Is there any quantitative evidence that the Mutual Gains Approach to negotiation actually produces "better" results?

Yes. There are dozens of studies - and many real-world cases described in numerous business school cases - that illustrate the “inefficiency” of most negotiated agreements in terms of missed opportunities to bring more value to the table, failures to understand the other party’s interests and perceived alternatives, stalled or delayed processes, and frayed relationships. Good academic reviews can be found in Thompson’s Heart and Mind of the Negotiator and Bazerman and Neale’s Negotiating Rationally. The Mutual Gains Approach provides a road map for effective preparation, value creation, dividing the value without destroying the relationship, and delivering on promises and commitments. Testimonials from groups CBI has trained suggest that using an MGA can produce gains of hundreds of thousands - and sometimes millions - of dollars.