Description

Most of the aspects of the world around us can be modelled by systems of strategically interacting agents. These agents are typically either cooperating or competing towards a common goal. The desire to understand such complex situations gave birth to the field of game theory. In the past 70 years or so, this field has definitely had a huge impact on the advancement of our modern society, from the point of view of economics, business, political sciences, computer science or even global nuclear strategies.

The modern mathematical game theory was initiated by John von Neumann, one of the most influential mathematicians of the 20th century. Together with Oskar Morgenstern, they wrote the first detailed text on the foundations of this theory. Later, many Nobel laureates in economics were acknowledged for their groundbreaking findings in game theory. These include John F. Nash (for developing the concept of Nash equilibrium), Lloyd Shapley (for contributions in cooperative games), John Harsanyi (for analysing games with incomplete information), Thomas Schelling and Robert Aumann (for evolutionary games and correlated equilibria, respectively), and many others.

The aim of this project is to develop an analytic intuition and skillset to understand various concepts in game theory. After getting familiar with some examples of classical games (such as the prisoner's dilemma and various combinatorial games), further topics which will be covered will include:

• Basic theory of zero-sum and general-sum games.
• Brouwer fixed point theorem, von Neumann's minimax theorems, Sperner's lemma.
• The notion of Nash equilibria.
• Fair division problems.

If time permits a selection from further topics will also be considered, such as

• Evolutionary games.
• The price of anarchy.
• Correlated equilibria.
• Stable matching problems.

Prerequisites

Main texts

• A.R. Karlin, Y. Peres, Game Theory, Alive, AMS, 2017.
• J. von Neumann, O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1944.