How would the other owner be an idiot? This is another common misconception in fantasy baseball. Just because a player was undrafted doesn't mean he doesn't have value or won't continue to play well. You should treat player values as fluid and don't be bitter that you didn't snag player x or y off the wire before someone else. The old "oh you just picked that guy off the wire a week ago, I could have had him for free" argument doesn't make any sense. Player x is no longer free, deal with it. Evaluate his worth from here on out, because your past actions are sunk costs and cannot be recovered.
Play for the rest of the season unless you're one of 3 leagues out there that use retroactive stats.
"If I have seen further it is by standing on the shoulders of giants."