Abstract This paper considers certain game-theoretic structures as relational structures A = «ng A, R 1 ,..., R n ang, where A is a non-empty set and each R j is a finitary relation on A for all j = 1,..., n . If for some recursive function g , A = Rng g and if for all j = 1,..., n if R j is an m -ary relation on A such that Dom( R j ) is an r.e. subset of A m = g ( N ) m , then we say that the game-theoretic structure A = A,R 1 ,…, R n > is recursively presented with index g . For A R a recursively presented structure let ф:Alt A g →Out A G be the task correspondence for A G that associates elements of an alternative space with elements of an outcome space. The Turing degree of the structure A G is defined to be deg( A G ):=(graph(ф)) and is the degree of complexity of a computable realization of the task of A G . If deg (graph(ф))≠0, then the realization of the task associated with the game A G is not recursive . The partial ordering of the Turing degrees is used to rank the minimal degrees of unsolvability (i.e. deg ( A G )≠0) associated with recursive presentations of the following game-theoretic structures: (i) γ G : Infinite stage Gale-Stewart games, (ii) U G prioric Banach-Mazur games, (iii) C G : Single-player choice functions, (iv) A G : Walrasian models of general equilibrium, (v) B G : N -person non-cooperative games in the sense of nash. The results of the paper are the following: Theorem 1. Sup{min(deg(γ G )),min(deg( U G ))}≤ 0' . Theorem 2. 0 ' C G )),min(deg( A G ))}. Collary. Sup{min(deg(γ U G )),min(deg( U G ))} B G )). From Theorem 1, minimal degrees of unsolvability for the structures γ G and U G are bounded by the interval ( 0,0 '] in D , the set of Turing degrees of reducibility; and thus by the results of Jockusch and Soare (1972) we have the characterization of minimal degrees for these structures as precisely the degrees of members of special recursively bounded П 0 1 -classes.