: 6.254 Game Theory with Engineering Applications Lecture 1: Introduction
Asu Ozdaglar MIT
February 2, 2010
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Game Theory: Lecture 1
Introduction
Optimization Theory: Optimize a single objective over a decision variable x Rn . i ui ( x ) subject to x X Rn . minimize Game Theory: Study of multi-person decision problems
Used in economics, political science, biology to understand
competition and cooperation among agents. role of threats/punishments in long term relations.
Models of adversarial behavior (strictly competitive strategic interactions, modeled as zero sum games).
Pursuit-evasion games.
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Game Theory: Lecture 1
Introduction
Game Theory (Continued):
Recent interest in networked-systems (communication and transportation networks, electricity markets).
Large-scale networks emerged from interconnections of smaller networks and their operation relies on various degrees of competition and cooperation. Online advertising on the Internet: Sponsored search auctions. Distributed control of competing heterogeneous users. Information evolution and belief propagation in social networks.
Model: n agents , each chooses some xi R, and has a utility function ui (x ), x Rn , or equivalently ui ( x i , x - i ) , x-i = (x1 , . . . , xi -1 , xi +1 , . . . , xn ).
What are the possible outcomes? Steady-state, stable operating point, characteristics? How do you get there (learning dynamics, computation of equilibrium)?
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Game Theory: Lecture 1
Introduction
Mechanism Design (MD): "Inverse Game Theory": Design of a game (or incentives) to achieve an objective (eg. system-wide goal or designer's selfish objective)
Optimization theory extended for systems in which there are independent agents not under direct control, and must be "coerced" through the use of incentives. Focal example: Internet Users' interest to skimp on congestion control ISP's interest to lie about routing information.
In Economics, MD is all about designing the right incentives. In CS/Engineering, focus is more on the design of efficient decentralized protocols that take into account incentives.
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Game Theory: Lecture 1
Course Information
Introduction to fundamentals of game theory and mechanism design. Emphasis on the foundations of the theory, mathematical tools; modeling issues and equilibrium notions in different environments. Motivations drawn from various applications: Engineered and networked systems: including distributed control of wireline and wireless communication networks, incentive-compatible and dynamic resource allocation, multi-agent systems, pricing and investment decisions in the Internet. Social models: including learning and dynamics over social and economic networks. Intended Audience: The course is geared towards Engineering-OR-CS students who need to use game-theoretical tools in their research. The course is also aimed at covering recent advances and open research areas in game theory.
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Game Theory: Lecture 1
Course Information
Prerequisites: A course in probability (6.041 equivalent) and mathematical maturity. A course in analysis (18.100 equivalent), and a course in optimization (6.251-6.255 equivalent) would be helpful but not required. Grading: 30 %: midterm 20 %: homeworks 50 %: project Project: Individual or groups of 2. Possible project types include but are not limited to: Read and report on 2-4 papers on a theoretical/application area related to game theory. An experimental study via implementation and simulation of a game/mechanism. Theoretical analysis of a game-theoretic model, which we have not covered in class and which has not been fully explored in the literature. As a starting point, check the reading list on the website.
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Game Theory: Lecture 1
Text and References
Main Text: Game Theory, by D. Fudenberg and J. Tirole, MIT Press, 1991. Other Useful References: The class notes available on the web. Algorithmic Game Theory, edited by N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani, Cambridge University Press, 2007. Auction Theory, by V. Krishna, Academic Press, 2002. Microeconomic Theory, by A. Mascolell, M. D. Whinston, and J. R. Green, Oxford University Press, 1995. A Course in Game Theory, by M.J. Osborne, A. Rubinstein, MIT Press, 1994. Game Theory, R. B. Myerson, Harvard University Press, 1991. The Theory of Learning in Games, by D. Fudenberg and D. Levine, MIT Press, 1999. Strategic Learning and its Limits, by H.P. Young, Oxford U Press, 2004. Individual Strategy and Social Structure: An Evolutionary Theory of Institutions, by H. P. Young, Princeton University Press, 1998. Dynamic Noncooperative Game Theory, by T. Basar and G. J. Olsder, 1999.
