# Game Theory - applied mathematics - how powerful is it?

• Dcase

#### Dcase

I am most familiar with this excellent reference [inside viewable on Amazon]:
Dynamic Noncooperative Game Theory (Classics in Applied Mathematics) (Paperback) by Tamer Basar, Geert Jan Olsder

Question 1 - Are there other recommended references?

The ArXiv groups Game theory with Computer programming.
http://arxiv.org/list/cs.GT/06

Many of the physics examples appear to involve the category of ‘Prisoner's dilemma’ games although Basar and Olsder devote a chapter to the category ‘Pursuit-evasion’ games. From my perspective, the latter appears to be more useful in physics.

Question 2 - What other game categories are used in physics?

Wikipedia [2nd paragraph] lists some areas of game theory applications:

“The field came into being with the 1944 classic Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. A major center for the development of game theory was RAND Corporation where it helped to define nuclear strategies. Game theory is now used in many diverse academic fields, ranging from biology and psychology to sociology and philosophy. Beginning in the 1970s, game theory has been applied to animal behavior, including species' development by natural selection. Because of games like the prisoner's dilemma, in which rational self-interest hurts everyone, game theory has been used in political science, ethics and philosophy. Finally, game theory has recently drawn attention from computer scientists because of its use in artificial intelligence and cybernetics.”
http://en.wikipedia.org/wiki/Game_theory

Question 3 - Could Archimedes use of minimal outer polygons and maximal inner polygons to approximate the circumference of a circle be an early example of game theory application?

Question 4 - Is Game theory powerful enough to unify all disciplines of mathematics?

Question 4 - Is Game theory powerful enough to unify all disciplines of mathematics?
I feel not. How can you think of unifying, say, Calculas and probability through game theory?

Hi ssd

This is done by discussing Stochastic Games in
part I: Finite Game and Static Noncooperative Infinite Games and
part II: Infinite dynamic games with discrete and continuous time
Vaious topologies are also discussed within this text of definitions and proofs:

'Dynamic Noncooperative Game Theory' (Classics in Applied Mathematics) (Paperback) by Tamer Basar, Geert Jan Olsder
[Search inside feature viewable on Amazon]
Basar is an engineer applying physics.
Olsder is a mathematician.

Much of work in robotitc algorithms is done with pursuit-evasion games.
See Nature editor's summary 25 January 2007 for
News and Views: Mathematical physics: On the right scent
Searching for the source of a smell is hampered by the absence of pervasive local cues that point the searcher in the right direction. A strategy based on maximal information could show the way.
Dominique Martinez doi:10.1038/445371a
and
Letter: 'Infotaxis' as a strategy for searching without gradients
Massimo Vergassola, Emmanuel Villermaux and Boris I. Shraiman
doi:10.1038/nature05464
[Nature 25 January 2007 Volume 445 Number 7126, pp339-458]
http://www.nature.com/nature/journal/v445/n7126/edsumm/e070125-10.html

I did not read the book though but highly interested to get it. Is an e-form of the book available? Does it really show that it is enough to have theories of probability to get theories of calculus and vice versa (I am really surprised)!

Last edited:
I do not think that the Basar and Olsder book [definitions and proofs] is available as an e-book.
The book should be in any university mathematical library.

Additional work by Basar of Olsder may be found on the web with a google search of their full names.

Their work is referenced in a 1,000 page book, available on the web:
Steven M. LaValle,
'Planning Algorithms',
Cambridge University Press, 2006
http://planning.cs.uiuc.edu/

This type of math has been awarded Nobel prizes in economics:
1994 - John C. Harsanyi, John F. Nash Jr., Reinhard Selten
2005 - Robert J. Aumann, Thomas C. Schelling

Only a minor tweak is needed for application in 'energy economics' such as robotics, communications, etc.

Rufus Isaacs with 'Differential Games I, II, III, IV' began this objective analysis in about 1954.
http://www.rand.org/pubs/authors/i/isaacs_rufus.html

Hi, of the things you mentioned I am familiar with only (may be a part of) Nash's work. Nash equilibrium was our text.

Surreal

This might be only an aside, but if we are talking about unification of mathematical disciplines using games theory, what about Conway's surreal numbers, and the extension to two player games?

ps: thanks to Dcase for the ref to LaValle on the web.

Hi ecurbian

1 - John Horton Conway has made major contributions to both game theory primarily cellular automata and physics through the monstrous moonshine. surreal numbers may also have a role.
http://en.wikipedia.org/wiki/John_Horton_Conway

2 - Paul-André Melliès [Google can translate French into English] uses Conway semantics in linking gane theory to traces used in physics.
http://www.pps.jussieu.fr/~mellies/

In the section 'Recent and vintage talks' are two that deal with game theory.

a - Functional boxings in string diagrams. Invited talk At CSL 2006; a discussion of game theory semantics beginning slide 64 of 88.

b - Asynchronous games: fully has supplements model of propositional linear logic. Wednesday talk At LiCS 2005; discusses strategies, lambda calculus and tensors in 55 slides.

3 - John von Neumann is credited with developing the discipline of game theory as well as contributing von Neumann algebras to physics.
http://en.wikipedia.org/wiki/John_von_Neumann

4 - John Forbes Nash, Jr shared the 1994 Economics Nobel for his Nash Equilibria in noncooperative games and "... the Nash embedding theorem, which showed that any abstract Riemannian manifold can be isometrically realized as a submanifold of Euclidean space. He also made contributions to the theory of nonlinear parabolic partial differential equations."
http://en.wikipedia.org/wiki/John_Forbes_Nash

5 - Terry Gannon (Author) in 'Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics' (Cambridge Monographs on Mathematical Physics) (Hardcover - 500p) briefly discusses the contributions of Conway, von Neumann and Nash in this area.

Life

Hi DCase

Conway, Von Neumann, Nash, of course I am strongly familiar with -- I met Conway, just once, he talked about packing oranges into a box. But, this guy Mellies is new to me, I can read French, but the papers I downloaded were in English anyway and very interesting, so I'll have to thank you again for the reference.

My (hobby) interest in Conway centres on the game of life. Building large scale systems such as general purpose computation or reproducing and evolving entities. But, related to this is Wolframs ideas about using cellular automata as a grounding for physics. I like this idea, (I have a copy of "A new kind of science" -- I am one of the probable few who actually read the whole thing) and I expect that it is possible to find something like that, but see the key problem as to how to calibrate a model. Given observed behaviour, how do you modify the model to account for discrepancies?

Does this sort of thing fall inside your interest of unification using games theory? For example, a formal infinitesimal cellular automata (Robinson's non standard plane for example) to give a foundation for differential equations.

Last edited:
ecurbian

I am not familiar with “... Robinson's non standard plane ...”
Could you provide a reference?

I have read some but not all of NKS by Wolfram.

I am familiar with the Game of Life by Conway, but am more interested in his Monstrous Moonshine conjecture proved by Borcherds.

Cellular automata from my perspective seem to be more consistent with Cooperative Games.
This may be the means to unify mathematics [along with Decision theory].
http://en.wikipedia.org/wiki/Decision_theory

However, I am presently more intrigued by the Noncooperative Game approach of Basar and Olsder who eventually use pursuit-evasion games in chapter 8 of their book.
This may be more powerful than cellular automata.
I am just not sure and have much to learn.