Auctions have provided a real-world arena in which to apply game theory. The theory has actually been applied to design auctions; most famously, the auctions for 3G wireless spectrum were designed along the principles of the theory.

Paul Klemperer has assembled several articles on the subject into a (fairly non-technical) book, and has provided the original articles online. For a more detailed approach, see this survey.

Even with the use of game theory, economists still had to use experiments with students to give detailed advice to regulators on the specific rules of spectrum auctions. See:

@ARTICLE{guala:shps01,

AUTHOR = “Francesco Guala”,

TITLE = “Building economic machines: The {FCC} {A}uctions”,

JOURNAL = “Studies in the History and Philosophy of Science”,

YEAR = “2001”,

volume = “32”,

number = “3”,

pages = “453–477″}

The reason is that game theory, and its applied offspring, auction theory, make assumptions about idealized participant behaviours and contexts, in order to make the math tractable. The real world is much messier, and these assumptions don’t hold. Even with the assumptions, the math is not all that tractable. I have long thought that a completely new approach to game theory is needed, for example using algebraic topology or differential geometry, to abstract away the detail.

Computational auction theory is now a major sub-discipline of artificial intelligence, as markets move online and as traders are automated. My own department is a leader in this area, for example, in automated auction mechanism design. See, eg, the pages of my colleague, Steve Phelps:

http://www.csc.liv.ac.uk/~sphelps/

Finally, a statistic to inspire or to scare you: the proportion of automated trades each week on the New York Stock Exchange is currently typically about 55% (by value), and some weeks has risen to over 70%. Source: NYSE press releases, eg:

http://www.nyse.com/Frameset.html?displayPage=/press/PressReleases.html

Most trading is now done by machines, although presumably still mostly for humans.