The primary motivators for both the Father and the Son in the context of the sacrifice of Jesus for humanity's sins can be understood as love and the desperate state of humanity.
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Love: The Christian doctrine emphasizes that God is love (1 John 4:8). The decision to offer Jesus as a sacrifice stems from a profound love for humanity, demonstrated in the willingness to provide a means for redemption and reconciliation.
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The Desperate State of Humanity: Humanity's need for salvation due to sin and separation from God underscores the necessity of the sacrifice. This state of desperation prompted both the Father's plan for redemption and the Son's acceptance of that role.
Thus, while other factors like mercy and pity may play a role in the broader narrative, the core motivators are love and the recognition of humanity's desperate situation.