On turing degrees of Walrasian models and a general impossibility result in the theory of decision-making

Abstract

M.O. Rabin of Harvard has remarked that ‘it is obvious that not all games that are considered in the theory of games are actually playable by human beings’. Following Rabin's remark, if we interpret H. Simon's concept of bounded rationality within Church's thesis, it is possible to ask which games are computationally rational in the sense that they can be realized by recursive procedures of computations that are effective in a well-defined mathematical sense. We will discuss the complexity of realizing certain types of game-theoretic structures in terms of the partial ordering associated with the set of Turing degrees of unsolvability, and show that the degrees of complexity for recursively presented Walrasian models of general equilibrium are not among the class of R.E. degrees. This result follows by showing that the minimal degrees for the class of recursively presented Walrasian models of general equilibrium are bounded below strictly away from 0′ in the Kleene-Mostowski hierarchy. As a consequence of this bound, it follows that non-trivial Walrasian models of general equilibrium are not recursively realizable by a class of general devices of artificial intelligence known as Turing machines. A by-product of this consequence is a generalization of Kramer's (1974) impossibility result on the realization of rational choice functions for finite state machines since the class of finite state machines is properly included in the class of Turing machines.