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Game Theory: Lecture 1
Strategic Form Games
Model for static games. Both matrix games and continuous games. Classical examples as well as examples from networking: "Selfish routing", resource allocation by market mechanisms, inter-domain routing across systems. autonomous Solution concepts:
Dominant and dominated strategies Elimination of strictly dominated strategies (iterated strict dominance). Elimination of never-best-responses (rationalizability). Nash equilibrium; pure and mixed strategies; mixed Nash equilibrium. Correlated equilibrium (Aumann).
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Game Theory: Lecture 1
Analysis of Static (Finite and Continuous) Games
Existence of a pure and mixed equilibrium
Nash Equilibrium: fixed point of best-response correspondences. Nash's theorem (for finite games, use fixed-point theorems to show existence of a mixed strategy Nash equilibrium) For continuous games: under convexity assumptions, can show existence of a pure strategy Nash equilibrium. For general continuous games, can show existence of a mixed strategy equilibrium. For discontinuous games (relevant in models of competition), existence of a mixed equilibrium established under some assumptions.
Uniqueness of an equilibrium using "strict diagonal concavity" assumptions
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Game Theory: Lecture 1
Games with Special Structure
Supermodular games: Instead of convexity, we have some order structure on the strategy sets of the players and conditions which guarantee "increase in strategies of the opponents of a player raises the desirability of playing a high strategy for this player."
Nice properties: Existence of a pure strategy equilibrium, convergence of simple greedy dynamics (strategy updates) to a pure strategy Nash equilibrium, lattice structure of the equilibrium set. Recent applications in wireless power control.
Potential games: Games that admit a "potential function" (as in physical systems) such that maximization with respect to subcomponents coincide with the maximization problem of each player.
Similar nice properties. Relation to congestion games: "Payoff of a player playing a strategy depends on the total number of players playing the same strategy" Recent applications in network design games.
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Game Theory: Lecture 1
Learning, Evolution, and Computation(Finite Games)
Learning:
Best-response dynamics, fictitious play (i.e., play best-response to empirical frequencies), dynamic fictitious play; convergence to Nash equilibrium. Regret-matching algorithms; convergence to correlated equilibrium.
Evolution:
Evolutionarily stable strategies. Replicator dynamics and convergence.
Computation of Equilibrium:
Zero-sum games. Nonzero-sum games. Algorithms that exploit polyhedral structure, Lemke-Howson algorithm; algorithms for finding fixed-points, Scarf's algorithm; exhaustive "smart" search etc.
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Game Theory: Lecture 1
Extensive Form Games and Repeated Games
Multi-stage games with perfect information:
Backward induction and subgame perfect equilibrium. Applications in bargaining games. Nash bargaining solution.
Repeated games:
Infinitely and finitely repeated games, sustaining desirable/cooperative outcomes (e.g. Prisoner's Dilemma) Trigger strategies, folk theorems Imperfect monitoring and perfect public equilibrium.
Stochastic games
Markov strategies and Markov perfect equilibrium.
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Game Theory: Lecture 1
Games with Incomplete Information and Introduction to Mechanisms
Static games with incomplete information.
Bayesian Nash Equilibrium Each player has private information (called his "type"). Players know the conditional distribution of types of other players.
Extensive form games with incomplete information
Perfect Bayesian Equilibrium
Applications in auctions:
Different auction formats (first-price, second-price auctions) Revenue and efficiency properties of different auction formats Can we design the "optimal" auction for a given objective?
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Game Theory: Lecture 1
Mechanism Design
Design of game forms to implement certain desirable outcomes.
e.g. to incentivize independent agents to reveal their types truthfully.
Mechanism as a mapping that maps "signals" from independent agents into allocations and payments (or transfers) Revelation principle, incentive compatibility Optimal Mechanisms (Myerson): Design a mechanism to maximize profits. Efficient Mechanisms (Vickrey-Clarke-Groves Mechanisms): Design a mechanism to maximize a "social" or system-wide objective.
Mechanisms in networks; distributed and online mechanisms. Mechanisms that operate with limited information
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Game Theory: Lecture 1
Network Effects and Games over Networks
Positive and negative externalities. Utility-based resource allocation: congestion control. Selfish routing. Wardrop and Nash equilibrium. Partially optimal routing. Network pricing: Combined pricing and traffic engineering.
Competition among service providers and implications on network performance.
Strategic network formation. Price of anarchy: Game-theory analogue of "approximation bounds"
Ratio of performance of "selfish" to performance of "social".
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