New Book on Nash

The Essential John Nash
Edited by Harold W. Kuhn and Sylvia Nasar (via Tyler Cowan)

Link to the book via Google Books

From the Introduction by Sylvia Nasar;

"Nash wrote his first major paper—his now-classic article on bargaining—while attending Albert Tucker’s weekly game theory seminar during his first year at Princeton, where he met von Neumann and Morgenstern. But he had come up with the basic idea as an undergraduate at Carnegie Tech—in the only economics course (international trade) he ever took.

Bargaining is an old problem in economics. Despite the rise of the marketplace with millions of buyers and sellers who never interact directly, one-on-one deals—between individuals, corporations, governments, or unions—still loom large in everyday economic life. Yet, before Nash, economists assumed that the outcome of a two-way bargaining was determined by psychology and was therefore outside the realm of economics. They had no formal framework for thinking about how parties to a bargain would interact or how they would split the pie.

Obviously, each participant in a negotiation expects to benefit more by cooperating than by acting alone. Equally obviously, the terms of the deal depend on the bargaining power of each. Beyond this, economists had little to add. No one had discovered principles by which to winnow unique predictions from a large number of potential outcomes. Little if any progress had been made since Edgeworth conceded, in 1881, “The general answer is . . . contract without competition is indeterminate.” In their opus, von Neumann and Morgenstern had suggested that “a real understanding” of bargaining lay in defining bilateral exchange as a “game of strategy.” But they, too, had come up empty. It is easy to see why: real-life negotiators have an overwhelming number of potential strategies to choose from—what offers to make, when to make them, what information, threats, or promises to communicate, and so on.

Nash took a novel tack: he simply finessed the process. He visualized a deal as the outcome of either a process of negotiation or else independent strategizing by individuals each pursuing his own interest. Instead of defining a solution directly, he asked what reasonable conditions any division of gains from a bargain would have to satisfy. He then posited four conditions and, using an ingenious mathematical argument, showed that, if the axioms held, a unique solution existed that maximized the product of the participants’ utilities. Essentially, he reasoned, how gains are divided reflects how much the deal is worth to each party and what other alternatives each has.

By formulating the bargaining problem simply and precisely, Nash showed that a unique solution exists for a large class of such problems. His approach has become the standard way of modeling the outcomes of negotiations in a huge theoretical literature spanning many fields, including labor-management bargaining and international trade agreements.

Since 1950, the Nash equilibrium—Nash’s Nobel-prize-winning idea—has become “the analytical structure for studying all situations of conflict and cooperation.”* Nash made his breakthrough at the start of his second year at Prince-ton, describing it to fellow graduate student David Gale. The latter immediately insisted Nash “plant a flag” by submitting the result as a note to the Proceedings of the National Academy of Sciences. In the note, “Equilibrium Points in n-Person Games,” Nash gives the general definition of equilibrium for a large class of games and provides a proof using the Kakutani fixed point theorem to establish that equilibria in randomized strategies must exist for any finite normal form game (see chapter 5).

After wrangling for months with Tucker, his thesis adviser, Nash provided an elegantly concise doctoral dissertation which contained another proof, using the Brouwer fixed point theorem (see chapter 6). In his thesis, “Non-Cooperative Games,” Nash drew the all-important distinction between non-cooperative and cooperative games, namely between games where players act on their own “without collaboration or communication with any of the others,” and ones where players have opportunities to share information, make deals, and join coalitions. Nash’s theory of games—especially his notion of equilibrium for such games (now known as Nash equilibrium)—significantly extended the boundaries of economics as a discipline."

Chapters 1 and 2 free online.