The associated games formula_153 begin with the random choice of an assignment formula_154; after this initial move, the game  formula_155 is played. The existence of a winning strategy for Eloise defines "positive satisfaction" (formula_156), and existence of a winning strategy for Abelard defines "negative satisfaction" (formula_157). At this level of generality, Game-theoretical Semantics can be replaced by an algebraic approach, "team semantics" (defined below). A definition of truth for IF sentences can be given, alternatively, by means of a translation into existential second-order logic. The translation generalizes the Skolemization procedure of first-order logic. Falsity is defined by a dual procedure called Kreiselization. Given an IF formula formula_23, we first define its skolemization relativized to a finite set formula_159 of variables. For every existential quantifier formula_1 occurring in formula_23, let formula_162 be a new function symbol (a "Skolem function"). We write formula_163 for the formula which is obtained substituting, in formula_23, all free occurrences of the variable formula_5 with the term formula_166. The Skolemization of formula_23 relative to formula_168, formula_169, is defined by the following inductive clauses:   If formula_23 is an IF sentence, its (unrelativized) Skolemization is defined as formula_179. Given an IF formula formula_23, associate, to each universal quantifier formula_2 occurring in it, a new function symbol formula_182 (a "Kreisel function"). Then, the Kreiselization formula_183 of formula_23 relative to a finite set of variables formula_185, is defined by the following inductive clauses:   If formula_23 is an IF sentence, its (unrelativized) Kreiselization is defined as formula_195. Given an IF sentence formula_23 with formula_197 existential quantifiers, a structure formula_49, and a list formula_199of formula_197 functions of appropriate arities, we denote as formula_201 the expansion of formula_49 which assigns the functions formula_199 as interpretations for the Skolem functions of formula_23. An IF sentence is true on a structure formula_49 (formula_206) iff there is a tuple formula_199 of functions such that formula_208.