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We consider a duopolistic power market in which firms bid their offer functions while abiding by capacity constraints. Minimal cost allocation decisions and marginal cost prices are determined through the solution of two optimization problems, the first a mixed-integer linear program (the unit commitment problem) and the second a related linear program (the economic dispatch problem) that has constraints derived from the solution of the commitmemt problem. Traditional marginal-cost pricing mechanisms have relied on using the dual variables associated with the supply-demand constraints from the dispatch problem. Unfortunately, such mechanisms fail to cover the cost of start-up and energy costs faced by dispatched generators. In existing markets, one attempt at a remedy is by providing every dispatched generator with an expost out of market settlement called a make-whole uplift. Given the presence of such an uplift mechanism, this paper seeks to characterize the nature of equilibria in duopolistic settings. The two main conclusions of our analysis are: 1) Pure Nash equilibria are identified when one firm can supply the entire demand; 2) Otherwise, it is shown that mixed-strategy Nash equilibria exist, and these equilibria can be characterized by two coupled differential equations. These results provide a framework for further analysis of electricity markets, and in particular the true value of make-whole uplift payments in a strategic setting. [ABSTRACT FROM PUBLISHER]

In recent work it is shown that importance sampling can be avoided in the particle filter through an innovation structure inspired by traditional nonlinear filtering combined with Mean-Field Game formalisms [9], [19]. The resulting feedback particle filter (FPF) offers significant variance improvements; in particular, the algorithm can be applied to systems that are not stable. The filter comes with an up-front computational cost to obtain the filter gain. This paper describes new representations and algorithms to compute the gain in the general multivariable setting. The main contributions are, (i) Theory surrounding the FPF is improved: Consistency is established in the multivariate setting, as well as well-posedness of the associated PDE to obtain the filter gain. (ii) The gain can be expressed as the gradient of a function, which is precisely the solution to Poisson's equation for a related MCMC diffusion (the Smoluchowski equation). This provides a bridge to MCMC as well as to approximate optimal filtering approaches such as TD-learning, which can in turn be used to approximate the gain. (iii) Motivated by a weak formulation of Poisson's equation, a Galerkin finite-element algorithm is proposed for approximation of the gain. Its performance is illustrated in numerical experiments. [ABSTRACT FROM PUBLISHER]

We consider a constrained Nash-Cournot oligopoly where the demand function is linear. While cost functions and capacities are public information, firms only have partial information regarding the demand function. Specifically, firms either know the intercept or the slope of the demand function and cannot observe aggregate output. We consider a learning process in which firms update their profit-maximizing quantities and their beliefs regarding the unknown demand function parameters, based on disparities between observed and estimated prices. A characterization of the mappings, corresponding to the fixed point of the learning process, is provided. This result paves the way for developing a Tikhonov regularization scheme that is shown to learn the correct equilibrium, in spite of the multiplicity of equilibria. Despite the absence of monotonicity of the gradient maps, we prove the convergence of constant and diminishing steplength distributed gradient schemes under a suitable caveat on the starting points. Notably, precise rate of convergence estimates are provided for the constant steplength schemes. [ABSTRACT FROM PUBLISHER]

This paper studies the phase transition in a heterogeneous mean-field oscillator game model using methods from bifurcation theory. In our earlier paper [1], we had obtained a coupled PDE model using mean-field approximation and described linear analysis of the PDEs that suggested possibility of a Hamiltonian Hopf bifurcation. In this paper, we simplify the analysis somewhat by relating the solutions of the PDE model to the solutions of a certain nonlinear eigenvalue problem. Both analysis and computations are much easier for the nonlinear eigenvalue problem. Apart from the bifurcation analysis that shows existence of a phase transition, we also describe a Lyapunov-Schmidt perturbation method to obtain asymptotic formulae for the small amplitude bifurcated solutions. For comparison, we also depict numerical solutions that are obtained using the continuation software AUTO. [ABSTRACT FROM PUBLISHER]

A new formulation of the particle filter for nonlinear filtering is presented, based on concepts from optimal control, and from the mean-field game theory. The optimal control is chosen so that the posterior distribution of a particle matches as closely as possible the posterior distribution of the true state given the observations: This is achieved by introducing a cost function, defined by the Kullback-Leibler (K-L) divergence between the actual posterior, and the posterior of any particle. [ABSTRACT FROM PUBLISHER]

We introduce exploration in the TD-learning algorithm to approximate the value function for a given policy. In this way we can modify the norm used for approximation, “zooming in” to a region of interest in the state space. We also provide extensions to SARSA to eliminate the need for numerical integration in policy improvement. Construction of the algorithm and its analysis build on recent general results concerning the spectral theory of Markov chains and positive operators. [ABSTRACT FROM PUBLISHER]