Existentially closed groups and determinacy (Q2367188)

\textit{M. Ziegler} [Word problems II, Stud. Logic Found. Math. 95, 449-576 (1980; Zbl 0451.20001)] and others have described games \(G\) for constructing groups. If \(G\) is such a game and \(\phi\) is a property, then we write \(G(\phi)\) for the form of \(G\) in which the second player wins if and only if the constructed group has property \(\phi\). Let us say that the property \(\phi\) is determined (with respect to \(G\)) if one of the two players has a winning strategy for \(G(\phi)\); let us say that \(G\) is wholly determined if every property \(\phi\) is determined. In their elegant recent book [Existentially closed groups (1988; Zbl 0646.20001)], \textit{G. Higman} and \textit{E. Scott} consider two games \(G\) for constructing groups, and they suggest [loc. cit., p. 85, line 10] that both games should be wholly determined. By Corollary 5.1 below, neither game is wholly determined. Both games can be generalized from groups to other types of structure; I give necessary and sufficient conditions for each game to be wholly determined, depending on the type of structure. It is known that groups fail to meet the conditions for either game.