Game Theory of Staying 6 Feet apart
In a day and age of a pandemic, there are guidelines to help prevent spread, such as wearing a mask or staying 6 feet apart. Adhering to these guidelines is a top priority for some, but for others, living life as if there is no pandemic seems to be the preferred course of action. So how do you ensure that someone doesn’t get too close to you while you’re in line at the grocery store, or step into your 6 foot bubble on the sidewalk? An article written by Jake J. Smith in the Kellogg Insight discusses the game theory of staying 6 feet apart. Smith’s article is based on the research of Alvaro Sandroni.
If you want to stay 6 feet away from other people around you, what is your best move? In the game that Sandroni lays out in his paper, there are two players and they each have an ideal distance they want to be from each other. At the exact same time, they each have to choose a spot to stand on a straight line. Since they both make this decision at the same time they don’t know where the other player is going to choose to stand. If someone near you also wants to be 6 feet away then the solution is simple, both you and the person stay 6 feet from each other. This is a Nash equilibrium as it is a best response for both players. However, if you want to keep 6 feet of distance but someone nearby is fine with 2 feet of distance, there is no pure-strategy Nash equilibrium. If you end up 2 feet away, the other person is happy but you are not, and if you end up 4 feet away you are both unhappy, so your best strategy would require some degree of randomization.
In Smith’s article, he gave the example of two players A and B, where player A’s optimal distance is 5 feet and player B’s optimal distance is 10 feet. Player A’s two strategies would be to go 10 feet to the right or left of where he thinks player B will go, and player B’s strategies would be to go 5 feet to the right or left of where he thinks player A will go. I drew a diagram to capture this game:
Players A and B would both want to randomize both of their strategies with a 50% chance of using each of their strategies. This isn’t necessarily the most practical way of thinking about how to stay 6 feet from someone since there are many more factors in real life that are not considered in this game, but it is fun to think about.