On dynamic games: stochastic games, search-dissimulation and information transmission.

Summary In this thesis, we study various models of dynamic games. These model decision processes taken by rational agents in strategic interactions and whose situation evolves over time. The first chapter is devoted to stochastic games. In these games, the current game depends on a state of nature, which evolves from one step to the next in a random manner depending on the current state and on the actions of the players, who observe these elements. We study communication properties between states, when the state space is in the form of an X ×Y product, and the players control the dynamics on their component of the state space. We show the existence of optimal strategies in any game repeated a sufficient number of times, i.e. the existence of the uniform value, under the assumption of strong communication on one side. On the other hand, we show the non-convergence of the value of the expected game, which implies the non-existence of the asymptotic value, under the hypothesis of weak communication on both sides. The next two chapters are devoted to models of search-dissimulation games. A searcher and a concealer act on a search space. The objective of the searcher is typically to find the concealer as quickly as possible, or to maximize the probability of finding it in a given time. The challenge is then to compute the optimal value and strategies of the players according to the geometry of the search space. In a patrol game, an attacker chooses a time and place to attack, while a patroller walks continuously. When the attack occurs, the patroller has a certain amount of time to locate the attacker. In a stochastic search-dissimulation game, the players are on a graph. The novelty of the model is that due to various events, at each step, some edges may not be available, so that the graph evolves randomly in time. Finally, the last chapter is devoted to a model of repeated games with incomplete information called dynamic information control. An advisor has private knowledge of the state of nature, which evolves randomly over time. Each day the advisor chooses how much information to reveal to an investor through messages. In turn, the investor chooses whether or not to invest in order to maximize his expected daily payout. If the investor does invest, the advisor receives a fixed commission from the investor. His objective is then to maximize the expected frequency of days in which the investment takes place. We are interested in a particular information disclosure strategy of the advisor called the gluttonous strategy. It is a stationary strategy with the property of minimizing the amount of information disclosed under the constraint of maximizing the advisor's current payout.