How to deal with 'complain about anything' workers? (Ranty...)

I work as a team leader overseeing 15 people. Most are great, help each other out, and contribute to get the job done. There's this one guy though who gets his duties done, but spends every free moment in the office complaining about everything. I used to see this as an opportunity to find problems early, but it's getting rediculous now. He complains that he's not on a particular task, but when switch to that task the following week still complains about it. He says he's discriminated because he wasn't asked to do overtime, but the tasks he is trained in arn't needed for overtime. He flips between saying he's leaving 'in a month', to complaining he isn't being trained in other skills.

Due to his task, he is paid more than everybody else on the team and has the most amount of downtime during the day. Many would be very happy with this arrangement.

I'm lost, I used to listen to each problem and try to resolve it but it seems pointless - It's always met by another complaint. I've explained to him the company policy about how he can escalate any problems to other managers too, no luck.

Any advice on how to manage this guy? (Aside from hoping he finally follows through with his 'threat' of leaving)

Tl;dr I oversee 15 people, one guy complains about everything. I can't work out how to deal with this guy.