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Logbook Algorithmic Game Theory - Winter 2022/23

• Lecture 26 -- 07.02.2023:
  Hedonic Games (HGs), stability concepts based on deviations of single or groups of agents, Additively Separable HGs and (non-)existence of the core, Friends and Enemies games and existence of the (strict) core.

  Literature:
  • Notes Hedonic Games.
  • For an overview on Hedonic Games see: Brandt, Conitzer, Endriss, Lang, Procaccia. Handbook of Computational Social Choice, 2016. (Chapter 15)



• Lecture 25 -- 02.02.2023:
  Envy-freeness up to any good (EFX), EFX existence for two agents, EFX existence under identical valuations, Maximin share (MMS), MMS allocations, 1/2-approximation algorithm for MMS.

  Literature:
  • Notes Indivisible Goods, pages 5-11.
  • Amanatidis, Birmpas, Filos-Ratsikas, Voudouris. Fair Division of Indivisible Goods: A Survey. IJCAI 2022.
  • Endriss. Trends in computational social choice. (Chapter 12)



• Lecture 24 -- 31.01.2023:
  Fair division of indivisible goods, envy-freeness up to one good (EF1), proportionality up to one good (PROP1), round-robin protocol, envy-cycle elimination, maximum Nash social welfare is EF1.

  Literature:
  • Notes Indivisible Goods, pages 1-5.
  • Amanatidis, Birmpas, Filos-Ratsikas, Voudouris. Fair Division of Indivisible Goods: A Survey. IJCAI 2022.
  • Endriss. Trends in computational social choice. (Chapter 12)



• Lecture 23 -- 26.01.2023:
  Selfridge-Conway protocol, Pareto optimality, Nash social welfare, envy-freeness existence, equitability.

  • Notes Cake-cutting, pages 6-11.
  • Brandt, Conitzer, Endriss, Lang, Procaccia. Handbook of Computational Social Choice, 2016. (Chapters 13.1 - 13.4)



• Lecture 22 -- 24.01.2023:
  Cake-cutting problem, proportionality, envy-freeness, relations between concepts, Cut-and-choose protocol, Dubin-Spanier protocol, Even-Paz protocol.

  • Notes Cake-cutting, pages 1-6.
  • Brandt, Conitzer, Endriss, Lang, Procaccia. Handbook of Computational Social Choice, 2016. (Chapters 13.1 - 13.4)



• Lecture 21 -- 19.01.2023:
  Random serial dictatorship, kidney exchange, stable matching, incentive compatibility of Deferred-Acceptence for the active side

  Material und Literature:
  • Slides Social Choice, pages 36-54.
  • Brandt, Conitzer, Endriss, Lang, Procaccia. Handbook of Computational Social Choice, 2016. (Chapters 14.1 and 14.2)
  • Nisan, Roughgarden, Tardos, Vazirani. Algorithmic Game Theory, 2007. (Chapter 10)
  • Roughgarden. Twenty Lectures on Algorithmic Game Theory, 2016 (Chapter 10)



• Lecture 20 -- 17.01.2023:
  House allocation, Top-Trading-Cycles, incentive compatibility and core

  Literature:
  • Slides Social Choice, pages 28-35.
  • Brandt, Conitzer, Endriss, Lang, Procaccia. Handbook of Computational Social Choice, 2016. (Chapters 14.3.2, 14.3.3)
  • Nisan, Roughgarden, Tardos, Vazirani. Algorithmic Game Theory, 2007. (Chapter 10)
  • Roughgarden. Twenty Lectures on Algorithmic Game Theory, 2016 (Chapter 9.4)



• Lecture 19 -- 12.01.2023:
  Four different social choice functions and their properties, Gibbard-Satterthwaite theorem, single-peaked preferences, and k-th order mechanisms

  Literature:
  • Slides Social Choice, pages 14-27.
  • Brandt, Conitzer, Endriss, Lang, Procaccia. Handbook of Computational Social Choice, 2016. (Chapter 2.8)
  • Nisan, Roughgarden, Tardos, Vazirani. Algorithmic Game Theory, 2007. (Chapter 9)



• Lecture 18 -- 10.01.2023:
  Voting, majority rule for 2 candidates, Condorcet paradox, plurality rule, incentive compatibility, dictatorships, Arrow theorem, properties of social choice functions

