If Jesus was God, does that mean He sacrificed Himself for Himself?

Jesus died for the sake of creating a covenant between God and man that man could keep (though this is a gross oversimplification). God recognized that there was no way that man could achieve salvation by his own merit or by following the law. So the Father sent Yehoshua to die, so that Yehoshua might become the mediator between God and man. God wanted a covenant with man. He did not die "for Himself" in the sense of God having any need or debt, but did die 'for Himself' in the sense of wanting to achieve God's purpose and create a way to redeem man. He did choose to die because he wished for man to have a covenant relationship with Him. God did not have to do this - but out of love, he chose to and desired to. As such, Jesus's death was not "for" God but for the sake of man, that mankind could have a relationship with God. "It was for me the day of vengeance; the year for me to redeem had come. I looked, but there was no one to help, I was appalled that no one gave support; so my own arm achieved salvation for me, and my own wrath sustained me." Yet in another sense it was 'for God' as God was pleased to be reconciled with man (Col 1:19-20.)

Isaiah 63:4-5 (This whole chapter is pertinent) Jesus's name (Yehoshua) actually means "Yahweh brings salvation" or "salvation comes from Yahweh". Jesus willingly submitted to the authority of His Father. He became the instrument of power not only in watching over Israel in the old testament, but in redeeming mankind in the new. Yehoshua then emptied Himself and took the form of a man. While He still had the spirit of God and was one with His father, He had taken an earthly body.

 Because Jesus was both our kinsmen (being human) and our redeemer (his sinless blood paying the price), God could then offer a new covenant with man. Jesus would be the mediator, and Christ's death/blood would allow it to be enforced. "For this reason Christ is the mediator of a new covenant, that those who are called may receive the promised eternal inheritance—now that he has died as a ransom to set them free from the sins committed under the first covenant. "In the case of a will it is necessary to prove the death of the one who made it, because a will is in force only when somebody has died; it never takes effect while the one who made it is living. This is why even the first covenant was not put into effect without blood...In fact, the law requires that nearly everything be cleansed with blood, and without the shedding of blood there is no forgiveness." Heb 9:15-18, 22 (Galatians 3:15-23 also discusses this).

 As to the why? For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life. - John 3:16 He redeemed us in order that the blessing given to Abraham might come to the Gentiles through Christ Jesus, so that by faith we might receive the promise of the Spirit. Galatians 3:14 Every verse about the 'why' tells us it is for our benefit. Certainly, God hates sin, and we could not stand before him otherwise. But, this is not a deficiency in His character that He needed to die to address or fix. Rather, because He is both Just and loving, the choice to send Jesus to die in our place allowed justice to be upheld while costing humans nothing. It upheld His promises, His justice, and His mercy.