Algorithmic game theory (NDMI098) - lecture

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Time of the lecture: Monday 9:00am, in the room S8.

Instructor: Martin Balko. E-mail: balko (AT) kam.mff.cuni.cz

Tutorials.

Information:	up

• 2/2, C+Ex, 5 E-Credits
• Annotation: An introduction to algorithmic game theory, a relatively new field whose objective is to study formal models of strategic environments and to design effective algorithms for them. This introductory course covers basic concepts and methods that are illustrated with several practical applications. To successfully pass the course, it is recommended to know basics from complexity theory.
• Throughout the winter term, I will keep posting lecture notes about topics covered so far.
• Literature:
  • Noam Nisan, Tim Roughgarden, Éva Tardos, and Vijay V. Vazirani, editors. Algorithmic game theory. Cambridge University Press, Cambridge, 2007.
  • Tim Roughgarden. Twenty lectures on algorithmic game theory. Cambridge University Press, Cambridge, 2016.
  • Kevin Leyton-Brown and Yoav Shoham. Essentials of game theory, volume 3 of Synthesis Lectures on Artificial Intelligence and Machine Learning. Morgan & Claypool Publishers, Williston, VT, 2008.
  • Jiří Matoušek and Bernd Gärtner. Understanding and Using Linear Programming. Springer-Verlag New York, Inc., 2006.
• Lecture notes: see the current run of the course.
  • The lecture notes are still under construction. If you notice any mistake or place for improvement, please, let me know by e-mail.

Lectures:	up

• First lecture (1.10.2018): We agreed on time of the lecture at the schedule of lectures and seminars of KAM and IÚUK.
• Second lecture (8.10.2018):
  • Introduction, information about the credit and the exam,
  • normal-form games, Nash equilibria (pure and mixed), examples of normal-form games,
  • Nash's theorem and its proof using Brouwer's fixed point theorem.
• Third lecture (15.10.2018):
  • Pareto optimality,
  • the Minimax theorem and its proof using the duality in linear programming,
  • the best response condition.
• Fourth lecture (22.10.2018):
  • Best response polyhedra,
  • brute-force algorithm to find all Nash equilibria,
  • best response polytopes,
  • vertex-enumeration algorithm to find all Nash equilibria.
• Fifth lecture (29.10.2018):
  • Lemke--Howson algorithm and a proof of its correctness,
  • complexity classes FNP and PPAD,
  • NASH being FNP-complete implies NP = coNP,
  • the END-OF-THE-LINE problem,
  • NASH is PPAD-complete (without proof).
• Sixth lecture (5.11.2018):
  • Epsilon-Nash equilibria,
  • quasi-polynomial time algorithm for finding epsilon-Nash equilibria,
  • correlated equilibria and their properties (without proofs),
  • linear program for finding correlated equilibria.
• Seventh lecture (12.11.2018):
  • Regret minimization, introduction of the formal model, external regret
  • large comparison classes cannot yield good bound on external regret,
  • greedy algorithm and its cumulative loss,
  • randomized greedy algorithm and its cumulative loss,
  • polynomial weights algorithm and its cumulative loss,
  • use of low-regret algorithms in game theory, no-regret dynamics,
  • cumulative loss of no-regret algorithms in zero sum games is (up to epsilon) at most the value of the game (proof at the tutorial).
• Eighth lecture (19.11.2018): The lecture was cancelled.
• Ninth lecture (26.11.2018):
  • No-regret dynamics,
  • proof of the Minimax theorem using regret minimization,
  • coarse correlated equilibria,
  • convergence to coarse correlated equilibria with no-regret dynamics,
  • internal regret and swap regret,
  • black-box reduction producing good bounds on swap regret using algorithms with good bounds on external regret,
  • no-swap-regret dynamics and its convergence to correlated equilibria.
• Tenth lecture (3.12.2018):
  • Mechanism design,
  • single-item auctions,
  • first-price auction, Vickrey auction,
  • DSIC property, Vickrey auction is awesome,
  • single-item environments, sponsored-search auctions,
  • statement of Myerson's lemma,
  • applications of Myerson's lemma (single-item auctions and sponsored-search auctions).
• Eleventh lecture (10.12.2018):
  • Revisiting basic definitions from mechanism design,
  • proof of Myerson's lemma,
  • Knapsack auctions,
  • almost optimal greedy algorithm for Knapsack auctions.
• Twelfth lecture (17.12.2018):
  • Revelation principle,
  • revenue maximization,
  • Bayesian model and expected revenue,
  • maximizing expected revenue = maximizing expected virtual social surplus,
  • maximizing revenue in single-item auctions.
• Thirteenth lecture (7.1.2019):
  • Recalling Bayesian model and maximization of expected revenue,
  • Bulow--Klemperer Theorem,
  • multi-parameter environments and their examples,
  • VCG mechanism,
  • showed solutions for Homework assignment 4.