David Warsh at economic principals, http://www.economicprincipals.com/issues/2012.10.21/1429.html has provided useful information about Lloyd Shapley from the book by Sylvia Nasar on Nash (A Beautiful Mind) and the book on the origins of game theory by Robert Leonard (Von Neumann, Morgenstern, and the Origins of Game Theory). Nasar emphasized the personal rivalry of Nash and Shapley, who both started in grad school in math at Princeton at the same time, and who also shared the same major professor, the late Albert W. Tucker, who coined the term "prisoner's dilemma" and was also co-developer of the Kuhn-Tucker theorem (not all of this is in either book or Warsh's post). Nasar emphasized this rivalry more from its egotistical aspects, who was smarter than whom, rather than its intellectual aspects, although Dave does refer to those.
These become clearer in the discussion from Leonard who recounts how Shapley was working at RAND in 1948 and successfully challenged von Neumann on a mathematical point. As a result, von Neumann reportedly offered Shapley a stipend to attend Princeton, which got him there to contest with Nash intellectually and personally. However, while Tucker was Shapley's major prof, his real mentor, von Neumann, was not formally a member of the math dept., but rather of the Institute for Advanced Studies nearby. In particular, he followed von Neumann in becoming a deep student and advocate of the use of cooperative game theory, which usually ends up being the study of coalition formation and interaction. This would lead him to develop the idea of the core in economic theory, as well as some of his other major ideas, including the Shapley value from his thesis, and his analysis with Shubik of power relations in groups. He would later coauthor a book with Robert Aumann on these matters in 1974, with Aumann publicly proclaiming him "the greatest game theorist of all time."
Now Nasar also reported on the rivalry of Nash with von Neumann himself. As with Shapley, much of this was personal and egotistical, who was the better mathematician than whom. They met once, and there are sharply conflicting reports about what transpired, although all agree that it was not a particularly friendly encounter. A strongly pro-Nash account appears in Nasar's book, while a strongly pro-von Neumann account appears in Machine Dreams by Phil Mirowski, who is a great fan of von Neumann's work (if not his person). While I was editor of JEBO, an author submitted a paper that claimed to determine which account was more accurate, but I demanded that this be removed from the paper, noting that the only primary sources for what happened were either dead or insane.
As it is, however, there was real intellectual substance to the debate as well, namely the rivalry between the cooperative and the non-cooperative approaches to game theory. Of course, the latter was advocated by Nash and formalized in his famous equilibrium that was the basis of his thesis. In the 1970s this would be the foundation for a major revival of game theory, with the Harsanyi and Selten extensions of it forming the basis of their co-receiving the Nobel with Nash in 1994, the first given for game theory. By and large, Nash's solution became far more cited than the earlier work of von Neumann or the later work of Shapley, and most would say that Nash "won," with the '94 prize, the book by Nasar, and the subsequent Oscar-winning movie starring Russell Crowe, simply reinforcing this victory.
However, along comes this prize for Shapley, and people like Warsh suddenly noticing that there have been no books on Shapley equivalent to that on Nash (who has been a more dramatic figure personally, although Shapley is plenty eccentric). There is more here than just people siding with Mirowski or the people (Harold Kuhn in particular) who were feeding Nasar with her information over von Neumann versus Nash, with so much emphasis on personalities. Without doubt non-cooperative game theory has been immensely important and useful. But it is probably now time for people to take more seriously its long discounted rival, cooperative game theory, once again.