Algorithms for Computing Solution Concepts in Game Theory

These are notes for the first five lectures of the course on Algorithmic Game Theory, given (starting November 2008) in the Weizmann Institute jointly by Uriel Feige, Robert Krauthgamer and Moni Naor. The lecture notes are not intended to provide a comprehensive view of solution concepts in game theory, but rather discuss some of the algorithmic aspects involved. Hence some of the definitions will not be given here in their full generality. More information can be found in the first three chapters of [NRTV07] and references therein. Each lecture in the course was roughly two hours, with a break in the middle. The experience of the author when teaching this material was as follows. The first lecture was intended to cover Section 1, but this turned out to be too optimistic. The notion of a mixed Nash was only touched upon briefly at the end of the lecture, and there was no time for the notion of a correlated equilibrium (which was deferred to lecture 5). The second lecture covered Section 2, except that the open questions mentioned in Section 2.2 were not discussed in class for lack of time. The third lecture covered sections 3.2 (an introduction to linear programming) and 3.3. Initially, I was planning to precede this by discussing known results on two-player win-lose games of partial information (see Section 3.1) but decided to drop this subject due to lack of time. The associated homework assignment partially touches upon this subject. The fourth lecture discussed mixed Nash equilibria. Towards the end of the lecture I started describing the Lemke-Howson algorithm, but did not finish in time. The fifth lecture started with the Lemke-Howson algorithm, which took roughly an hour to describe (longer than I expected). Then the class PPAD was discussed (including showing the proof of Sperner’s Lemma in two dimensions, not appearing in the lecture notes), finishing with correlated equilibria (the end of Section 1), and a short mention that it can be computed using linear programming. Theorem 3.7 was omitted due to lack of time. I would have liked to have an additional lecture covering parts of Chapter 4 in [NRTV07] (iterative algorithms for regret minimization), which among other things explains how to (approximately) solve two-player zero-sum games in certain situations in which the game is not in standard form, and provides an alternative proof to the minimax theorem. However, I decided to drop this subject since the other subjects already took up the five lectures devoted to this part of the course. Homework assignments as given in the course are also included. They are intended to be an integral part of the course. The first question in assignment 2 turned out to be too difficult, especially as it was given with an incorrect hint. The first question in assignment 5 could have been given after lecture 4. The notes are made accessible on the web for the use of students in the course, and for noncommercial use of others who may be interested. To help make the notes a more reliable source of information for future courses, please send comments to [email protected].