The purpose of the story is implied. This story has the structure and feel of a fairy tale or a fable, but it is set in a more modern context. Given this juxtaposition and that the fairy tale structure does not lead to a happy ending, the indications are that the fairy tale is unrealistic and/or such notions of heroes and happy endings can not be successful in the modern world which, as described in this story, revolves more around money than love or true happiness.
The beginning of the story sounds like the beginning of a fairy tale:
There was a woman who was beautiful, who started with all the advantages, yet she had no luck. She married for love, and the love turned to dust.
As the story progresses, we learn that the mother is not the typical "damsel in distress." She is incapable of loving her children and she equates luck with money; she equates good fortune/happiness with money. Her husband is nonexistent, so there simply is no love and certainly no financial support from him in her life. Her son, Paul, attempts to find luck/money for his mother. Thus, Paul attempts to make his mother happy and to satisfy her needs. When Paul's winnings do not really change his mother's sad state of mind, the conclusion is that Paul, the knight on his horse, could not save the damsel (his mother). The fairy tale and the honorable knight are doomed to fail in a world where the damsel is superficial and concerned only with money. Paul is literally riding a horse to nowhere.
The other conclusion is that the relationship of roles Paul and his mother were trying to play was also doomed to failure. Paul becomes the breadwinner of the family, essentially replacing his father. But Paul can not "satisfy" his mother's needs the way his father can, so Paul can not completely "save" his mother and restore love and happiness to her life.
Freudian analysis (Oedipus Complex) aside, one of the general themes (or purposes) of the story is that money can not buy happiness, especially for a woman who had psychological issues and could not bring herself to completely love her children. When Paul's mother gets the $5,000, the voices in the house become more frequent ("There must be more money!"), symbolizing the futility of Paul's quest in the world that he lives.