Berry (JASA, 1974): optimal fixed sample size rules for squared error loss plus cost when there are 2 or more independent Bernoulli means.

Shapiro (JASA, 1985): fixed and sequential rules for the same problem with more general distributions and loss functions.

Rekab (IEEE Reliability, 1993): ad hoc fixed and sequential rules using squared error loss

There is apparently no previous work on the general bivariate polynomial problem.

Allocation Rules Outline General Setup