Why is "saving face" helpful and productive as a strategy for moving along diplomatic negotiations that are otherwise caught in stalemate?
Allowing a defeated or cornered opponent a "face saving" way out, in which he or she is allowed to accept defeat without being overtly humiliated, is used to minimize the hard feelings that would otherwise result, with the attendant risks that entails.
When Britain and France emerged victorious at the end of World War I, they forced the defeated Germany to sign the Treaty of Versaille, which required the Germans to relinquish large territories populated mainly by German-speaking people, to pay large reparations to the victors, and to live with tight restrictions on its ability to rebuild its armed forces -- a symbol of national pride in many nations. A result -- attending economic conditions throughout Europe and the world certainly contributed a great deal -- was the bitterness and humiliation felt by the German people, and which made them susceptible to a demogogue who preached of national rejuvenation and revenge against those who were responsible for Germany's plight (i.e., Adolf Hitler).
Contrast that with the approach General Douglas MacArthur insisted upon using with the defeated Japanese at the end of World War I. While many Americans, and others who suffered under brutal Japanese occupation for years, clamored, literally, for the Emperor of Japan's head, General MacArthur argued instead for the restoration of the emporer and that respect be shown this living symbol of Japanese pride. The emperor is a greatly respected figure in Japanese society, and MacArthur understood that rebuilding Japan into a modern, democratic country would require Japanese participation. That participation would not be forthcoming if the Japanese people felt humiliated and degraded by their defeat and by the U.S. occupation. By allowing Japan to "save face," General MacArthur was returning to the Japanese a measure of national pride.
Allowing the other party to a negotiation to "save face" serves the same purpose as was demonstrated with Japan's evolution into a liberal democracy lacking imperial ambitions. By treating the adversary in a negotiation with respect, the stronger party is more likely to receive cooperation and respect in return. That serves both parties' interests.