Computational systems able to collaborate with human improvisers are live algorithms: systems that can cooperate proactively, on an equal basis, with musicians. This definition is an ideal, and it raises fundamental questions about creativity and group interaction, how these might be computationally modelled. Can musicians and computers relate to each other as equivalent partners? Could an audience recognise this relationship? Live algorithms offer the prospect of a new understanding of real- time creative practice. This chapter explores this from various theoretical perspectives. The aims of live algorithm research are contrasted with existing practices in interactive computer music. Social psychology and pragmatics are explored, recognising that music-making, with or without computers, is a social praxis. We tentatively suggest how such perspectives can be represented formally, with the prospect of algorithmic encoding, and we identify improvisation systems currently in use that indicate the future promise of this field.