Differentiating generated and human-written content is increasingly difficult. We examine how an incentive to convey humanness and task characteristics shape this human vs AI race across five studies. In Study 1-2 (n=530 and n=610) humans and a large language model (LLM) wrote relationship advice or relationship descriptions, either with or without instructions to sound human. New participants (n=428 and n=408) judged each text's source. Instructions to sound human were only effective for the LLM, reducing the human advantage. Study 3 (n=360 and n=350) showed that these effects persist when writers were instructed to avoid sounding like an LLM. Study 4 (n=219) tested empathy as mechanism of humanness and concluded that LLMs can produce empathy without humanness and humanness without empathy. Finally, computational text analysis (Study 5) indicated that LLMs become more human-like by applying an implicit representation of humanness to mimic stochastic empathy.