A few of John Nash's ideas, developed while he was a graduate student at Princeton from 1948 to 1950, transformed the field of game theory, and led to major developments in economic, political, and biological theories. Nash subsequently turned to other areas of mathematics, where he made enormous contributions. In his early 30s, he started to experience delusions and was diagnosed a "paranoid schizophrenic". He spent time in mental hospitals, enduring treatments painful to read about. His "schizophrenic" episodes were interspersed with periods of "enforced rationality" (his own term (Nash 1995, 278)). Eventually, in his 60s, the latter periods came to dominate the former. In 1994 he shared with Reinhard Selten and John Harsanyi the Nobel Prize in Economics (officially, the "Bank of Sweden Prize in Economic Sciences in Honor of Alfred Nobel").
A biography of Nash has the potential to enlighten the reader in several directions. What is game theory? What was the significance of Nash's contributions? How did he come upon these ideas? What was the significance of his subsequent work in mathematics? How did he come to be diagnosed as a "paranoid schizophrenic"? How did society treat such a person in the 1960s and 1970s? Is there any relation between Nash's creativity and his subsequent "schizophrenic" behavior?
Sylvia Nasar's A beautiful mind (Simon and Schuster, 1998) explores some of these questions, and at its best provides considerable enlightenment. Its greatest success is a discussion of Nash's "illness": the treatments he had to endure, the support of his friends, his ambivalence to his return to "rationality". Nasar went to considerable lengths to find out what happened to Nash during this period; her discussion is sensitive and thought-provoking. Her style is reportorial rather than analytical, but she raises many significant issues.
Nasar also does a good job of exposing both the Econometric Society's handling of Nash's nomination to be a Fellow, and the machinations behind the award of his Nobel prize. If I interpret footnote 101 in Chapter 48 correctly, the credit for unmasking the Nobel committee is due to a reporter for a Swedish newspaper, but Nasar conducted many additional interviews and learned a lot more about what happened.
The book is much less successful in conveying the significance of game theory, and of Nash's contribution. Worse, many pages in the first half of the book are devoted to reports of gossip, about both Nash and other game theorists and mathematicians. Some of this gossip is probably true, but it is very largely unrelated to any interesting questions. In the words of John Milnor, it is "a drastic violation of the privacy of its subject" (1998, p. 1330).
Nasar's main source of information was a large number of interviews (the jacket says "hundreds"). Nash himself did not cooperate—the biography is "unauthorized"—though his ex-wife and constant companion, Alicia Nash, and other family members and friends did. Nasar seems to have gone to some lengths to double-check stories she was told, but inevitably sometimes her sample of sources is one-sided. And even when it isn't, depending on people's memories of events 30 or 40 years ago is problematic, especially when these memories concern a person whose personality evoked strong reactions.
Nasar's technique is to uncover all the information she can, and uncritically present it. Because she is an expert at ferreting out information, the result is that there are many details. She seems too often to paraphrase what someone told her, without subjecting it to careful evaluation.
She writes, for example, that in high school Nash "often [succeeded] in showing, after a teacher had struggled to produce a laborious, lengthy proof, that the proof could be accomplished in two or three steps" (p. 34; my emphasis). We can be pretty sure that most, if not all of the italicized words are exaggerations. No proofs in high school math are laborious and lengthy; no teacher "often" "struggles" with such a proof. Perhaps a teacher once gave an argument that Nash pointed out could be shortened.
As another example, she recounts as if they were facts some of the many stories about John von Neumann's exceptional mental abilities. By contrast, Poundstone (1992, p. 32) writes:
"Separating fact from legend with von Neumann is a maddening task. He was a genius, a practical joker, and a raconteur, all of which tended to produce stories that sound too pat, too anecdote-like, to be true. I ran the best-known anecdotes by several surviving associates. All had heard of the stories; few could supply any specifics—except to tell new anecdotes. In general the people who actually knew von Neumann were less skeptical of the stories than I was. So all right: whether streamlined in the retelling or not, these stories give an idea of how von Neumann was perceived by others and possibly himself."
That's a perfect introduction: von Neumann was a genius, and the stories are fun to hear, but how much of them is true is impossible to tell. Could he really divide two eight-digit numbers in his head at the age of six? [Note] That doesn't seem likely. Had he mastered calculus by the age of eight? Quite possibly.
Certainly Nasar had little alternative but to start by interviewing people who know Nash. But it would have been wise for her to have treated what people told her more critically. The book contains too much of her sources, too little of her.
Her own voice comes through most strongly in a few fictionalized passages. These passages do not fit well with the rest of the book; they don't fit well beside the sections based closely on her sources. For example, at the start of Chapter 38 she describes Nash and Alicia's departure from New York in 1959 as if she had been a witness. Unlike most other passages in the book, this one has no footnote, and its claims—what they watched as the Queen Mary set sail, the fact that they were both lost in thought, for example—appear to be purely conjectures on Nasar's part.
