Ioannis Kordonis

On Stability and LQ Control of MJLS With a Markov Chain With General State Space

We study the mean square stability and the LQ control of discrete time Markov Jump Linear Systems where the Markov chain has a general state space. The mean square stability is characterized by the spectral radius of an operator describing the evolution of the second moment of the state vector. Two equivalent tests for the mean square stability are obtained based on the existence of a positive definite solution to a Lyapunov equation and a uniformity result respectively. An algorithm for testing the mean square stability is also developed based on the uniformity result. The finite and infinite horizon LQ problems are considered and their solutions are characterized by appropriate Riccati integral equations. An application to Networked Control Systems (NCS) is finally presented and a simple example is studied via simulation.

LQ Nash Games With Random Entrance: An Infinite Horizon Major Player and Minor Players of Finite Horizons

We study Dynamic Games with randomly entering players, staying in the game for different lengths of time. Particularly, a class of Discrete Time Linear Quadratic (LQ) Games, involving a major player who has an infinite time horizon and a random number of minor players is considered. The number of the new minor players, entering at some instant of time, is random and it is described by a Markov chain. The problem of the characterization of a Nash equilibrium, consisting of Linear Feedback Strategies, is reformulated as a set of coupled finite and infinite horizon LQ optimal control problems for Markov Jump Linear Systems (MJLS). Sufficient conditions characterizing Nash equilibrium are then derived. The problem of Games involving a large number of minor players is then addressed using a Mean Field (MF) approach and asymptotic ε-Nash equilibrium results are derived. Sufficient conditions for the existence of a MF Nash equilibrium are finally stated.

Dynamic Games among Agents with Partial Information of the Structure of the Interactions Graph: Decision Making and Complexity Issues

We consider dynamic games on large networks, motivated by structural and decision making issues pertaining in the area of Systems of Systems. The players participating in the game do not know the network structure and the characteristics of the dynamics and costs of the players involved. Instead, they know some local characteristics of the topology, as well as a statistical description of the network. An approximate equilibrium concept is introduced and a complexity notion that describes the minimum amount of structural and feedback information needed for the players in order to behave approximately in Nash equilibrium, is defined. An example of a Linear Quadratic game on a ring is finally studied and an asymptotic upper bound for the complexity of the game is derived.

Games on large random interaction structures: Information and complexity aspects

Games on large structures of interacting agents are considered. The participants of the game do not have a full knowledge of the interaction structure or the characteristics of the other players. Instead of that, an ensemble of possible interaction structures as well as a probability measure on that ensemble are assumed to be a common knowledge among the players. Furthermore, we assume that the agents have also local information. Specifically, they know the characteristics of some players, important for them. A new notion of equilibrium, describing approximate Nash equilibrium with high probability, is introduced. A concept of complexity of a game is also defined, as the minimum amount of information needed, in order to play almost optimally. Some special cases are then analyzed. Particularly, games on random graphs are considered and are shown to be simple, under high connectivity assumptions. Games on rings, under quadratic and non quadratic cost functions, are finally studied. Bounds on the complexity of the ring games are derived.

Cheating in adaptive games motivated by electricity markets

We study Dynamic Game situations with incomplete structural information, motivated by problems arising in electricity market modeling. Some Adaptive strategies are considered as an expression of the Bounded Rationality of the participants of the game. The Adaptive strategies are typically not in Nash equilibrium. Thus, in order to assess those strategies, two criteria are stated: Firstly, how far the cost of each player is from the cost of her best response in the sense of the Nash equilibrium. Secondly, we consider the case where the first player follows the adaptive strategy and the second player implements the best response to the first player. Then, the criterion depends on the difference of the cost of the first player comparing with the cost in case where both players follow their adaptive control laws. This difference may be positive or negative. We then examine a smaller class of strategies, called the pretender strategies, where each player acts as if she had different, not real, preferences. It turns out that under certain technical conditions, if only one player is pretending, she can achieve the same cost as if she were Stackelberg leader. The situation where all the players are pretending is then considered. The effects of adaptation and cheating, when the number of players in the game becomes large, is examined in a simple example.

