|
We begin the lecture with an introduction to normal-form games and Nash's classical result on the existence of equilibrium, along with examples. Next, we analyze the set of Nash equilibria by studying a system of multilinear
equations. In general, the set of Nash equilibria is finite, and we explore lower- and upper-bound results for the number of Nash equilibria using tools from convex and algebraic geometry. We also examine cases where the set is
infinite and possesses a rich algebro-geometric structure.
To generalize Nash equilibria, various concepts have been introduced in game theory. One such concept is correlated equilibria, which form a convex polytope within the probability simplex. We investigate some combinatorial
properties of this polytope and examine where the Nash equilibria lie on it. Another generalization is dependency equilibria, which have an algebro-geometric model known as the Spohn variety. This framework allows us to explore
connections to Nash equilibria and provides insights into Pareto-optimal equilibria, such as in the Prisoner's Dilemma.
Lastly, after examining existing equilibrium concepts, we introduce a new one: conditional independence equilibria. This concept marks the first intersection between game theory and algebraic statistics. Here, we model a game
using discrete undirected graphical models, where the vertices represent players as discrete random variables, and the edges capture dependencies in their choices.
Throughout the lecture, we aim to utilize mathematical software, including the GameTheory package in Macaulay2.
Keywords: equilibrium, Nash, semialgebraic set, correlated, polytope, dependency, variety, conditional independence, graphical model
Prerequisites: Knowledge of basic algebraic geometry is helpful but not required.
Remarks and notes: Everyone, including non-algebraists, is welcome! We will work through examples using mathematical software as well.
|