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We present a new denotational model for the untyped&lgr;-calculus, using the techniques of game semantics. Thestrategies used are innocent in the sense of Hyland and Ong(Inform. and Comput., to appear) and Nickau (HereditarilySequential Functionals: A Game-Theoretic Approach to Sequentiality,Shaker-Verlag, 1996. Dissertation, UniversitätGesamthochschule Siegen, Shaker-Verlag, 1996), but the traditionaldistinction between "question" and "answer" moves is removed. Wefirst construct models D and DREC as global sections of a reflexiveobject in the categories A and AREC of arenas and innocent andrecursive innocent strategies, respectively. We show that these aresensible &lgr;&eegr;-algebras but are neither extensionalnor universal. We then introduce a new representation of innocentstrategies in an economical form. We show a strong connexionbetween the economical form of the denotation of a term in the gamemodels and a variable-free form of the Nakajima tree of theterm. Using this we show that the definable elements of DREC areprecisely what we call effectively almost-everywhere copycat(EAC) strategies. The category AEAC with these strategies asmorphisms gives rise to a λη-model DEAC which we showhas the same expressive power as D, i.e. the equational theory ofDEAC is the maximal consistent sensible theory H*. We show that themodel DEAC is sensible, order-extensional and universal (i.e. everystrategy is the denotation of some λ-term). To our knowledgethis is the first syntax-free model of the untypedλ-calculus with the universality property. Copyright 2002Elsevier Science B.V.