Pretending in Dynamic Games, Alternative Outcomes and Application to Electricity Markets

This work studies dynamic game situations with incomplete structural information, motivated by problems arising in electricity market modeling. Some adaptive/learning strategies are considered as an expression of the bounded rationality of the participants of the game. The adaptive strategies are typically not in Nash equilibrium. Thus, the possibility of manipulation appears. That is, a player may use the dynamic rule of the opponent in order to manipulate her. We focus on a smaller class of manipulating strategies, called pretending strategies, where each player acts as if she had different, not real, preferences. It turns out that under certain technical conditions, if only one player pretends, she can achieve the same cost as if she were the Stackelberg leader. The situation where all the players are pretending is then considered, and an auxiliary game, called pretenders’ game, is introduced. A class of quadratic games is then studied, and several relations among pretending and Stackelberg leadership are derived. A linear quadratic environmental game is also studied. We then study some competitive electricity market models. Particularly, a supply function model and the market mechanism described in Rasouli and Teneketzis (electricity pooling markets with strategic producers possessing asymmetric information ii: inelastic demand, arXiv: 1404.5539, 2014) are considered. It turns out that pretending may increase competition or cooperation and in some cases pretending may cause behaviors making the system not working at all.

Stochastic stability in Max-Product and Max-Plus Systems with Markovian Jumps

We study Max-Product and Max-Plus Systems with Markovian Jumps and focus on stochastic stability problems. At first, a Lyapunov function is derived for the asymptotically stable deterministic Max-Product Systems. This Lyapunov function is then adjusted to derive sufficient conditions for the stochastic stability of Max-Product systems with Markovian Jumps. Many step Lyapunov functions are then used to derive necessary and sufficient conditions for stochastic stability. The results for the Max-Product systems are then applied to Max-Plus systems with Markovian Jumps, using an isomorphism and almost sure bounds for the asymptotic behavior of the state are obtained. A numerical example illustrating the application of the stability results on a production system is also given.

Games on Large Networks: Information and Complexity

In this work, we study Static and Dynamic Games on Large Networks of interacting agents, assuming that the players have some statistical description of the interaction graph, as well as some local information. Inspired by Statistical Physics, we consider statistical ensembles of games and define a Probabilistic Approximate equilibrium notion for such ensembles. A Necessary Information Complexity notion is introduced to quantify the minimum amount of information needed for the existence of a Probabilistic Approximate equilibrium. We then focus on some special classes of games for which it is possible to derive upper and/or lower bounds for the complexity. At first, static and dynamic games on random graphs are studied and their complexity is determined as a function of the graph connectivity. In the low complexity case, we compute Probabilistic Approximate equilibrium strategies. We then consider static games on lattices and derive upper and lower bounds for the complexity, using contraction mapping ideas. A LQ game on a large ring is also studied numerically. Using a reduction technique, approximate equilibrium strategies are computed and it turns out that the complexity is relatively low.

Network design for fast convergence to the nash equilibrium in a class of repeated games

This work studies the problem of designing a Network such that a set of dynamic rules, in a class of repeated games, converges quickly to the Nash equilibrium. Particularly a very simple class of repeated games with mean field interactions is considered and we assume that the actions of the participants are determined using some simple myopic gradient based dynamic rules. The information about the actions of the other players is transmitted through a Network, using a consensus type dynamics. The speed of the convergence to equilibrium is characterized, using the Lyapunov equation involving a Laplacian like matrix. A topology optimization problem for the communication graph is then stated and an algorithm, based on the effects of new edges to the speed of convergence, is proposed. Numerical results are also given.

Effects of Players’ Random Participation to the Stability in LQ Games

We consider a linear quadratic (LQ) game with randomly arriving players, staying in the game for a random period of time. The Nash equilibrium of the game is characterized by a set of coupled Riccati-type equations for Markovian jump linear systems (MJLS), and the existence of a Nash equilibrium is proved using Brouwer’s fixed point theorem. We then consider the game, in the limit as the number of players becomes large, assuming a partially Kantian behavior. We then focus on the effects of the random entrance, random exit, and partial Kantian cooperation to the stability of the overall system. Some numerical results are also presented. It turns out that in the noncooperative case, the overall system tends to become more unstable as the number of players increases and tends to stabilize as the expected time horizon increases. In the partially cooperative case, an explicit relation of the expected time horizon of each player with the minimum amount of cooperation, sufficient to stabilize the closed loop system, is derived.