I played as Keeper for the first time last night, and something I had a problem with is that the investigators set up an Infinite Duke Chain, effectively running the key items back and forth by using Duke as their action and speed through the clues by passing everything back and forth AND bypassing kleptomania.

I didn't see anything stopping this from happening. My assumption would be that Actions on a card can only be used once a turn, but I can't find anything which would back that up.

Read the card again.

"Action: Once per round give Duke..."
-> "Once per round"

Imho that states that the Action itself can be used once per round. Not "by this player", but really, once once.
Next round (not turn!) some one else can send Duke running through the mansion.

 So is kleptomania really bypass-able by using duke to shuffle things around? Or just dropping the item and having someone else pick it up? It seems silly, but I guess the madness isn't being absolutely greedy; you just need to steal things, you don't have to care about holding onto them forever.

Yeah, it looks like those are both ways to get around the kleptomania issue. The only drawback is that they both use an action, while trading items is usually free.

By the way, does Duke require line of sight?

Pigeonbane said:

No.

My guess is that he's got a pretty strong sense of smell ...

... and I don't envy him when he's hanging around any shoggoths.