Shall our lack of faith make God lose faith in us?

In our daily life, cooperation always yields the best results for partners involved in zero-sum games. This is the findings of Game Theory in dealing with the so-called Prisoner's Dilemma. However, situations may arise in which a partner will defect, causing losses on the other party. Assuming the partners are locked up in a long-term relation, not a one-shot partnership. How should you proceed if your partner betrays you? Game theory comes up with a highly effective strategy call "Tit for Tat" to deal with such situations. Simply put,

1. To begin with, cooperate.
2. Retaliate only if provoked.
3. Be quick to forgive.

Therefore, you should always be faithful to start with. Once your friend/partner breaks his promise, you retaliate by NOT keeping your promise. When he repents, keep your promise again. This is very human. However, in our relationship with God, will God apply this human "tit for tat" strategy? Of course not. It is because God and us are unequal partners. There is NO way to sustain this relation if God adopts a "tit for tat" strategy. We would all be long dead.

Now, let me turn to the first rhetorical question in Romans 3. The other three rhetorical questions come about as a result of logical fallacy: post hoc ergo propter hoc.
For what if some did not believe? shall their unbelief make the faith of God without effect?
God forbid: yea, let God be true, but every man a liar; as it is written, That thou mightest be justified in thy sayings, and mightest overcome when thou art judged (Romans 3:3-4).

Given that Jews quote from authorities to prove their points. For the moment, we put aside the question whether the readers, in this case the Roman Christians, would be convinced by such a practice or not. To begin with, let us take a look at the scripture Paul quotes to support his position: that God is faithful in spite of our disbelief.
There are many proof-texts to demonstrate God's faithfulness. For example, Paul could have quoted a more famous and direct passage from Isaiah:
For as the rain cometh down, and the snow from heaven, and returneth not thither, but watereth the earth, and maketh it bring forth and bud, that it may give seed to the sower, and bread to the eater:
So shall my word be that goeth forth out of my mouth: it shall not return unto me void, but it shall accomplish that which I please, and it shall prosper in the thing whereto I sent it (Isaiah 55:10-11).
Unexpectedly, Paul  quotes a penitential psalm of David.
Here is the background of the psalm. In his adultery with Bathsheba, David had tried every means to cover up his track. All his efforts were frustrated by Uriah, Bathsheba's husband, who seemed to have known the affair. At last, he sent Uriah to his death. When God sent Nathan the prophet to reproach him, David immediately repented (2 Samuel 11-12). Psalm 51 was supposed to be written in this circumstance. Here, Paul quotes the Septuagint version word for word.
Ὅπως ἂν δικαιωθῇς ἐν τοῖς λόγοις σου καὶ νικήσεις ἐν τῷ κρίνεσθαί σε (Psalm 50:6, LXX; Psalm 51:4 KJV, Romans 3:4b).
Why did Paul choose this penitential psalm? How relevant was such a quotation? What was the underlying, implicit assumption Paul had in mind?

In this psalm, David confessed his sins and acknowledged the righteousness of God's judgment. When Paul quoted it, did he mean to say that we had sinned when we did not believe in God's oracles? How did we sin? What had we done? Then, is our lack of faith a sin?
Not doing our obligations is a sin of omission. When we do not believe in God's promise, have we done something wrong? Is believing in God an obligation?
God respects our free will, doesn't He? Don't we have the freedom not to believe?
It seems that, for Paul, not believing in God is a sin. This reminds me of the gospel we read yesterday. So, Paul and John reached a consensus here.
For God sent not his Son into the world to condemn the world; but that the world through him might be saved.
He that believeth on him is not condemned: but he that believeth not is condemned already, because he hath not believed in the name of the only begotten Son of God (John 3:17-18).

Who will believe that God has written a blank cheque to pay for all our debts? He has issued the cheque whether you believe it or not. God remains true. I think this is what Paul meant when he said,
God forbid: yea, let God be true, but every man a liar (Romans 3:4a)
Has Paul been too sweepingly harsh in declaring that "every man a liar"? There must be some people who believe, including Paul for example. So, this is only rhetoric. But the logic does not follow. The 3 pieces of texts do not form a logical unity.
How does a penitential psalm proves that God is true and every man a liar?
How does the fact that God is true and every man a liar prove that God is faithful despite man's lack of faith?

Let me take a further step back. Paul was writing about the advantages of the Chosen People in that the oracles of God are committed to them (Romans 3:2). Then he talked about some people who did not believe in the oracles, yet God kept His faithfulness. He proved this by quoting David's penitential psalm. Now, the whole argument begins to make sense. Paul must have Nathan's Oracle (2 Samuel 7:12-16) in mind when he wrote the oracles of God were committed to the Chosen People. In the oracle, God promised David a throne, a kingdom which would last forever. This oracle in no ordinary prophecy. It is truly a great advantage for the Chosen People. However, this oracle carries with it certain warnings.  
... If he commit iniquity, I will chasten him with the rod of men, and with the stripes of the children of men:
But my mercy shall not depart away from him (2 Samuel 7:14b-15a)
So, when David committed adultery with Bathsheba and murdered Uriah, he disregarded the warnings of the Oracle. His actions were interpreted as a disbelief in the promises of God. Yet, God would not retract His oath (Romans 3:3). God remained true. As a king, David represented the whole people of Israel. David was supposed to be an exemplar. And yet David failed. Then nobody would be exempted. Everybody would be a liar like David. The word "liar" makes sense because this was what David tried to do to cover his sinful track. That is why Paul said God is true and every man a liar (Romans 3:4a). Then Paul quoted David's confession to prove his point. Through the case of King David (God be true and every man a liar), Paul proves the general case for all men (God remains faithful despite men's lack of faith). Therefore, the first rhetorical question in Romans 3 is best understood in the context of Nathan's Oracle and David's adultery with Bathsheba.

Dear Lord, I praise You for Your wonderful work in turning evil into good. In Your unfailing love we trust. Amen.