David Blackwell’s Enduring Legacy

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Wednesday, Nov. 15, 2023

David Blackwell’s Enduring Legacy

by Sophia Chen (science communicator in residence, Fall 2022)

In 1976, investment banker John C. Bogle introduced the first index fund available for the general public to purchase. As the story goes, when Bogle was an undergraduate at Princeton University in 1951, he conducted a study that found that stock market indexes, which track the collective price of a group of stocks, tended to grow more than most actively managed funds. Two and a half decades later, as the founder and CEO of Vanguard, Bogle introduced a fund that tracked the price of the S&P 500 index based on that insight.

Investors distrusted Bogle’s first index fund, as Vanguard raised only $11 million for its initial public offering, around one-tenth of its goal. But they eventually came around. Today, index funds have a reputation for delivering reliable returns. People have invested trillions of dollars in index funds, which now make up more than 40% of the money invested in mutual funds and exchange-traded funds.

But this reputation isn’t a fluke. “There’s a theorem that backs up the use of index funds,” says mathematician Rakesh Vohra of the University of Pennsylvania. Assuming that buying an index fund is a zero-sum game — where the seller’s loss is the buyer’s gain, and vice versa — the theorem guarantees that “the average growth rate of your portfolio is going to match the average growth rate of the best stock,” he says.

The theorem goes by many names, as researchers have discovered and rediscovered it in different contexts. Some call it the experts’ theorem: for example, given two recommendations from two experts to buy two different stocks, the theorem lays out a method to combine the two recommendations that will perform almost as well as the best of these two experts. For this story, we will call it Blackwell’s approachability theorem — to highlight David Blackwell, the UC Berkeley mathematician who proved it and published it in 1956.

Stated more abstractly, Blackwell’s approachability theorem presents an algorithm to win a type of game. In a simplified version of the game, a player chooses numbers from the interval [-1,1] with the goal of nudging the average toward zero, while a second player tries to prevent the first player from doing so. In each round, Player 2 offers Player 1 a choice of two numbers, one negative and one positive from the interval. Player 1 chooses one of the numbers. A “toy” version of the theorem guarantees that if Player 1 always chooses the number whose sign is the opposite of the current average, then the average of Player 1’s numbers converges to zero. Thus, as the number of rounds increases, the theorem guarantees that Player 1 will always be able to win the game.

“In other words, no matter what the second player does, the first player has a strategy for pushing the average point to the origin,” says Vohra. More generally, the game applies to any situation where one player aims to encourage the average to converge to any number, not just 0, provided Player 2 always provides options on both sides of that number. The number might correspond to the number of dollars the player seeks to pay for some product, for example.

Blackwell’s approachability theorem applies beyond the domain of scalar numbers in this game and applies to higher dimensions, where both players are choosing vectors rather than single numbers. Player 2 offers Player 1 a choice of two vectors, one from above a hyperplane depending on the current average, and another from below. The theorem offers Player 1 a strategy for nudging the average vector toward zero. In this case, Player 1’s target average vector might correspond to a payoff in multiple currencies. “You can think of the vector as the payoff not being just a few dollars, but five oranges, two apples, 27 bananas,” says Vohra. More generally, the theorem prescribes an algorithm for the first player to choose a vector each round that guarantees that the average will converge to the first player’s desired vector.

In fact, many modern technologies make use of Blackwell’s approachability theorem to operate. The theorem, for example, underpins a form of machine learning known as reinforcement learning. Researchers often use reinforcement learning to train robots to learn how to move autonomously. In reinforcement learning, the robot receives rewards when it behaves in a desired way, and it receives punishments when it does not. The robot’s learning algorithm can be thought of as a Blackwell game.

“Blackwell’s approachability theorem gives us a framework for decision-making in dynamic settings with very low information,” says Vohra.

Researchers are still finding new applications of the theorem. Negin Golrezaei, who works in operations research at MIT, has recently cast a common problem in online marketplaces in terms of this type of game.

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