Extremum Seeking for Nash Games in Financial and Energy Markets

Abstract

In economic networks, behaviors that seem intelligently conceived (for the most part) emerge without a coordinating entity, but as a result of multiple agents pursuing self-interested strategies. Such behaviors have traditionally been interpreted using paradigms of non-cooperative game theory. However, standard game theory usually assumes that the players know the models of their own payoff functions and the actions of the other players. Relying on the method of extremum seeking, we design algorithms that do not need any modeling information and where, instead, the players employ only the measurements of their own payoff values. The extremum seeking algorithms are proved to converge to the Nash equilibria of the underlying non-cooperative games. In other words, extremum seeking allows the player to learn its Nash strategy. Extremum seeking algorithms are not restricted to games with a limited number of players but are, in fact, applicable to games with uncountably many players. While in finite games each player employing extremum seeking must employ a distinct probing frequency, a remarkable situation arises in games with uncountably many players - we show that, as long as any frequency is employed by countably many players, convergence to the Nash equilibrium is guaranteed. Such large (for all practical purposes uncountable) games arise in future energy trading markets involving households that own plug-in hybrid electric vehicles, whose battery capacity is used for the storage of energy during periods of excess production from wind and solar sources and for selling energy back to the grid, at a price, or in quantity, determined by a controller pursuing profit maximization for the household with the help of an extremum seeking algorithm.