The discussions of game theory are disappointing; they seem unlikely to convey the essence of game theory clearly to a neophyte. On page 96, Nasar gets off to a shaky start in a discussion of the notion of Nash equilibrium by claiming that the "entire edifice of game theory rests on two theorems: von Neumann's min-max theorem of 1928 and Nash's equilibrium theorem of 1950". This statement errs in several respects. First, von Neumann's theorem plays an extremely minor role in modern game theory (even though it has great historical significance). Second, while the idea of a Nash equilibrium is absolutely central to game theory, the theorem is much less significant. (It does not, as Nasar claims on page 97, show that "every noncooperative game ... has at least one Nash equilibrium point", but only that every finite noncooperative game has a Nash equilibrium.) Third, a significant part of game theory is concerned with coalitional games, in which the notion of Nash equilibrium is not directly relevant.
The subsequent paragraphs draw upon Dixit and Nalebuff's (1991) excellent description. However, when Nasar departs from their words, the descriptions become a bit garbled. What does it mean to say that "Players look for a set of choices such that each person's strategy is best for him when all others are playing their best strategies"? Each player independently chooses a single strategy—no player chooses a set of strategies. Further, each player's best strategy depends on the others' strategies—so what does "when all others are playing their best strategies" mean? (This phrase should be replaced by "given the other players' strategies".)
In the following paragraph, Nasar claims that the search for equilibrium "should begin by looking for dominant strategies and eliminating dominated ones". It's true that when searching for a Nash equilibrium it's a good idea to first eliminate dominated strategies and find any dominant strategy. But it seems that this is not what Nasar means. The following sentences remark that in general there are no dominated strategies, so "one must turn to Nash's construct". Thus the intended meaning of the sentence seems to be that when looking for the likely outcome of a game one should start by eliminating dominated strategies, and, if this doesn't yield a single outcome, one should turn to the notion of Nash equilibrium. The impression that this gives is surely misleading. The ideas behind the elimination of dominated strategies and the notion of Nash equilibrium are different; we don't try one concept and then, if it doesn't work, turn to the other.
In the next paragraph on Nash equilibrium, Nasar mentions mixed strategies without explaining them. Her subsequent claim that games without Nash equilibria are "relatively rare" conflicts with her earlier (erroneous) claim that Nash showed that all games have Nash equilibria.
A few figures would definitely help; it is amazing that a book about Nash the game theorist lacks a single two-by-two strategic game! A few such examples would make the idea of Nash equilibrium transparent. Even the Prisoner's Dilemma gets only a verbal description. And how can anyone understand the game of Hex without a diagram of the board?
The discussion of Prisoner's Dilemma on pages 118–122 repeats the definition of a Nash equilibrium as a strategy profile in which each player follows "his best strategy assuming that the other players will follow their best strategy." Again the second "best" is confusing; in addition, the last word should be "strategies". But of course the Prisoner's Dilemma has little to do with Nash equilibrium—each player has a dominant action, so her best action is independent of the other player's action. (Nash's work definitely led people to think about the Prisoner's Dilemma, but that is another issue.)
The next paragraph mentions Dresher and Flood for the first time in the text in connection with the game, without making clear that they are the "two RAND mathematicians" mentioned on the previous page. A few paragraphs later, Nasar reports without comment Nash's criticism of Dresher and Flood's experiment; at a minimum it should be pointed out that Nash was absolutely correct!
To ask that Nasar describe to us Nash's mathematical work outside game theory seems, to a game theorist at least, more demanding. Nevertheless, what is written should at least be accurate. In a review of Nasar's book, John Milnor writes "Mathematical statements and proper names are sometimes a bit garbled, but the astute reader can usually figure out what is meant" (Milnor, p. 1329). Probably that's true if the astute reader is Milnor (a distinguished mathematician). But for the merely normally astute reader, it's tough....
The book is a bit rough around the edges. There are many errors and oddities, of which I will give a small sample. On page 20 Nasar writes that at the time when Nash's ideas were becoming influential, Nash himself remained "shrouded in obscurity". (She returns to this point on page 354.) One piece of evidence she cites is the absence of a biographical sketch of him in The New Palgrave Dictionary of Economics. But the reason for this absence is not Nash's obscurity, but the policy of the editors of The New Palgrave to restrict biographical sketches of living economists to those born before 1916. (Nash was born in 1928.)
On page 58, Nasar writes that Solomon Lefschetz made the Annals of Mathematics into "the most revered mathematical journal in the world". But the attached footnote cites Lefschetz's obituary in the New York Times as saying only that he developed the Annals of Mathematics into "one of the world's foremost mathematical journals" (my italics).
On page 107 Nasar writes that of the new tools developed at the RAND Corporation, including operations research, linear programming, dynamic programming, and systems analysis, "game theory was far and away the most sophisticated." Even without the "far and away", many researchers would strongly disagree with this claim. On page 111, she claims that Lloyd Shapley, who was maintaining a twenty-five-hour sleep cycle (page 101) was "rarely seen before midafternoon", two pieces of information that appear to be inconsistent. On page 122, she quotes a passage from Luce and Raiffa's (1957) landmark book—but she quotes not directly from the book, but from the passage as quoted by Poundstone (1992) (and omits a word in the process).
