the original game, and then propose a distributed algorithm to seek the $epsilon$-Nash equilibrium, where the projection is then of a standard form in quadratic programming. With the help of the existing developed methods for solving quadratic programming, we show the convergence of the proposed algorithm, and also discuss the computational cost issue related to the approximation. Furthermore, based on the exponential convergence of the algorithm, we estimate the approximation accuracy related to $epsilon$. Additionally, we investigate the computational cost saved by approximation on numerical examples.
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This paper shows the existence of $mathcal{O}(frac{1}{n^gamma})$-Nash equilibria in $n$-player noncooperative aggregative games where the players cost functions depend only on their own action and the average of all the players actions, and is lower semicontinuous in the former while $gamma$-H{o}lder continuous in the latter. Neither the action sets nor the cost functions need to be convex. For an important class of aggregative games which includes congestion games with $gamma$ being 1, a proximal best-reply algorithm is used to construct an $mathcal{O}(frac{1}{n})$-Nash equilibria with at most $mathcal{O}(n^3)$ iterations. These results are applied in a numerical example of demand-side management of the electricity system. The asymptotic performance of the algorithm is illustrated when $n$ tends to infinity.
With the proliferation of distributed generators and energy storage systems, traditional passive consumers in power systems have been gradually evolving into the so-called prosumers, i.e., proactive consumers, which can both produce and consume power. To encourage energy exchange among prosumers, energy sharing is increasingly adopted, which is usually formulated as a generalized Nash game (GNG). In this paper, a distributed approach is proposed to seek the Generalized Nash equilibrium (GNE) of the energy sharing game. To this end, we convert the GNG into an equivalent optimization problem. A Krasnoselski{v{i}}-Mann iteration type algorithm is thereby devised to solve the problem and consequently find the GNE in a distributed manner. The convergence of the proposed algorithm is proved rigorously based on the nonexpansive operator theory. The performance of the algorithm is validated by experiments with three prosumers, and the scalability is tested by simulations using 123 prosumers.
We propose a fully asynchronous networked aggregative game (Asy-NAG) where each player minimizes a cost function that depends on its local action and the aggregate of all players actions. In sharp contrast to the existing NAGs, each player in our Asy-NAG can compute an estimate of the aggregate action at any wall-clock time by only using (possibly stale) information from nearby players of a directed network. Such an asynchronous update does not require any coordination among players. Moreover, we design a novel distributed algorithm with an aggressive mechanism for each player to adaptively adjust the optimization stepsize per update. Particularly, the slow players in terms of updating their estimates smartly increase their stepsizes to catch up with the fast ones. Then, we develop an augmented system approach to address the asynchronicity and the information delays between players, and rigorously show the convergence to a Nash equilibrium of the Asy-NAG via a perturbed coordinate algorithm which is also of independent interest. Finally, we evaluate the performance of the distributed algorithm through numerical simulations.
We introduce a novel class of Nash equilibrium seeking dynamics for non-cooperative games with a finite number of players, where the convergence to the Nash equilibrium is bounded by a KL function with a settling time that can be upper bounded by a positive constant that is independent of the initial conditions of the players, and which can be prescribed a priori by the system designer. The dynamics are model-free, in the sense that the mathematical forms of the cost functions of the players are unknown. Instead, in order to update its own action, each player needs to have access only to real-time evaluations of its own cost, as well as to auxiliary states of neighboring players characterized by a communication graph. Stability and convergence properties are established for both potential games and strongly monotone games. Numerical examples are presented to illustrate our theoretical results.
This paper considers a networked aggregative game (NAG) where the players are distributed over a communication network. By only communicating with a subset of players, the goal of each player in the NAG is to minimize an individual cost function that depends on its own action and the aggregate of all the players actions. To this end, we design a novel distributed algorithm that jointly exploits the ideas of the consensus algorithm and the conditional projection descent. Under strongly monotone assumption on the pseudo-gradient mapping, the proposed algorithm with fixed step-sizes is proved to converge linearly to the unique Nash equilibrium of the NAG. Then the theoretical results are validated by numerical experiments.
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