Colin, I think I can say something about your 2nd question:

   (3) u[i] (b,v,p) = -p[i] if x[i] = F, else v[i]-p[i], 

   (ii) how do I specify the role of the Boolean condition in u[i]?

Caution: I don't know what R^n_+ means.  Is that R^n with infinity
added on?  I'll pretend it's just R^n.  Then b,v and p are elements of
R^n, and u is a function

u: R^n x R^n x R^n ---> bool^n

with u[i] the i-th coordinate of u, itself a function

u[i]: R^n x R^n x R^n ---> bool

You can see from your definition (3) that u[i] isn't well-defined,
because the x on the right isn't an argument on the left.  So just
make x an argument: 

u[i] (b,v,p,x) = - p[i] if x[i] = F, else v[i] - p[i]

-- 
Best,
Bill 

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