This text, over 7 years in the making (the main author sadly passed away before completion), is a 1,000 page definitive guide to game theory-- from an applied mathematical perspective. The audience is claimed to be undergrads, grads and researchers, but the math tends to the graduate level unless you've taken and done well in group theory, linear programming and linear algebra, for example, in undergrad, or via self study. There ARE numerous exercises, some undergrad level (most MBA type in applied), but the wonderful exercises pale in comparison to the more formal proofs and research value of this text.
The bib, citations and notes are a cornucopia of VERY CURRENT research in countless fields, and when the publisher's hype calls this a reference work for researchers, it is right!
Game theory used to be almost a stepchild of probability and statistics, because even deterministic games were handled stochastically (games without dice, like chess) for a long time, due to complexity and covariates. Then, zero sum concepts, cooperation vs. competition, derivative trading, dynamical systems such as the rabbit vs. coyote models, etc. gradually made mathematicians begin to think that game theory might be an important and even broader field of math.
Until Ruse and Conway! Those two geniuses, ala the Matrix, Tron, The 13th floor, Avatar, etc. began to posit (especially Conway) that math ITSELF is a SUBSET of game theory, and at its extreme, the essence of an actual unifying field theory. Conway explained this in terms that non math pros can understand-- this fine text takes those ideas (in the sense of the almost limitless applications of game theory) to a much more advanced level of math. If you read Conway (try: On Numbers and Games) you'll see right away how coordinate systems, axioms, integers, etc. can all be rule and positional elements of gaming in Conway's world-- eg. the first rule of any game is to learn the rules. The "board" can then be seen as the axioms, including in math as well as complexity theories and computing.
Many community colleges are creating courses on gaming-- some video game oriented, some more general and math oriented so students can go on in many other related fields, including even Python programming for AI. This text can supplement and generalize that approach, but is really more appropriate for a targeted game theory/math course (or a series of up to 4 courses as outlined by the authors in the intro). Since it is the first of its kind that is up to date and very complete, I'm guessing that it will change curriculum design as well as create new courses.
For example, students on an engineering, economics, math, computer science, and many other tracks could easily take this as an elective OR as a main track with a number of courses, supplemented by more advanced group theory, analysis, linear algebra, differential equations, etc. Everything from search engines to cryptography are employing techniques from this field, including quantum computing. In my opinion linear algebra is a must, as matrices, especially due to many new numeric computing methods, are growing exponentially in use in these fields. If you're new to this field, don't confuse linear algebra (which isn't necessarily linear OR algebra) with linear programming-- finding corner solutions using inequalities, etc. There are NUMEROUS mappings in GT as well as vector payoffs, and Nash theory itself requires significant linear algebra, so don't try this without it! Some topology also is included, but at advanced undergrad, not math major levels.
Even a field as "light" on the surface as game programming involves, a step deeper, inverse kinematics, physics, cellular automata, beziers, tesselation, projective geometry, quaternions, and much more. The growing breadth and depth of sims themselves (including modeling) reflects Conway's insights about the "Game of Life" (in general as well as in CA) and the ubiquitous applications of game theory. As he states, it is as diverse as math itself if you're willing to consider math one of its subsets!
Highly recommended. The writing is journal quality but not obtuse or show off. The proofs are too brief in some sections, as the authors literally cover the ENTIRE field, but they see this and give ample sources from which to glean more detail. This should be on the shelf of every mathematician as well as those dozens of applied fields to which GT is now being applied, from stock trading to econ, biology, chemistry, physics, ecology and of course human psych!
A less obvious audience might be good math readers who have tired of the 20,000 books on why we make the choices we make, and are ready for the real meat behind decision theory and utility functions, both competitive and cooperative. More than those popular sci titles, this fine volume gives an "aha" mathematically about the hundreds of games we see in front of our eyes daily, but don't necessarily recognize them as such.
This field has advanced by leaps in the last 5 years alone, and it is really great to have a timely and thorough update right now on the whole discipline. There ARE great texts today on each of the aspects, but not nearly as extensive or current. The text is not cheap, but you could spend three times this much adding up the topics with other separate texts, and still not be as current or complete. Other texts (in similar fields) over 1,000 pages are going for over $180 and more when very current, and although I certainly understand budgets, this one is worth it for the amount of work that went into it.
ONE IMPORTANT MISS: This book has nada/nothing/zero on Go! In fact, there really is very little info on non-chance combinatorial games in general, except a small section on zero sum games that also doesn't mention go. In that section the authors state that "no cooperative effort is possible in this category of games." From a utility function view, they are right since there has to be a winner and a loser, but there is a TON of cooperation possible in go-- including the choice of whether to carve out non competitive territory, or go at it tooth and nail! It's not surprising that Western mathematicians don't spend a lot of time on go, even though it is perhaps the most mathematically complex game known, and most single digit kyus like myself can beat the best computer programs available, unlike in chess. In fact, a large Western international chess meeting might have over 5,000 attendees, and you're lucky to get 400 for go in the West!
The authors state that they only cover combinatorials (zero sums) for historical reference, as their math is now so relatively simple. Not so with go! With 31 million seconds in a year and trillions of teraflops of processing per second, it would still take over 10^200 years to brute force the moves-- much longer than the current 1,000 trillion years left until the estimated end of the known universe! Tablebases for endings are similarly weak compared to chess. There isn't even a listing for "go" or "combinatorial games" in the index. Need to fill this gap? I recommend: Nowakowski (Games of No Chance (Mathematical Sciences Research Institute Publications) and Albert (Lessons in Play: An Introduction to Combinatorial Game Theory) both available resonably used from time to time.
FORMAT NOTE: Game theory books are generally written by computer scientists, economists and/or mathematicians. This text is by three mathematicians, so the rigor and math are extensive (a very good thing). That means that some of the formulas are slaughtered on e-readers. The problem with this for teachers is that the index here is very weak. Since this is from Cambridge but translated from Hebrew, there is no web support for indexing (although we are working with Cambridge to post a free e-index on their site right now for teachers, from their Adobe version). For example there is a whole chapter on Chess, and extensive examples that could at least apply to Go from a zero sum perspective, but no index mention of either-- and scanning 1,000 pages visually is, well, tough! Cambridge emailed us that they would send a free e-copy to any teacher ordering the text, but you have to deal with many layers in the UK to make this happen! We'll keep you posted.
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