  Literature:
  • Slides Social Choice, pages 1-14.
  • Brandt, Conitzer, Endriss, Lang, Procaccia. Handbook of Computational Social Choice, 2016. (Chapters 1.2, 2.1, 2.2)
  • Nisan, Roughgarden, Tardos, Vazirani. Algorithmic Game Theory, 2007. (Chapter 9)



• Lecture 17 -- 20.12.2022:
  Optimal mechanisms for single-parameter domains, virtual values, regular distributions, interpretation with reservation prices, Vickrey vs. optimal auctions, Bulow-Klemperer theorem

  Literature:
  • Slides Designing Incentive-Compatible Mechanisms, pages 76-94.
  • Myerson. Optimal Auction Design. Math. Oper. Res. 6(1):58-83, 1981.
  • Bulow, Klemperer. Auctions versus Negotiations. Amer. Econom. Rev. 86(1):180-194, 1996.
  • Kirkegaard. A Short Proof of the Bulow-Klemperer Auctions vs. Negotiations Result. Econom. Theory 28(2):449-452, 2006.
  • Roughgarden. Twenty Lectures on Algorithmic Game Theory, 2016. (Chapters 5 and 6)



• Lecture 16 -- 15.12.2022:
  Myersons Lemma (remaining proof), revelation principle (without proof), approximation algorithms and mechanism design, knapsack auction, Greedy algorithm is monotone and 2-approximate, standard FPTAS with (1+ε)-approximation is non-monotone, adjustment to monotone FPTAS

  • Slides Designing Incentive-Compatible Mechanisms, pages 34-75.
  • Mu'Alem, Nisan. Truthful Approximation Mechanisms for Restricted Combinatorial Auctions. Games Econom. Behav. 64(2):612-631, 2008.
  • Briest, Krysta, Vöcking. Approximation Mechanisms for Utilitarian Mechanism Design. STOC 2005.
  • Nisan, Roughgarden, Tardos, Vazirani. Algorithmic Game Theory, 2007. (Chapter 9)
  • Roughgarden. Twenty Lectures on Algorithmic Game Theory, 2016. (Chapter 4)



• Lecture 15 -- 13.12.2022:
  Direct Characterization, Affine maximizers, Roberts Theorem, single-parameter domain, monotonicity, Myersons Lemma (Proof IC ⇒ monotonocity).

  Literature:
  • Slides Designing Incentive-Compatible Mechanisms, pages 21-34.
  • Myerson. Optimal Auction Design. Math. Oper. Res. 6(1):58-83, 1981.
  • Nisan, Roughgarden, Tardos, Vazirani. Algorithmic Game Theory, 2007. (Chapter 9)
  • Roughgarden. Twenty Lectures on Algorithmic Game Theory, 2016. (Chapter 3)



• Lecture 14 -- 08.12.2022:
  Vickrey second-price auction, incentive compatibility, VCG mechanisms, Clarke rule, examples

  Literature:
  • Slides Designing Incentive-Compatible Mechanisms, pages 1-20.
  • Nisan, Roughgarden, Tardos, Vazirani. Algorithmic Game Theory, 2007. (Chapter 9)



• Lecture 13 -- 06.12.2022:
  Price of stability, cost sharing games with equal sharing, price of anarchy and stability of equal-sharing games. Equal-sharing vs. arbitrary sharing.

  Literature:
  • Slides Prices of Anarchy and Stability, pages 38-57.
  • Nisan, Roughgarden, Tardos, Vazirani. Algorithmic Game Theory, 2007. (Chapter 19)
  • Anshelevich, Dasgupta, Kleinberg, Roughgarden, Tardos, Wexler. The Price of Stability for Network Design with Fair Cost Allocation. SIAM J. Comput. 38(4):1602-1623, 2008.
  • Anshelevich, Dasgupta, Tardos, Wexler. Near-Optimal Network Design with Selfish Agents. Theory Comput. 4(1):77-109, 2008.
  • Hoefer. Non-cooperative Tree Creation. Algorithmica 53(1):104-131, 2009.



