"Algorithmic Game Theory" (henceforth referred to as AGT)
is edited by Noam Nisan, Tim Roughgarden,
Eva Tardos, and Vijay Vazirani, with a foreword by Christos
Papadimitriou.  Although I am not an expert in algorithmic game theory, 
I used AGT for a class on algorithmic game theory immediately after the book was released. 
As essentially all of the obvious experts have written a chapter, 
this apparently qualifies me for writing a review.


Perhaps the most important fact to know about AGT is that it
is the first and only text on algorithmic game theory, and currently has no competitors. 
AGT contains 29 chapters written by experts in the area.
The chapters are divided into four sections: computing in games,
algorithmic mechanism design, quantifying the inefficiency of
equilibrium, and additional topics. As best as I can tell as a non-expert, 
the claim by Christos Papadimitriou
in the foreword that "the book is a good snapshot of the state of
research in algorithmic game theory circa 2007" is correct. For those
areas where the research is more mature, such as inefficiency of
equilibrium results, the chapters could be reasonably useful for many
years. Other chapters will presumably become obsolete more quickly as
the current fast pace of research either produces significantly more
results, or leaves the chapters as an historical curiosities.

I have had several inquiries as to my opinion on the usefulness of AGT for
teaching, and I will address the rest of my comments to this point. 
The preface of AGT states that "The text is pitched at a beginning graduate 
student in computer science ...  and that authors' goal was 
"a book that can be used as a textbook for a wide variety of courses ..."
I teach an algorithmic topics course to roughly this targeted audience, 
computer science PhD students who are not primarily mathematically/theoretically oriented.
The first chapter of each of the first three sections of AGT 
is an introduction to the section written by
some subcollection of the editors. These introductory chapters are
well written and provide a nice, readable, introduction to the section.
But other than the three introductory chapters, most of
the rest of the chapters of AGT are primarily surveys, not tutorials. There
is generally more emphasis on cramming in many results than on explaining
the basic concepts well. The writing is generally at the level that one would expect
for conference publications in computer science. In fact, it is not hard
to find portions cut and pasted from conference papers.
So the exposition is generally intended for fellow researchers, and not
students. Also, the chapters haven't been debugged at anything close to
the level that one would hope for in a teaching text. Perhaps the most
common problem is underspecified definitions. So often one has
to follow a proof with several possible definitions in mind, and for
each statement in the proof, determine which definition
would make this statement true. And there are a lot of just
plain minor errors. Although the bugs slowed me down some in
preparing lectures, dealing with bugs is a pretty standard skill
for a researcher. But surmounting frequent bugs it is a lot to ask of a
casual student who might read the book.  

I found many of the chapters in AGT not interesting for
teaching purposes. For example, there are many chapters that are long on
definitions/concepts, but are short on interesting applications of the
definitions/concepts. There is plenty interesting stuff in AGT for
teaching, but it took a lot of time to extract the good stuff from the
rest. The book is not ordered with the intention that a prefix of
the chapters is a reasonable course. The preface also doesn't provide
any guidance or suggestions as to topics to cover. You can check my 
web page for some hints as to the chapters that I found most 
appropriate for teaching.  I also suggest looking at the web
pages of the editors, and Christos Papadimitriou. These pages contain
pointers to algorithmic game theory courses that they teach. From these,
you can get a feel for the topics that they think are most important and
that they teach. Many of these course homepages also contain very good
class notes. These notes are generally much more useful for teaching than AGT. 
In fact, when I assigned students to give presentations
on some of the additional topics chapters, most of them admitted that they
found notes and teaching lectures using Google that they preferred to the chapters in AGT.


So in summary, AGT is a fine snapshot of algorithmic game theory research
circa 2007. And probably this was effectively the main collective goal
of the editors and chapter authors. They should be commended for
spending the considerable time that it takes to write such surveys,
particularly given the small relatively small rewards. However, I don't
really recommend AGT for teaching purposes, at least not for students
who not destined to become mathematical researchers. 
I believe that there is a demand for a real teaching textbook on algorithmic 
game theory if anyone is thinking about writing one.