What are good mathematics books to learn decision theory?

Question

I am looking for a set of mathematics books to understand proofs in decision theory. Examples would be proofs of utility representations and social choice (im-)possibility results. I found many textbooks on both decision theory and social choice, some of which contain proofs which contain references to mathematical results.

However, I could not find a collection of mathematical results commonly used in these fields. Usual "Mathematics for Economists" textbooks fall short. What is the standard reference/textbook researchers use to find the results they employ?

Answer 1

I do not know about social choice, but for utility representations I think the two most cited books are "Convex analysis" by Rockafellar and "Infinite Dimensional Analysis: A Hitchhiker's Guide" by Alliprantis and Border. They contain most (if not all) results on convex analysis and functional analysis used by economists.

Answer 2

Decisions with Multiple Objectives: Preferences and Value Tradeoffs by Ralph L Keeney

Answer 3

For decision theory:

• Gilboa, I. (2009). Theory of decision under uncertainty. Cambridge University Press.
• Parmigiani, G., & Inoue, L. (2009). Decision theory: Principles and approaches. John Wiley & Sons.
• Peterson, M. (2009). An introduction to decision theory. Cambridge University Press.

For social choice theory, see my answer to this question : Good books to learn social choice theory.