Interesting idea for sure. I am just thinking, intuitively couldn't I 'game' the mediator by overstating my preference and requirements to achieve a more favorable outcome?

Thank you. Yes, you could inflate your BATNA, but then you risk the other side rejecting the agreement when a mutually beneficial agreement was possible if you had been honest.

This kind of property in a negotiation system, where honesty is rewarded and dishonesty can backfire, is called “incentive compatibility.” I’m not claiming my approach is formally incentive compatible, but it is directionally so.