#### Well, not exactly. But it does offer a new way of thinking about the problem.

How close is too close?

In an age of social distancing, it’s an ever-present question. And, unfortunately, opinions differ. The impatient customer behind you in the grocery store line may have no problem violating your sacrosanct six-foot bubble, for instance.

But even for reasons that have nothing to do with the pandemic, we are often at odds on how close we want to be to each other. Just ask the introvert who finds herself hiding from an extroverted roommate, or the parent who craves a moment alone from the child who only wants to cuddle.

To Alvaro Sandroni, a professor of managerial economics and decision sciences at the Kellogg School, these conundrums call to mind a well-known parable from 19th century German philosopher Arthur Schopenhauer. In his “hedgehog’s dilemma,” Schopenhauer imagined a group of hedgehogs trying to keep warm on a cold day. “And so they start to get close to each other to warm themselves,” Sandroni explains, “but then they spike each other. And so they move apart because of the spikes. But then they get cold again.”

Schopenhauer saw this as a metaphor for human intimacy: we all crave connection, yet the closer we get, the more likely we are to hurt one another. “He thought that there was no solution,” says Sandroni. “You just trade one kind of unhappiness for another.”

But what if there was a way to make at least one person a bit less unhappy?

In a recent paper, Sandroni used game theory to take a fresh look at this age-old problem. What he found largely confirms Schopenhauer’s pessimistic conclusion: Unless both parties prefer to stay the exact same distance apart, there’s no simple strategy for “solving” this problem. One person will always wish that she would have chosen some other position.

However, Sandroni’s math does uncover an intriguing new wrinkle. If the person who prefers to stay farther away can somehow randomize her position relative to her counterpart—say, if there’s a 50–50 chance that she’ll be 10 feet to the left or 10 feet to the right—then her counterpart will have to split the difference, giving her some of the space she craves. And in this scenario, both parties will have done the best they can do under the circumstances.

While randomizing one’s position may not be possible in the real world, Sandroni’s findings nonetheless offer some guidance for those looking to keep their distance from someone who prefers closeness. “The idea is that in games like this, you have to be unpredictable.”

## Modeling People’s Preferences for Distance

In the paper, Sandroni lays out a game with two players. The setup is simple: Each player has some optimal amount of distance they want to remain from the other; the closer they end up to that optimal distance, the happier they will be. Both players know each other’s optimal distance, as well as their own. Each player must unilaterally choose where to stand on a straight line. They make this decision at the exact same time, and there’s only one round, meaning they can’t simply wait and see what the other player does before choosing their position.

However, because each player knows the other’s preference for closeness, they can each make a guess as to what the other will do.

Sandroni used mathematics to find the “solution” to this game—that is, the position that each player would ideally choose, given the other player’s choice.

“The idea is that in games like this, you have to be unpredictable.”

In game theory, there are two kinds of solutions. Some games have what theorists call a “pure” strategy solution, where one can define the precise move each player would make in order to achieve the best outcome possible, given what the other player does. For instance, tic-tac-toe has a pure-strategy solution; you could write an algorithm that always wins or draws, no matter what moves your opponent makes.

In the paper, Sandroni proves that the distance game only has a pure solution when both players prefer being the exact same distance apart. But as long as one person wants to be close, and the other wants to be further away, there’s always some “better” distance that at least one player could have chosen. “Because if you got to decide, you would decide to be closer,” he explains. “But if I got to decide, I would decide to be further away. This was Schopenhauer’s point.”

However, some games have the second kind of solution, known as a “mixed” strategy. Mixed-strategy solutions don’t dictate exactly what each player should do, but rather assign probabilities to a number of possible moves.

Rock-paper-scissors is one such game: the mixed-strategy solution is that both players choose between rock, paper, or scissors completely randomly. Choosing randomly is optimal because all three choices have some other single choice that always beats them. “You don’t want to reveal your strategy, because if you do, the other person wins,” Sandroni says.

By randomizing, you become unpredictable, meaning that your opponent only has a one in three chance of beating you, no matter what strategy she chooses.

The same idea is behind the mixed-strategy solution to the hedgehog’s dilemma.

## Why Randomizing Your Position—Or Staying Put—Is Your Best Move

Imagine that you’re one of the players in Sandroni’s game. You prefer to stay 10 feet away from your opponent, but your opponent prefers to be just 5 feet away from you.

In this case, Sandroni explains, your optimal strategy is to randomly choose between going 10 feet to the left of where you expect your opponent to be and 10 feet to the right. Your opponent, meanwhile, will simply go to the place halfway between those two spots.

Proving that this is the unique solution to the game was a formidable mathematical task. But the intuition is straightforward.

For the person who prefers to be further away, the optimal strategy is analogous to rock-paper-scissors: stay unpredictable so that your opponent cannot pin you down.

If the first player’s position is unpredictable, that makes it risky for the second player to guess where she will go. After all, if she bets the first player will go 10 feet to the left, but guesses wrong, she could end up 15 feet away, a terrible outcome. Instead, she’s better off hedging her bets and going exactly halfway between the first player’s two possible options (since she will have inferred that the first player will randomize between these positions). That way, she’ll only be 10 feet away, no matter which direction the first player chooses.

“So the best thing I can do is to stay put in the middle,” Sandroni explains.

“So this is what I call ‘oppression by randomization,’ in that you get whatever you want and I have no say, basically.”

Of course, this “solution” doesn’t mean both parties are equally happy with the outcome. The second player is guaranteed to be disappointed with her final position, which is much further from the other player than she would like. But given that the first player is unpredictable, staying halfway between the two possible spots is nonetheless the best choice that the second player can make.

“So this is what I call ‘oppression by randomization,’ in that you get whatever you want and I have no say, basically,” Sandroni summarizes. “That’s how this whole thing ends.”

Sandroni points out that this solution makes the game fundamentally different from games like rock-paper-scissors. Whereas randomizing in rock-paper-scissors gives every player an equal probability of winning, in the distance game, randomizing allows one player to essentially corner the other—even though the rules of the game grant both players the same amount of power.

## A Parable for Social Distancing

Sandroni sees the study as a sort of mathematical parable. “It’s not meant to be something that you’re going to take off the shelf and apply,” he says.

Indeed, the findings have limited practical use, unless we’re willing to set aside pesky constraints of time and space to imagine that people can instantly apparate from place to place and somehow randomize their position relative to someone else.

But in the era of social distancing, Sandroni can see something like this game playing out in real life.

“Imagine a movie star, or somebody that people just want to be close to,” he says. When that person ventures to the grocery store, adoring fans want to come up close for a selfie—but the celebrity wants to maintain a safe social distance.

“So what would that person do? You can’t just stay there,” he says. “The only way to do it will be to be unpredictable”—perhaps by moving erratically around the grocery store, for example.

The general lesson for socially wary people (or hedgehogs) looking to keep their distance in a touchy-feely world? “You would like people not to know where you are.”

Jake J. Smith is a research editor at *Kellogg Insight*.

Sandroni, Alvaro. 2019. “The hedgehog’s dilemma.” *Journal of Mathematical Economics*.

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*This article first appeared in www.insight.kellogg.northwestern.edu*

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