Dear Vincent

In May 1950, a 21-year-old mathematics student at Princeton University submitted a PhD thesis that was only 27 pages long. His name was John Forbes Nash Jr. The dissertation was titled Non-Cooperative Games. Its central result established that every finite game has at least one equilibrium when mixed strategies are allowed. The idea was simple to state: a group of players reaches a Nash equilibrium when no individual can improve their outcome by changing their strategy alone, assuming everyone else keeps their choices unchanged. Consider two competing businesses deciding their prices. Each company's best decision depends on what its competitor does. An equilibrium occurs when neither has an incentive to change its own decision unilaterally. Whether such an equilibrium is desirable for customers or society is a separate question. Nash turned this observation into a general mathematical theorem, using fixed-point methods. The thesis also contained an analysis of a three-person poker game, illustrating how the abstract theory could be applied. In 1994, forty-four years after submitting his dissertation, Nash shared the Nobel Prize in Economic Sciences for his contributions to non-cooperative game theory. What interests me most is the contrast between the modest size of the dissertation and the generality of its central result. A theorem does not become important because its proof occupies hundreds of pages. Its value lies in what it establishes, how carefully it is proved, and the new questions it allows mathematicians to ask. Nash's 27-page thesis is an extraordinary example.