Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
MS&E 336
Ramesh Johari
Lecture 1: Dynamic games
April 4, 2007
1 The General Denition
A dynamic game (or extensive game, or game in extensive form) consists of:
A set of players N ;
A set H of sequences (called histories) such that:
1. H;
2. If (a1 , . .
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
7
Calculus of Variations
Ref: Evans, Sections 8.1, 8.2, 8.4
7.1
Motivation
The calculus of variations is a technique in which a partial dierential equation can be reformulated as a minimization problem. In the previous section, we saw an example of this t
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
MS&E 336
Ramesh Johari
Lecture 15: Calibration
May 30, 2007
Calibration is a concept that tries to formalize a notion of quality for forecasters. For example,
suppose a weatherman predicts each day whether the it will rain, or be sunny. Typically forecast
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
MS&E 336
Ramesh Johari
Lecture 14: Approachability and regret minimization
May 23, 2007
In this lecture we use Blackwells approachability theorem to formulate both external and internal regret minimizing algorithms. Our study is based primarily on the alg
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
MS&E 336
Ramesh Johari
Lecture 13: Blackwells approachability theorem
May 21, 2007
In this lecture we formulate and prove the celebrated approachability theorem of Blackwell,
which extends von Neumanns minimax theorem to zerosum games with vectorvalued
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
MS&E 336
Ramesh Johari
Lecture 11: The multiplicative weights algorithm
May 9, 2007
This lecture is based on the corresponding paper of Freund and Schapire [2], though with some
differences in notation and analysis. We introduce and study the multiplicati
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
MS&E 336
Ramesh Johari
Lecture 10: Learning in games
May 9, 2007
This lecture introduces our study of learning in games. We rst give a conceptual overview of
the possible approaches to studying learning in repeated games; in particular, we distinguish bet
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
MS&E 336
Ramesh Johari
Lecture 8: Supermodular games
April 30, 2007
In this lecture, we develop the theory of supermodular games; key references are the papers
of Topkis [7], Vives [8], and Milgrom and Roberts [3]. Our development closely follows that of
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
MS&E 336
Ramesh Johari
Lecture 9: Adaptive learning
May 7, 2007
In this lecture, we develop the notion of adaptive learning as proposed by Milgrom and
Roberts [1]. Although the learning denition they give is of interest in its own right, it primarily
deri
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
MS&E 336
Ramesh Johari
Lecture 7: Fictitious play: Examples and convergence
April 25, 2007
In the last lecture, we dened several variants of ctitious play. In this lecture we will discuss
some examples that reveal the behavior of ctitious play, and then r
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
MS&E 336
Ramesh Johari
Lecture 6: Fictitious play
April 23, 2007
In this lecture we dene several variants of the ctitious play dynamic, and study some of the
properties of ctitious play.
Throughout the section we consider a nite N player game, where each
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
MS&E 336
Ramesh Johari
Lecture 4: Stochastic games
April 16, 2007
In this lecture we dene stochastic games and Markov perfect equilibrium.
1 Stochastic Games
A (discounted) stochastic game with N players consists of the following elements.
1. A state spac
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
MS&E 336
Ramesh Johari
Lecture 3: Reputation and payoff bounds
April 9, 2007
In the last lecture, we saw an example of reputation analysis in repeated interaction (a sequential entry game) with one longlived player and many shortlived players, where the
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
MS&E 336
Ramesh Johari
Lecture 2: A sequential entry game
April 9, 2007
In this draft we study a sequential entry game rst studied by Kreps and Wilson [2] and Milgrom and Roberts [3]. A similar analysis was used by the same authors to demonstrate that coo
Topics in Game Theory with Engineering Applications
MS&E 336

Spring 2014
3
3.1
Conservation Laws
Motivation
Example 1. (Burgers' Equation) Consider the initialvalue problem for Burgers' equation, a firstorder quasilinear equation of the form ut + uux = 0 u(x, 0) = (x). This equation models wave motion, where u(x, t) is the h