This paper discusses certain key issues of the effectiveness of certain mathematical constructions within mathematical economics and the theory of games. By effective, we mean recursive constructions and more generally, constructions that are recursively enumerable. This is a more precise notion than the unspecified concept of construction usually found in constructive analysis (e.g. Bishop’s formulation). The advantage to our commitment to recursive constructions is that with the precision of the interpretation of effective, qualitative and quantitative measures of complexity can be obtained which are usually left unspecified in the latter frameworks.
Among the results presented is a proof that for the case of totally discrete Hurwiczian resource allocation mechanisms of the form \(\pi=\langle E,M,A\langle h,g\rangle\rangle\) having binary quadratics as outcome and performance criteria, there is an R.E. (recursively enumerable) class \(\alpha=\{\pi_ j\}_{j<\omega}\) of such mechanisms whose realization is uniformly sub-recursive in NP-complete complexity. The complexity of this class is uniformly less than a class of \(p\)-stage multilinear programs: \[ {\mathcal L}=\{\langle\{A^ i_ j\}_{i\in I},\{x^ i_ j\}_{i\in I},b,\{c^ i_ j\}_{i\in I}\rangle\}_{j\in J}. \] The level of complexity for the Hurwiczian mechanisms \(\pi=\langle E,M,A\langle h,g\rangle\rangle\) that are binary quadratic as stipulated, is bounded by the complexity of the following ordinary problem: If \(\langle a_ 1,\dots,a_ n\rangle\) is an arbitrary sequence of integers, is \(\langle a_ 1\theta,\dots,a_ n\theta\rangle\) a solution to the integral equation: \[ \int^{2\pi}_ 0\left(\prod^ n_{j=1}(\cos(a_ j\theta))\right)d\theta. \]