Robustness of solutions in game theory : Values and strategies in partially observable, perturbed, stochastic, and infinite games

Game Theory is the mathematical formalization of social dynamics - systems where agents interact over time and the evolution of the state of the system depends on the decisions of every player. This thesis takes the perspective of a single player and focuses on what they can guarantee in the worst case over the behavior of other players. In other words, we consider that the objective of every other player in the game is exactly the opposite to the player. We focus on sustained interactions over time, where the players repeatedly obtain quantitative rewards over time, and they are interested in maximizing their long-term performance. Formally, this thesis focuses on zero-sum games with the liminf average objective. Two fundamental questions that Game Theory aims to answer are the following. 1. How much can a player guarantee to obtain after the interaction? 2. How to act in order to obtain the previously mentioned guarantee? These questions are formalized by the concepts of "value" and "optimal strategies". We study their properties on games that exhibit one or more of the following properties. 1. Partial Observation: the players can not perfectly observe the current state of the system during the game. We consider the model of (finite) Partially Observable Markov Decision Processes and prove that finite-memory strategies are sufficient to approximately guarantee the value. 2. Perturbed Description: the formal description of the game is perturbed by a small parameter. We consider the model of (finite) Perturbed Matrix Games, and provide algorithms to check various robustness properties and to compute the parameterized value and optimal strategies. 3. Stochastic Transitions: the actions of the players determine the behavior of the evolution of the system, described as a probability distribution over the next state. We consider the model of (finite) Perturbed Stochastic Games and provide formulas for the marginal value. 4. Infinite States: the system can be in infinitely many states. We consider the model of Random Dynamic Games on a class of infinite graphs, prove the existence of the value, and quantify the concentration of finite-horizon values.