On page 150 she claims that, in "an essay on Nash's mathematical work", Milnor (1995) "adopts the ... view ... that Nash's work in game theory was trivial compared with his subsequent work in pure mathematics." I don't see how Milnor's paper can be interpreted in this way. He starts by noting that "to some" the work that earned Nash the Nobel prize "may seem the least of his achievements". But then he immediately writes that
"Nevertheless, I applaud the wisdom of the selection committee in making this award. It is notoriously difficult to apply precise mathematical methods in the social sciences, yet the ideas in Nash's thesis are simple and rigorous, and provide a firm background, not only for economic theory but also for research in evolutionary biology, and more generally for the study of any situation in which human or nonhuman beings face competition or conflict."
He goes on to approvingly quote Ordeshook (1986, 118), who writes that "The concept of a Nash equilibrium n-tuple is perhaps the most important idea in noncooperative game theory." Approximately 2/3 of the remainder of Milnor's paper is devoted to Nash's work in game theory, and 1/3 to his work in pure mathematics. In Nasar's quotation from the paper on page 150, she omits two sections (without indicating their omission), one of which is critical. Milnor writes "Evidently, Nash's theory was not a finished answer to the problem of understanding competitive situations." Nasar omits the sentence that follows: "Rather, it was a starting point, which has led to much further study during the intervening years". At a minimum the omission changes the tone of the paragraph. How can the paper be interpreted as "adopting the view" that Nash's work on game theory is trivial?
At the start of Chapter 38, Nasar describes Nash and Alicia's crossing of the Atlantic in July 1959 as "restful". When checking the footnote, one learns that the source for this adjective is Nash himself, in a postcard to his mother; given his state of mind at the time, it's not clear that this description should be taken literally. The next sentence claims that the "beauty of Paris overwhelmed them just as it had a year earlier, 'verdure everywhere ... with the giant blue Paris pigeons bolting above it, two by two'." The implication is that that's what Nash or Alicia wrote or said. But no—the attached footnote cites the book Paris Journal 1944–1965 by Janet Flanner (1966). It is an entry for April 7, 1959 about the arrival of Spring; it has no connection to Nash or Alicia, whom, as far as I can tell, the book does not mention. A minor point ... but I don't understand the narrative. The quotation is not related to Nash and Alicia, so why include it? A reader who doesn't consult the footnote (like all the others, annoyingly at the end of the book instead of at the bottom of the page) will presumably think that the quote is from one of them; a reader who does consult the footnote will probably also assume that Flanner is writing about Nash. (Only a slightly deranged reader who obtains Flanner's book and finds the quotation on page 418 (which isn't easy, because Nasar doesn't give a page reference and the index isn't helpful) will learn the truth.)
On page 375 Nasar claims that the "latest generation of texts used in top graduate schools today all recast the basic theories of the firm and the consumer ... in terms of strategic games." Not so—these theories are definitely not game-theoretic.
There are many errors like these. Do they matter? I can see an argument that they do not: what matters is the big picture, and the details don't alter it. But they are indicative of a sloppiness that worries me; many are more than trivial "slips". Further, I can notice errors only in material that I know something about, and a good deal of the book covers material about which I previously knew nothing. Are there as many lapses and mistaken interpretations in the remaining material? Are there more serious ones?
Just as Nash managed to perfectly express his pathbreaking ideas in the 28 pages of his thesis (1950), so he eloquently discusses his life in his terrific four and a half page autobiographical essay (1995). In fact, I am tempted to argue that these pages suffice.
- Dixit, Avinash, and Barry Nalebuff (1991), Thinking strategically. New York: Norton.
- Flanner (Genêt), Janet (1966), Paris Journal, 1944–1965. London: Victor Gollancz.
- Luce, R. Duncan, and Howard Raiffa (1957), Games and decisions. New York: John Wiley and Sons.
- Milnor, John W. (1995), "A Nobel prize for John Nash", Mathematical Intelligencer 17 (3), 11–17.
- Milnor, John W. (1998), "John Nash and 'A beautiful mind'", Notices of the American Mathematical Society 45, 1329–1332.
- Nash, John F., Jr. (1950), Doctoral dissertation, unpublished.
- Nash, John F., Jr. (1995), "John F. Nash, Jr.", pp. 275–279 in Les prix Nobel 1994, Stockholm: Almqvist and Wiksell.
- Ordeshook, Peter C. (1986), Game theory and political theory. Cambridge: Cambridge University Press.
- Poundstone, William (1992), Prisoner's dilemma. New York: Doubleday.
Nasar (p. 80) attributes this story to Poundstone (1992), but in fact he does not report the age at which von Neumann is said to have had this skill; Halmos (1973, 383) is one source of the story. [Return to text]