1. E.B. Tylor believed that sacrifice evolved through three stages that usually focused on the ritual participant trying to win the favor of the gods: gift giving, which obligated the receiver (the god) to act in kind, homage, this is similar to gift giving in that the giver seeks to gain the goodwill and protection of the god, and abnegation or renunciation, which expresses only self denial without an expectation of "in kind" from the god(s). For Tylor, renunciation was the highest form of sacrifice.
W. Robertson Smith focused on how sacrifice secured a social bond. He pointed out that with a sacrificial ritual, a sacred feast or meal was usually included. The meal served to establish a union between the god and the participants, and also strengthened the bonds between participants.
The most important function of sacrifice is thought to be expiation, which is the making of amends or atonement for transgression. These sacrificial rites assume some offense against the sacred that must be repaired by ridding the community of the pollution. Life can NOT be restored simply by good works, but must also include self-sacrifice. Life must be offered in order for life to be preserved. The community is defiled and must reconcile with the sacred.