A constructive proof of Ky Fan's coincidence theorem
To show the existence of solutions in economic problems, game theoretical problem and other fields in mathematics and engineering, one usually refers to the fixed point theorems of Brouwer and Kakutani or to one of the related theorems. It is well known for researchers in these fields that these fixed point theorems can be constructively proofed through simplicial algorithms. But more recent developments in these research areas led to the result that the theorem of Brouwer and Kakutani (or related solution methods/algorithms) is no longer sufficient, and researchers used the coincidence theorem by Ky Fan to prove the existence of solutions in given problems. In the paper Talman and Yang show that Fan's coincidence theorem can be constructively proved via a simplicial algorithm as well. The advantage of this proof is given by the fact that the constructive proof is also of practical use in the sense that also numerical methods can be used to compute equilibria in the above mentioned economic problems (at least approximately). This then enables the researcher to evaluate and to analyze the policy implications of the models under investigation. The relevance of the proof is that the boundary condition differs considerably from those made in Brouwer's theorem and its extensions. And furthermore, the assumptions made in the theorem with respect to the separation conditions and boundary conditions are weaker than the corresponding assumptions in a fixed point theorem.
scientific article; zbMATH DE number 5550542
scientific article