Decision theory references for advanced undergrad/early grad students? I'm studying measure theoretic stochastic calculus, and I was hoping to pick up some knowledge of decision theory along the way.
I'm very happy with Rudin or Karatzas in level of rigor, and I was hoping to find a game theory or decision theory (mainly decision theory) reference at this level as well (with good exercises, if possible). I want this because 1) I feel I need to build my math ability by working on reasonably tough books, and 2) I might get bored if it's too easy, although I like to think I'm a patient person.
Also, if it's not too much trouble, supposing I finish my introduction and am interested in learning more, could I also have a suggestion for a followup book.
 A: I have a couple of suggestions.  I'm a big fan of Osborne and Rubinstien's  A Course in Game Theory, but that may not have enough decision theory.  You might also try The Economics of Risk and Time By Christian Gollier.  More hardcore is work by Peter Fishburn.
A: I recommend Ok, Real Analysis with Economic Applications. It's the next level up from (baby) Rudin in terms of mathematical sophistication, but at the same time much more leisurely, and proofs are instructive rather than clever, with lots of exercises. It covers a variety of standard decision theories, and a bit of game theory, but these are presented as illustrations of the preceding mathematics. For example, the basic case of standard expected utility theory is used to illustrate separating hyperplane theorems. More complex decision theories illustrate three chapters of convexity theory. 
A disadvantage is that there's no measure theory. Not for want of trying, I've never been able to find a good textbook measure theoretic treatment of decision theory. I think you just have to do graduate courses or equivalent on measure theory and functional analysis to get the background then read a brief treatment of expected utility theory. Popular is Kreps, Notes on the Theory of Choice, which is a wonderful, breezy treatment going from basics to advanced very quickly, with proofs which are more like sketches of sketches. E.g. "(You won't have a chance of doing this if you don't know the weak topology.)
Ah, I may just have found the missing link: Efe Ok (NYU Economics) has a textbook in progress on his website called Probability Theory with Economic Applications. Chapter F is just what is needed: weak convergence, Prohorov Metric etc. taking you to chapter G decision theory under both risk and uncertainty (the latter doesn't seem to be available yet). But you could read chapter F then understand Kreps in the meantime.
A: If you're interested in material to do with Bayesian Decision Theory, Jim Smith's Bayesian Decision Analysis could be interesting. 
For a very complete, rigorous treatment as well as some advanced topics in Game Theory, there is also Maschler, Zolan and Zamir's Game Theory. 
Good luck with your studies! 
