Θεωρία παιγνίων και εφαρμογές - Αγγελοπούλου, Ιουλία (Εθνικό Μετσόβιο Πολυτεχνείο. Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών, 2014)
This thesis is a study on game theory and its applications. The first chapter is an introduction to game theory showing the basic elements that constitute it, and some significant developments that have played a crucial role in its development. Based on different criteria, games can be classified into various categories. Here, we have separate the games based on the way they are represented in two categories, in normal form games and extensive form games. The second chapter deals with the normal form games or strategic games, that are games where all players move simultaneously. These games are usually represented by a matrix. The players’ strategies can are either pure strategies (when players have decided the strategy that they will use before the game's beginning) or mixed strategies (where players choose their strategies randomly). Then, we focus on the solution of these games by introducing the method of iterated deletion of strictly dominated strategies and the concept of Nash equilibrium in pure and mixed strategies. Then, we examine the possibility that players might make a mistake in choosing their strategies and we introduce the notion of a trembling hand perfect Nash equilibrium. Finally, we consider the case of incomplete information games in which some players do not know all the information about their opponents. The third chapter refers to the extensive form games or dynamic games. These are games in which players take their decisions sequentially. These games are represented by game trees and distinguished in games with perfect information or imperfect information. In the case of games with perfect information, all players know the history of the game, so when it is their turn to move is able to identify in which node of the tree he is. To solve these games we use the concept of Nash equilibrium, which can be determined by using the method of backward induction. In games with imperfect information, there are players who do not know all the moves that have occurred in the game, so when it is their turn to move is not able to determine exactly in which node of their information set they are. To ensure the principle of sequential rationality in those games we study a more powerful notion, known as subgame perfect Nash equilibrium. However, there are games that they have not any subgame. Then, in order to achieve a solution that it will ensure that the players are sequentially rational we introduce the concept of sequential equilibrium. In the fourth chapter we study auctions with complete information using tools from game theory. In particular, we discuss the first price sealed bid and the second price sealed bid auctions, and we describe how these auctions can be written as a normal form game.