Game theory was indeed, developed around the time of World War II. The example generally cited is the dilemma facing American Air Force General George Kenney in New Guinea.
There were two east-west routes across New Guinea for Japanese convoys, the cloudy northern route, and the sunny southern route. A convoy used the southern route, one of two things would happen: The convoy would be destroyed if Kenney directed his planes that way, and unscathed if he chose to patrol the northern route.
A convoy using the northern route would be "damaged," but some were likely to survive if attacked. (Of course, they could be "unscathed" if Kenney patrolled the southern route.)
The Japanese opted for the northern route, "some damage, some survival" versus the all-or-nothing southern route. Anticipating this, Kenney patrolled the northern route. And the "saddle point" result of "some damage" was achieved.
A similar decision faced Eisenhower during the invasion of Italy. The two choices were Rome, and Salerno (southern Italy). In the latter location, fighter cover would ensure at least an American beachhead, even against German opposition. If the landing was at Rome, it would either shorten the Italian campaign by a year (capture Rome in 1943 instead of 1944 if unopposed), or be destroyed by German resistance with no fighter protection (a real setback).
Eisenhower opted for the safer choice (sure beachhead establishment, longer slog up the peninsula). Germany's "Smiling Albert" Kesselring anticipated this and pushed German forces to the south, mostly bypassing Rome along the way.