D. A. Carson provides a good introduction when he argues that the following two propositions are both taught and exemplified in the Bible:
- God is absolutely sovereign, but his sovereignty never functions in Scripture to reduce human responsibility.
- Human beings are responsible creatures—that is, they choose, they believe, they disobey, they respond, and there is moral significance in their choices; but human responsibility never functions in Scripture to diminish God's sovereignty or to make God absolutely contingent.
Carson rightly argues that "We tend to use one to diminish the other; we tend to emphasize one at the expense of the other. But responsible reading of the Scripture prohibits such reductionism."
"Hundreds of passages," he suggests, "could be explored to demonstrate that the Bible assumes both that God is sovereign and that people are responsible for their actions. As hard as it is for many people in the Western world to come to terms with both truths at the same time, it takes a great deal of interpretative ingenuity to argue that the Bible does not support them."
Carson briefly works through a number of representative passages: Genesis 50:19-20; Leviticus 20:7-8; 1 Kings 11:11-13, 29-39; 12:1-15 (cf. 2 Kings 10:15; 11:4) 2 Samuel 24; Isaiah 10:5-19; John 6:37-40; Philippians 2:12-13; Acts 18:9-10; and Acts 4:23-30. I'd encourage readers to study each passage in context and see if they comport with Carson's two statements above.
After looking at Acts 4:23-30, Carson makes this telling comment:
Christians who may deny compatibilism on front after front become compatibilists (knowing or otherwise) when they think about the cross. There is no alternative, except to deny the faith. And if we are prepared to be compatibilists when we think about the cross—that is, to accept both of the propositions I set out at the head of this chapter as true, as they are applied to the cross—it is only a very small step to understanding that compatibilism is taught or presupposed everywhere in the Bible.
Elsewhere he writes, "At Calvary, all Christians have to concede the truth of these two statements [above], or they give up their claim to be Christians."
I especially appreciate Carson's conclusion as he locates the deepest foundation of compatibilism:
So I am driven to see not only that compatibilism is itself taught in the Bible, but that it is tied to the very nature of God; and on the other hand, I am driven to see that my ignorance about many aspects of God's nature is precisely that same ignorance that instructs me not to follow the whims of many contemporary philosophers and deny that compatibilism is possible. The mystery of providence is in the first instance not located in debates about decrees, free will, the place of Satan, and the like. It is located in the doctrine of God.
Carson's popular-level writings on compatibilism can be found in chapter 9 ("A Sovereign and Personal God") of A Call to Spiritual Reformation: Priorities from Paul and His Prayers (Grand Rapids: Baker, 1992), 145-66; and chapter 11 "(The Mystery of Providence") of How Long, O Lord? Reflections on Suffering and Evil (Grand Rapids: Baker, 1990), 199-228. For a more technical treatment (based on his doctoral dissertation), see Carson's Divine Sovereignty and Human Responsibility: Biblical Perspective in Tension.
For more technical discussions on the philosophical nature of freedom and responsibility, see the chapters in John Feinberg's No One Like Him. Among the best things I've read—accessible but philosophically informed—are the relevant chapters in John Frame's The Doctrine of God.
For a brief overview of passages on God's absolute sovereignty, see this post.