Winman
You start out quoting Romans 9:10-12 but you stopped too soon. You should have included at least verse 13. The passage then reads as follows [NASB]:
Perhaps I am incorrect but it appears that GOD, through the Apostle Paul, is stating, not the arbitrariness of election, but that election, HIS choice of some to salvation, was according to HIS purpose. Admittedly verse 13 which you omitted is difficult to comprehend; but then who can comprehend the mind of GOD.
Now is GOD constrained to explain his actions to mankind? Absolutely not. There is much in the book of Job that I do not understand but one thing stands out clearly to me. After Job finally stops his complaints against and questioning of GOD what does GOD tell Job. NOTHING. HE simply asks: Where were you Job? GOD owes man nothing. However, in this case GOD, through the Apostle Paul, does provide some explanation, using an episode from the history of Israel.
Consider Verses 14-18 [NASB]:
14. What shall we say then? There is no injustice with God, is there? May it never be!
15. For He says to Moses, “I will have mercy on whom I have mercy, and I will have compassion on whom i have compassion.”
16. So then it does not depend on the man who wills or the man who runs, but on God who has mercy.
17. For the Scripture says to Pharaoh, “For this very purpose i raised you up, to demonstrate my power in you, and that my name might be proclaimed throughout the whole earth.”
18. So then He has mercy on whom He desires, and He hardens whom He desires.
Note particularly verses 15 & 16. GOD through the Apostle Paul is using the history of the release of Israel in bondage in Egypt to make the point that HE alone determines who will receive mercy. Verse 15 tells us that God choses those to whom he will extend mercy, those to whom HIS purpose of salvation is made manifest. Verse 16 tells us that the actions of man are completely irrelevant. Mercy does not depend upon man but only upon GOD.
Again turning to Romans we read:
Romans 8:28-30
28. And we know that God causes all things to work together for good to those who love God, to those who are called according to His purpose.
29. For whom He foreknew, He also predestined to become conformed to the image of His Son, that He might be the first-born among many brethren;
30. and whom He predestined, these He also called; and whom He called, these He also justified; and whom He justified, these He also glorified.
There are other Scripture which clearly show the purpose of GOD in the election of some to salvation.
Ephesians 1:3-7 [NASB]
3. Blessed be the God and Father of our Lord Jesus Christ, who has blessed us with every spiritual blessing in the heavenly places in Christ,
4. just as He chose us in Him before the foundation of the world, that we should be holy and blameless before Him. In love
5. He predestined us to adoption as sons through Jesus Christ to Himself, according to the kind intention of His will,
6. to the praise of the glory of His grace, which He freely bestowed on us in the Beloved.
7. In Him we have redemption through His blood, the forgiveness of our trespasses, according to the riches of His grace,
2 Timothy 1:8-10 [NASB}
8. Therefore do not be ashamed of the testimony of our Lord, or of me His prisoner; but join with me in suffering for the gospel according to the power of God,
9. who has saved us, and called us with a holy calling, not according to our works, but according to His own purpose and grace which was granted us in Christ Jesus from all eternity,
10. but now has been revealed by the appearing of our Savior Christ Jesus, who abolished death, and brought life and immortality to light through the gospel,
From this sampling of many such Scripture we see the affirmation of the purpose of GOD in Salvation:
1. It is God the Father who foreknew and chose His people before the foundation of the world [Ephesians 1:4].
2. It is God the Father who predestined that His people would be conformed to the image of His Son that they might be adopted as His children [Ephesians 1: 5] and become heirs and joint heirs with Jesus Christ [Romans 8: 16, 17].
3. It is God the Son who humbles Himself, take upon Himself the form of man, Jesus Christ, and sheds His blood on a Roman cross to pay the penalty for the sins of those whom the Father has chosen to salvation [John 17; Philippians 2:6-10].
4. It is God the Holy Spirit, who applies the work of the Jesus Christ to those chosen by God unto salvation so that none are lost [John 6: 37, 39, 40, 44].
5. It is the Triune God who, in time, will regenerate, call, justify, and glorify those whom He chose to Salvation before the foundation of the world, therefore, they can never fall away or lose that Salvation [Romans 8:28-30].