For example, in the 100-player restaurant game, there are 2.
Even if we start out knowing only our own preferences and we can’t communicate our strategies before the game, it won’t take too many rounds of missed connections and solitary dinners before we thoroughly understand each other’s preferences and, hopefully, find our way to one or the other equilibrium.
But imagine if the dinner plans involved 100 people, each of whom has decided preferences about which others he would like to dine with, and none of whom knows anyone else’s preferences.
For example, if 100 of us are each choosing one of two routes for our morning commute, you probably don’t care of them.
Economists could use such arguments to justify why Nash equilibrium might be attainable for particular games.
Nash proved in 1950 that even large, complicated games like this one do always have an equilibrium (at least, if the concept of a strategy is broadened to allow random choices, such as you choosing the Chinese restaurant with 60 percent probability).
But Nash — who died in a car crash in 2015 — gave no recipe for how to calculate such an equilibrium.
But the new result implies that such justifications must be made on a case-by-case basis; there’s no killer argument that will cover all games all the time.
What’s more, even though many games that have evolved along with civilization may be amenable to such simplifications, the Internet era is giving rise to all kinds of new many-player games, from dating sites to online stock trading.
“They use these equilibrium concepts, and they’re analyzing them as if people will be at equilibrium, but there isn’t always a satisfying explanation of why people will be at Nash equilibrium as opposed to just groping around for one.” If people play a game only once, it is often unreasonable to expect them to find an equilibrium.