• Lecture 12 -- 01.12.2022:
  Price of anarchy in affine congestion games, definition (λ,μ)-smoothness and proof template, price of anarchy for coarse-correlated equilibria, tight games, limits and intuition of smoothness

  Literature:
  • Slides Prices of Anarchy and Stability, pages 17-37.
  • Awerbuch, Azar, Epstein. The Price of Routing Unsplittable Flow. SIAM J. Comput. 42(1):160–177, 2013.
  • Christodoulou, Koutsoupias. The Price of Anarchy of Finite Congestion Games. STOC 2005.
  • Blum, Hajiaghayi, Ligett, Roth. Regret Minimization and the Price of Total Anarchy. STOC 2008.
  • Roughgarden. Intrinsic Robustness of the Price of Anarchy. J. ACM 62(5):32:1–32:42, 2015.
  • Roughgarden. Twenty Lectures on Algorithmic Game Theory, 2016. (Chapter 14)



• Lecture 11 -- 29.11.2022:
  Wardrop games, Wardrop equilibria, Braess paradox, prices of anarchy and stability, price of anarchy for linear latency functions, existence and uniqueness of Wardrop equilibria.

  Literature:
  • Slides Prices of Anarchy and Stability, pages 1-16.
  • Roughgarden. Twenty Lectures on Algorithmic Game Theory, 2016. (Chapter 11)
  • Beckmann, McGuire, Winsten. Studies in the Economics of Transportation. Yale Univ. Press, 1956.
  • Roughgarden, Tardos. How bad is selfish routing? J. ACM 49(2):236-259, 2002.
  • Correa, Schulz, Stier-Moses. A Geometric Approach to the Price Anarchy in Nonatomic Congestion Games. Games Econom. Behav. 64(2):457-469, 2008.



• Lecture 10 -- 24.11.2022:
  Correlated equilibrium, definition, swap regret, no-swap-regret learning leads to correlated equilibria, master algorithm, consensus distribution, proof of correctness.

  Literature:
  • Slides Learning and Correlated Equilibria, pages 39-55.
  • Blum, Mansour. From External to Internal Regret. J. Machine Learning Res. 8:1307-1324, 2007.
  • Roughgarden. Twenty Lectures on Algorithmic Game Theory, 2016. (Chapter 18)



• Lecture 9 -- 22.11.2022:
  No-regret learning in 2-player zero-sum games, proof of Minimax Theorem, convergence on average in hindsight vs. pointwise convergence of actual strategies, adaptive RWM algorithm with convergence time

  Literature:
  • Slides Learning and Correlated Equilibria, pages 20-38.
  • Freund, Schapire. Adaptive Game Playing using Multiplicative Weights. Games Econ. Behav. 29:79–103, 1999.
  • Nisan, Roughgarden, Tardos, Vazirani. Algorithmic Game Theory, 2007. (Chapter 4)
  • Roughgarden. Twenty Lectures on Algorithmic Game Theory, 2016. (Chapter 17)



• Lecture 8 -- 17.11.2022:
  Recap stable matching, correctness of 2-phase improvement sequences, weak acyclicity, randomized better-response dynamics. Expert problem, Greedy is bad, definition no-regret algorithm, Randomized Weighted Majority with regret guarantee, application in games leads to coarse-correlated equilibra

  Literature:
  • Slides Pure Nash Equilibria, pages 64-70.
  • Slides Learning and Correlated Equilibria, pages 1-19.
  • Littlestone, Warmuth. The Weighted Majority Algorithm. Inf. Comput. 108(2):212–261, 1994.
  • Nisan, Roughgarden, Tardos, Vazirani. Algorithmic Game Theory, 2007. (Chapter 4)
  • Roughgarden. Twenty Lectures on Algorithmic Game Theory, 2016. (Chapter 17)



• Lecture 7 -- 15.11.2022:
  Stable matching, ordinal potential, correlated stable matching is ordinal potential game; general stable matching has cyclic improvement sequences, Deferred-Acceptance algorithm, construction of short improvement sequences

  Literature:
  • Slides Pure Nash Equilibria, pages 45-66.
  • Abraham, Levavi, Manlove, O'Malley. The Stable Roommates Problem with Globally Ranked Pairs. Internet Math. 5(4):493-515, 2008.
  • Mathieu. Self-Stabilization in Preference-Based Systems. Peer-to-Peer Networking and Applications 1(2):104-121, 2008.
  • Roth, Vande Vate. Random Paths to Stability. Econometrica 58(6):1475-1480, 1990.
  • Ackermann, Goldberg, Mirrokni, Röglin, Vöcking. Uncoordinated Two-Sided Matching Markets. SIAM J. Comput. 40(1):92-106, 2011



• Lecture 6 -- 03.11.2022:
  PLS, PLS-reductions, MaxCut, PLS-completeness of computing pure Nash equilibria in congestion games

  Literature:
  • Slides Pure Nash Equilibria, pages 32-45.
  • Johnson, Papadimitriou, Yannakakis. How easy is local search? J. Computer & System Sciences 37(1):79-100, 1988.
  • Fabrikant, Papadimitriou, Talwar. The Complexity of Pure Nash Equilibra. STOC 2004.
  • Roughgarden. Twenty Lectures on Algorithmic Game Theory, 2016. (Chapter 19)
  • Ackermann, Röglin, Vöcking. On the Impact of Combinatorial Structure in Congestion Games. J. ACM 55(6), 2008.



