Prince Oblonsky, Anna's brother and Dolly's husband, is a great guy to have at a party: he loves a good joke, he makes other people feel good about themselves, and he mingles with the best of them. Prince Oblonsky, however, is not such a great guy to go to if you need any kind of serious emotional depth. After all, after doing his best to arrange his own sister's divorce, he immediately starts imagining what kind of jokes he can make out of the whole thing to his society buds.
Oblonsky is an example of what you get when you have a character that's heavy on sentiment but light on real emotion. Consider Part 1, Chapter 11, when he gets choked up over Dolly's anger at him when he cheats on her. But in the same breath, he starts feeling for his mistress, the governess. He can't settle on any one feeling for very long, because none of his feelings are that meaningful:
Stepan Arkadyevitch was a truthful man in his relations with himself. He was incapable of deceiving himself and persuading himself that he repented of his conduct. He could not at this date repent of the fact that he, a handsome, susceptible man of thirty-four, was not in love with his wife, the mother of five living and two dead children, and only a year younger than himself. All he repented of was that he had not succeeded better in hiding it from his wife. But he felt all the difficulty of his position and was sorry for his wife, his children, and himself. Possibly he might have managed to conceal his sins better from his wife if he had anticipated that the knowledge of them would have had such an effect on her. He had never clearly thought out the subject, but he had vaguely conceived that his wife must long ago have suspected him of being unfaithful to her, and shut her eyes to the fact. He had even supposed that she, a worn-out woman no longer young or good-looking, and in no way remarkable or interesting, merely a good mother, ought from a sense of fairness to take an indulgent view. It had turned out quite the other way. (1.2.1)
Nothing particularly bad happens to Oblonsky, despite all of his adultery—he gets the job that he had been hoping for at the end (in Part 8, Chapter 2), and he and his wife continue their dysfunctional marriage. It's only characters with real emotional depth that change over the course of the novel—for better or for worse.
Oblonsky's character is fixed, which is what makes him such a great contrast for characters like Anna and Levin. It all comes down to literary algebra: it's much easier to measure a variable (i.e., a character who changes) when you've got a constant (i.e., a character who stays the same). Because Oblonsky doesn't change over the course of the novel, we're forced to wonder, why can he sleep around without ruining his life, unlike Anna? Why does his adultery not hurt his career at all, while Karenin's cheating wife ruins Karenin's career? Minor characters like Oblonsky serve as comparisons to major characters, so we can see their issues even more clearly.