How practically relevant is game theory? I usually don't care too much about the practical relevance of nice mathematics :-) But this time, as I am looking to find some areas where I can apply maths and possibly collaborate with non-mathematicians who are experts in other fields, I wonder whether game theoretical methods really have practical relevance. Are there institutions (policy makers, administrative entities, enterprises, other organisations) who are using game theory for practical analysis and for strategic decisions?
If so, I would be very happy to get some hints/references to application case studies or the like, if they exist.
Thank you!
 A: The following is an opinion regarding your first question ("has game theory any practical relevance?") by Ariel Rubinstein, a famous game theorist. I do not agree with him, but he has an interesting argument :
From an article in the Frankfurter Allgemeine (hope this is not infringing on anyone's copyright...)

"I have devoted most of my life to economic theory and game theory. I believe that I would like to do some good for humankind and, in particular, for the people in Israel, the country where I was born and where I make my home. I would like to make an impact and redress injustices. Ostensibly, all this should motivate me to utilize my professional knowledge in order to bring some relief to the world. But, the thing is, that is not how I feel. [...]
The heart of game theory is not empirical science. It does not study how people actually behave in strategic situations. It is doubtful whether it is even possible to generalize about the way people will behave in a situation like the Hide and Seek Game. After all, people are diverse. [...]
Game theory is written in a mathematical language. [...] Personally, the nearly magical connection between the symbols and the words in game theory is what captivated me. But there are also disadvantages: The formal language greatly limits the audience that really understands it; the abstraction blurs factors that natural thought takes into account and the formality creates an illusion that the theory is scientific.
Game theory fascinates me. It addresses the roots of human thought in strategic situations. However, the use of concepts from natural language, together with the use of ostensibly “scientific” tools,  tempt people to turn to game theory for answers to questions such as: How should a system of justice be built? Should a state maintain a system of nuclear deterrence? Which coalition should be formed in a parliamentary regime? Nearly every book on game theory begins with the sentence: “Game theory is relevant to …” and is followed by an endless list of fields, such as nuclear strategy, financial markets, the world of butterflies and flowers, and intimate situations between men and women. Articles citing game theory as a source for resolving the world’s problems are frequently published in the daily press. But after nearly forty years of engaging in this field, I have yet to find even a single application of game theory in my daily life. [...]
In my view, game theory is a collection of fables and proverbs. Implementing a model from game theory is just as likely as implementing a fable. A good fable enables us to see a situation in life from a new angle and perhaps influence our action or judgment one day. But it would be absurd to say that “The Emperor’s New Clothes” predicts the path of Berlusconi [...]
The search for the practical meaning of game theory derives from the perception that academic teaching and research directly benefit society. This is not my worldview. Research universities, particularly in the fields of the humanities and social sciences, are part of a cultural fabric. Culture is gauged by how interesting and challenging it is, and not by the benefit it brings. I believe that game theory is part of the culture that ponders the way we think. This is an ideal that can be achieved in many ways – literature, art, brain research and yes, game theory too. If someone also finds a practical use for game theory, that would be great. But in my view, universities are supposed to be “God’s little acre,” where society fosters what is interesting, intriguing, aesthetic and intellectually challenging, and not necessarily what is directly beneficial.

I think Rubinstein develops these ideas in yet another paper or interview but I could not find it (in this other paper, he notably argued that if game theory has ever been useful to anyone, it is to the wealthy and to the powerfull and that, as a consequence, it did not help foster anything like "social justice"). Anyone has a clue on that?
A: One of the best examples is the application of Game Theory to the theory of auctions.  Prominent game theorists were on both sides of the electromagnetic spectrum auctions, and one got a knighthood out of it.  http://www.bristol.ac.uk/efm/people/kenneth-g-binmore/
The questions of how to design the auction, and how best to bid in the auction, is classic applied game theory, with millions and perhaps billions of dollars at stake.
http://www.cramton.umd.edu/papers2000-2004/01hte-spectrum-auctions.pdf
A: Here's a good application, coming from my university (the only reason I know if it):
When does compromise prevent more pollution?
Since it's definitely out of my wheelhouse, I'll let the paper speak for itself. From the abstract:

We look at a game theoretic scenario in which an industrial firm has the option of voluntarily controlling the pollutants it originates. A regulating agency sets the tax rate on corporate profits as well as a Pigouvian pollution tax: preset percentages of projected cleanup cost which must be covered by the firm.

And a little more, from the conclusion:

The application of our results would theoretically be to inform policy makers of the likely outcomes of their decisions. The regulator gets to choose the game that will be played, and then gets to make the rst move. If there is a preferred outcome, especially if it is preferred on the basis of non-monetary goals, then the policy may be set in such a way as to make that outcome more likely

That's the only example I know of, off the top of my head, but hopefully it gives you an idea of just how often game theory can be applied to actual situations. I do not know if any policy-makers actually utilized anything from the paper, although I imagine otherwise. The point is that it certainly can be used for the purposes you describe.
