We study some theoretical topics on the theory of Dynamic Games, having as motivation
and possible application area the modeling of Electricity Markets and the Smart Grid. The
thesis is divided into three parts.
First Part: At first, some results on the theory of Markov Jump Linear Systems (MJLS),
in which the Markov chain has a general state space are presented, extending the existing
literature for discrete (finite or countably infinite) state space. Particularly, the mean square
stability of the MJLS is characterized and the Linear Quadratic (LQ) control problems for
the finite and infinite time horizon are solved, using appropriate Riccati type equations.
We then analyze Dynamic Games in which there is a random entrance of players.
Particularly, we consider an infinite time horizon player called the major player interacting
with a random number of minor players having finite time horizons, the entrance of whom is
governed by a Markov chain. The analysis is made in a LQ framework. The Nash equilibria
are characterized using a set of coupled Riccati type equations for MJLS. An emphasis is
paid on the large number of players case, in which the Mean Field (MF) approximation is
used.
Second Part: In this part, Static and Dynamic games involving agents interacting on a
large graph are studied. We assume that the players do not know the graph of interactions
precisely nor the other players preferences. Instead, we assume that each player possesses
statistical information about the network of interactions, as well as some local information.
Some notions from the Statistical Physics domain are modified to define a Probabilistic
Approximate Nash (PAN) equilibrium concept. Furthermore, we define an informational
complexity notion. Some special cases are then analyzed, involving Static and LQ games on
Erdos-Renyi Random Graphs or Small World Networks, Static Quadratic games on Lattices
and LQ games on rings.
Third Part: In the last part of the thesis, the possibility of cheating Dynamic rules (such
as learning or adaptation), when applied to Repeated or Dynamic Game situations with
incomplete structural information, is studied. An example of such a game situation is the
repeated reaction of the energy producing firms, where each one does not know precisely
the production cost of its opponents.
At first, two criteria to assess the Dynamic rules are stated. Then, we concentrate
to a subclass of cheating strategies, called pretenders strategies and study some possible
outcomes, when a player or all the players are pretending. If only one player pretends and
there is enough uncertainty the outcome would be the same as if the pretending player
were the Stackelberg leader. Furthermore, in games with a large number of equivalent
players, the gain from pretending is small and the optimal pretended values are close to the
actual. Finally, we study applications to Electricity Market models. Cases where pretending
enhances cooperation or competition are identified.
(EN)
National Technical University of Athens
