One problem with various Nash equilibria is that they are not always unique. Selten applied stronger conditions to reduce the number of possible equilibria and to eliminate equilibria that are unreasonable economically. In 1965 he introduced the concept of “subgame perfection,” his term for this winnowing down of possible equilibria. “At that time I did not suspect that it often would be quoted, almost exclusively for the definition of subgame perfectness,” Selten wrote in his Nobel autobiography.1
An example of subgame perfection is Selten’s “chain store paradox.” Imagine that firm A has a number of chain stores in various locations and that firm B contemplates entering in one or more of these locations. If firm A threatens a price war, then firm B might be dissuaded from entering, not just in a particular market, but in any of A’s markets. In that case, it could well be worthwhile for A to threaten and, indeed, to carry out a price war in a single market. Knowing this, B does not enter. This is a Nash equilibrium. But Selten also saw that another Nash equilibrium was for B to enter. The reason: B would realize that A would have losses in each market in which B entered and A carried out a price war. These losses, cumulatively, would not be worthwhile. By looking forward and reasoning backward, B realizes that A will not carry on a price war, and therefore B enters. Which Nash equilibrium is the “right” one? Selten argued that it is the one where B enters because B thought through the whole sequence and realized that, from A’s viewpoint, a price war would be irrational. B’s strategy of entry and A’s strategy of avoiding a price war are “subgame perfect.”
Interestingly, this discussion of game theory was carried on long after economist Aaron Director, in 1953, offered a similar argument that “predatory pricing” is an irrational strategy. John S. McGee, following Director’s hunch, examined the trial transcripts of U.S. v. Standard Oil New Jersey, which was thought to be one of the best-documented cases of “predatory pricing”—that is, pricing below a rival’s costs to put the rival out of business. Sure enough, McGee found no evidence that Standard Oil of New Jersey had ever engaged in predatory pricing.2 Indeed, the U.S. Supreme Court has accepted this reasoning and, in judging the antitrust laws, has given wide latitude to firms to cut prices in a price war.3
Selten realized, though, that there can be situations in which even the requirement of subgame perfection is insufficient. This led him to his next major contribution, the “trembling-hand” equilibrium. Imagine that each “player” thinks there is a small probability that a mistake will occur—that is, that someone’s hand will tremble. A Nash equilibrium in a game is “trembling-hand perfect” if it obtains even with small probabilities of such mistakes.
Selten was born in Breslau, Germany, now the city of Wrocław, Poland. Growing up half-Jewish, he learned an important lesson from the virulent anti-Semitism he saw around him. As Selten put it, “I had to learn to trust my own judgment rather than official propaganda or public opinion.”4
Selten became interested in game theory after reading about it in Fortune in the late 1940s. He earned his master’s degree in mathematics at the University of Frankfurt in 1957 and his Ph.D. at the same school in 1961. He was a full professor of economics at the Free University of Berlin from 1969 to 1972. This was a period of student “unrest,” which, wrote Selten, “made teaching difficult and sometimes impossible.” For other reasons, though, he moved to the University of Bielefed in 1972 and stayed there until 1984, when he moved to the University of Bonn. He often collaborated with fellow future Nobel winner John Harsanyi.
Available online at: http://nobelprize.org/nobel_prizes/economics/laureates/1994/selten-autobio.html.
John S. McGee, “Predatory Price Cutting: The Standard Oil (N.J.) Case,” Journal of Law and Economics 1 (October 1958): 137–169; see esp. p. 138, fn. 2, where McGee thanks Director for his strictly logical reasoning that predatory pricing makes no sense.
In Matsushita Electric Industrial Co. v. Zenith Radio Corp. (1986), the U.S. Supreme Court stated that “predatory pricing schemes are rarely tried, and even more rarely successful.”