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

Alain A. Lewis
1992 Mathematical Social Sciences
games is comprised of several types of relational structures that are used to model diverse kinds of game-theoretic phenomena. We consider game-theoretic structures to be of the form: (// = where A is a nonempty set and the R-'s are relations of finite arity on A J for all j = 1,...,n. -2- The use of relational structures of the algebraic kind of Ai = is most likely unfamiliar to most readers in the theory of games. However, it is not very difficult to show that the normal form games of von
more » ... ann [1932] can be reexpressed as relational structures of the form /jf = . Formally, we will state ^ V 1 n this simple fact as the following result: Representation Lemma: If T = is an N-person von Neumann game in normal form, then T can be reexpressed as a relational structure ^(r) = where Dom(R^(r)) = A(r) for i = 1,2. Proof; Suppose we are given a game of the form r = . We can then define the domain A(r)df: = N X T, where T = {t} and t = = n S . JEN ^ Next, let be given a pair of relations <\$,(|J> such that Dom(i) = Dom((i^) = A(r) with the defining conditions for all pairs (j,t) e N X T = A(r) given by \$(j,t) = *. and Dom( with R<,(r) = 5 and R"(r) = cj^ recovers algebraically the structure r = and hence uCi'^) is the desired reexpression of r as claimed. Given an arbitrary recursive presentation of a ^me-theoretic structure y^ with index g, we ask whether or not (/(_o °^" ^ recursively realized. When we say that QX.^ °^" ^ recursively realized we mean that the task of//:" can be executed by a Turing machine. Associated with every model 'X a i^ '^o"'® \&B}^ of (/{^g* i-e-generate winning strategies, equilibrium points, optimal choices, stable outcomes, etc. Typically, the task of U'i is the form of a correspondence \$: Alt OL ->■ Out .-JY \$ acts a space of alternatives to a space of outcomes for (^g« Let degC/f ) be the Turing degree of computational complexity of the realization of \$. Then (^" i^ ^ recursively realizable game-theoretic structure if and only if deg(^ ): = degCgraphC'J')) is recursive, in O which case the task of (7"^ g is realizable by a Turing machine. By Rabin's theorem (Rabin [1957]) there exists a recursively presented Gale-Stewart game T with no effectively computable winning strategy, and thus with no Turing realization since deg(r ) is not recursive. In Lewis [l985a] we have shown that single person g^mes against nature having the form of representable choice functions such that (A) = {x e A: Vy G A X > y} for >: X^' -> {1,0} and A e P(X) for some compact convex set in X CIFP, are such that deg(". _) is not recursive when g is an index of a presentation of /O, that is recursive. In the present paper, we * 10