On Independence For Non-Additive Measures, With a Fubini Theorem
Recent models of decision making represent agents' beliefs by non-additive set-functions. An important technical question which arises in applications to diverse areas of economics is how to define independence of such set-functions. After arguing that the straightforward generalization of independence does not in general yield a unique product, in this work I show that, while Fubini's theorem is in general false if additivity is not granted, it is true when a certain type of function is being integrated. For these functions the iterated integrals coincide with the integral with respect to products which satisfy a certain property, strictly stronger than independence. I show that most of the assumptions made in these results are very close to being necessary. In general the mentioned property is still not strong enough to uniquely define a product. On the other hand I discuss some proposals which have been made in the literature, and I show that unicity can however be obtained when the product is assumed to be a belief function. Moreover I show that the unique product thus obtained has an intuitive justification when the marginals are distributions induced by random correspondences. Finally I use the results in the paper to discuss the question of randomization in decision models with non-additive beliefs.
This paper is a modified version of chapter 4 of my doctoral dissertation at UC Berkeley. I wish to thank my adviser Bob Anderson, Massimo Marinacci, Chris Shannon and especially Marco Scarsini for helpful comments and discussion. The usual disclaimer applies. Financial support from an Alfred P. Sloan Doctoral Dissertation Fellowship is gratefully acknowledged. Published as Ghirardato, Paolo. "On independence for non-additive measures, with a Fubini theorem." journal of economic theory 73, no. 2 (1997): 261-291.
Submitted - sswp940.pdf