
Understanding the difficult in computing equilibria in strategic settings is one of the primary goals of algorithmic game theory. We know from Nash's theorem that an equilibrium typically exists, but how does one find an equilibrium? The pursuit of this goal has lead not only to interesting and surprising algorithmic techniques, but also to deep new results in computational complexity. Our goal in this course is to survey, in some depth, the techniques that go into both the design of exact and approximation algorithms, as well as determining the hardness of equilibrium computation. In some sense, this is an advanced course in algorithms and complexity, though we will focus on the problem of equilibrium computation. Please note that this is a reading course, and hence all participants in the course will be presenting the topics and papers we cover. 

A good background in linear algebra as well as basic knowledge of computational complexity will be required. 

Classes will be held Tue/Thu 9:3011 am in D405.
Classes begin on August 9th. As this is a reading course, there will not be any formal evaluation. Our primary reference materials will be the conference or (preferably) journal versions of papers, or the relevant chapters from the AGT book by Nisan et al. We may supplement these by surveys, course notes from other courses, etc. 

Here is a partial list of topics and papers we plan to cover:



Umang: Course details. Introduction to games, and a simple support enumeration algorithm for computing equilibria in bimatrix games.  
Umang: Equilibrium computation in twoplayer zerosum games, and a proof of Nash's theorem.  
Umang: The LemkeHowson algorithm for equilibrium computation in twoplayer symmetric games.  
Gunjan: Playing large games using simple strategies, EC 2003.  
Phani: Rank1 bimatrix games: a homeomorphism and a polynomial time algorithm, STOC 2011.  
Somnath: A proof of Brouwer's Fixed Point Theorem via Sperner's Lemma.  
Umang: Constant rank bimatrix games are PPADhard, STOC 2014.  
Umang continues. Gunjan: Approximating Nash Equilibria and Dense Bipartite Subgraphs via an Approximate Version of Caratheodory's Theorem, STOC 2015.  
Gunjan continues.  
Phani: Nash and Correlated Equilibria: Some Complexity Considerations. Games and Econ. Behav, 1989  
Somnath: Membership in PPAD of 2Nash and 2DSperner. Refs: Section 4 from A note on the LemkeHowson algorithm, Math. Prog. Study 1974, and Theorem 3 from On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence, JCSS 1994  
Umang: A polynomialtime algorithm for market equilibria in Fisher Markets with Linear Utilities, and a PTAS for ArrowDebreu markets with linear utilities. Refs: Section 5.1, 5.2 from AGT book, and Auction Algorithms for Market Equilibrium, Math. Oper. Res. 2006.  
Umang: PTAS for ArrowDebreu markets, continued.  
Umang: A discrete tatonnementbased algorithm for exchange markets satisfying weak gross substitutes. Ref: Market Equilibrium via the Excess Demand Function, STOC '05  
Umang: The Separation Lemma, and convex programming for market equilibria in Fisher Markets with concave utilities. Refs: Lemma 5 from On the Stability of the Competitive Equilibrium II, Econometrica 1959, and Sec. 1, 2, 4 from Efficient Computation of Equilibrium Prices for Markets with Leontief Utilities, ICALP '04.  
Phani: Market equilibria in polynomial time for a fixed number of goods or agents. Devanur and Kannan, FOCS '08  
Somnath: The complexity of equilibria: Hardness results for economies via a correspondence with games. Codenotti et al., Theor. Comput. Sci., 2008 


The papers below are of varying levels of difficulty (which of course depends on your own background). Feel free to suggest papers outside of this list also.