• Lecture 5 -- 01.11.2022:
  Singleton congestion games, convergence time of improvement sequences in singleton congestion games, approximative Nash equilibria, symmetric congestion games, α-bounded jump, convergence time of the relative ε-dynamics

  Literature:
  • Slides Pure Nash Equilibria, pages 18-31.
  • Ieong, McGrew, Nudelman, Shoham, Sun. Fast and Compact: A Simple Class of Congestion Games. AAAI 2005.
  • Matroid games are discussed here.
  • Chien, Sinclair. Convergence to Approximate Nash Equilibria in Congestion Games. Games Econom. Behav. 71(2):315-327, 2011.
  • Roughgarden. Twenty Lectures on Algorithmic Game Theory, 2016. (Chapter 16)



• Lecture 4 -- 27.10.2022:
  Congestion games, Rosenthal's potential function, exact potential games, singleton congestion games.

  Literature:
  • Slides Pure Nash Equilibria, pages 1-18.
  • Monderer, Shapley. Potential Games. Games and Economic Behavior 14:1124-1143, 1996.



• Lecture 3 -- 25.10.2022:
  PPAD, END-OF-LINE problem, Nash is in PPAD, triangle coloring, Sperner's Lemma. 2-player zero-sum games, optimal strategies, value of the game (Minimax Theorem), linear programs and strong duality, efficient computation of mixed Nash equilibria

  Literature:
  • Slides Strategic Games and Nash Equilibrium, pages 28-61.
  • Slides for Sperner's Lemma
  • Sperner's Lemma in German: Video from the 80s, general result on Wikipedia.
  • Owen. Game Theory, 2001. (Chapter 1+2).
  • Nisan, Roughgarden, Tardos, Vazirani. Algorithmic Game Theory, 2007. (Chapter 1.4)
  • von Neumann, Morgenstern. Theory of Games and Economic Behavior, 1944
  • More on linear optimization, duality, and algorithms:
    Cormen, Leiserson, Rivest, Stein. Introduction to Algorithms, 3rd edition. MIT Press, 2009. (Chapter 29)



• Lecture 2 -- 20.10.2022:
  Example games and mixed Nash equilibria, existence of mixed Nash equilibria (Nash theorem), proof with Brouwer's fixed point theorem, complexity class PPAD, END-OF-LINE problem, Nash is in PPAD, triangle coloring

  Literature:
  • Slides Strategic Games and Nash Equilibrium, pages 14-38.
  • Nisan, Roughgarden, Tardos, Vazirani. Algorithmic Game Theory, 2007. (Chapter 2)
  • Nash. Non-cooperative Games. Annals of Math. 54:195-259, 1951.
  
  In the lecture we only sketched the proof how to find mixed Nash equilibria by solving an END-OF-LINE problem, thereby showing that the problem is in in PPAD. Finding mixed Nash equilibria has also been shown to be PPAD-complete. Hence, by finding an approximate Nash equilibrium we can also solve END-OF-LINE or arbitrary Brouwer fixed point problems in polynomial time. This holds even for Nash equilibria in games with 2 players.
  • Goldberg, Daskalakis, Papadimitriou. The Complexity of Computing a Nash Equilibrium. SIAM J. Comput. 39(1):195-259, 2009.
  • Chen, Deng, Teng. Settling the Complexity of Computing Two-Player Nash Equilibria. J. ACM 56(3), 2009.



• Lecture 1 -- 18.10.2022:
  Basic concepts of game theory, strategic games, dominant strategies, dominant strategy equilibrium, best response strategy, pure Nash equilibrium, mixed strategy, mixed best response, mixed Nash equilibrium.

  The material is standard and can be found in virtually every book on game theory.

  Literature:
  • Slides General Information and Strategic Games and Nash Equilibrium, pages 1-14.
  • Owen. Game Theory, 2001. (Chapter 1)
  • Nisan, Roughgarden, Tardos, Vazirani. Algorithmic Game Theory, 2007. (Chapter 1)