Avraham's refusal to accept anything from the King of Sdom lest the King of Sdom claim, "Ani he'esharti es Avraham," I have made Avraham rich, strikes some of the meforshim as difficult to fathom. It was only Avraham, acting independently, who was able to turn the tide of battle and rout the enemy where Sdom and others had failed. The King of Sdom was stuck literally begging Avraham for his kingdom back. Surely for him to claim that he enriched Avraham would strike anyone as ludicrous.
I'm a cynical reader so this question doesn't really bother me thay much. Watch or read the news put out by any major media news outlet and you will see truth distorted into falsehood and falsehood peddled as truth on a nighly basis, and the gullible, ignorant public buys it hook, line, and sinker. But I'll take the question as is only to give you the Shem m'Shmuel's answer. There are two ways to win over an enemy: 1) crush them with brute strength and force; 2) overpower them with love and kindness so that they cannot help but give up their animosity. Avraham was the paragon of chessed. He won over those who disagreed with him through his generosity. Yet, in our parsha he was forced to go to war to save Lot. Had he taken riches in the process, his image would have been forever tarnished. When people looked at Avraham they would no longer see a man who earned success through chessed, but rather they would see a man who earned success through warfare. "Ani he'esharti es Avraham": the King of Sdom would say that it was Avraham's becoming like him -- a warrior -- that brought Avraham true riches and success -- not Avraham's love and kindness. Avraham preferred to give up the riches and spoils of war and remain true to his own identity rather than risk that identity for the sake of material possessions.