In A Beautiful Mind, Nash talks about all his friends wanting to pick up the blonde girl but for the best outcome they should all go after her friends. What did this inspire him to write? Was it the Nash Equilibrium?
In the movie A Beautiful Mind, mathematical genius John Nash boils down competitive behavior to the simple terms of sexual conquest. Trying to pick up women at a bar, Nash and several other Princeton graduate students are captivated by a beautiful blonde surrounded by other pretty girls.
“Recall the lessons of Adam Smith, father of modern economics,” says one of Nash’s cohorts. “Individual ambition serves the common good. Every man for himself!”
In a flash of revelation, the brilliant Nash disagrees.
“Adam Smith needs revision,” he says. “If we all go for the blonde, we block each other. So then we go for her friends, but they will all give us the cold shoulder because nobody likes to be second choice. But what if no one goes for the blonde? We don’t get in each other’s way, and we don’t insult the other girls. That’s the only way we win.”
Suddenly more excited about economic theory than sexual opportunity, Nash lapses into an intellectual reverie that summarizes his Nobel Prize-winning dissertation. “Adam Smith said the best result comes from everyone in the group doing what’s best for himself. Incomplete! Incomplete! Because the best result will come from everyone in the group doing what’s best for himself and the group. Adam Smith was wrong!”
Did it really happen that way? The real John Nash could not be reached for comment on this story, but University of Richmond economist Jonathan B. Wight sees at least one flaw in the Hollywood version — not in the movie’s portrayal of Nash, but in Nash’s understanding of Adam Smith.
Wight argues that Smith would be appalled by the suggestion that he advocated selfishness. In his recent novel, Saving Adam Smith, Wight brings the father of modern economics back to life to set the record straight by making the critical distinction between selfishness and self-interest tempered by self-restraint.
Adam Smith would have readily agreed with Nash’s “original” idea. In fact, centuries before Nash devised his equilibrium for non-cooperative games, Smith came up with his own equilibrium for the game of life: “Superior prudence,” he said, “is the best head joined to the best heart.” But over the years, economics instructors have edited out Smith’s “moral sentiments” — leaving only the impression that the “invisible hand” of free markets can magically convert individual greed into mutual benefit.
wolfhnd wrote:Can the lecture continue with some basic math behind the theory?
I heard this tale in India. A hat seller, on waking from a nap under a tree, found that a group of monkeys had taken all his hats to the top of the tree. In exasperation he took off his own hat and flung it to the ground. The monkeys, known for their imitative urge, hurled down the hats, which the hat seller promptly collected.
Half a century later his grandson, also a hat seller, set down his wares under the same tree for a nap. On waking, he was dismayed to discover that monkeys had taken all his hats to the treetop. Then he remembered his grandfather's story, so he threw his own hat to the ground. But, mysteriously, none of the monkeys threw any hats, and only one monkey came down. It took the hat on the ground firmly in hand, walked up to the hat seller, gave him a slap and said, "You think only you have a grandfather?"
This story illustrates an important distinction between ordinary decision theory and game theory. In the latter, what is rational for one player may depend on what is rational for the other player. For Lucy to get her decision right, she must put herself in Pete's shoes and think about what he must be thinking. But he will be thinking about what she is thinking, leading to an infinite regression. Game theorists describe this situation by saying that "rationality is common knowledge among the players." In other words, Lucy and Pete are rational, they each know that the other is rational, they each know that the other knows, and so on.
The assumption that rationality is common knowledge is so pervasive in game theory that it is rarely stated explicitly. Yet it can run us into problems. In some games that are played over time, such as repeated rounds of Prisoner's Dilemma, players can make moves that are incompatible with this assumption.
I believe that the assumption that rationality is common knowledge is the source of the conflict between logic and intuition and that, in the case of Traveler's Dilemma, the intuition is right and awaiting validation by a better logic. The problem is akin to what happened in early set theory. At that time, mathematicians took for granted the existence of a universal set-a set that contained everything. The universal set seemed extremely natural and obvious, yet ultimately several paradoxes of set theory were traced to the assumption that it existed, which mathematicians now know is flawed. In my opinion, the common knowledge of rationality assumed by game theorists faces a similar demise. -K.B.
Giacomo wrote:Speaking of chess, here's Conway's Angel problem. It is played on an infinite chessboard.
Can the Devil, who removes one square per move from an infinite chessboard, strand the Angel, who can jump up to 1000 squares per move?
Everytime I read one of your posts I wish I could go back in time and study mathematics in university instead of Economics. Although reading game theory note again was quite the pleasure. Thanks for the free education Giacomo.
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