If You Wanted to Find Someone in an Amusement Park, Should You Roam Around or Stand Still?
It's a conundrum rendered obsolete in the age of cellphones, but still interesting to think about: If you and a friend were at an amusement park (or some other large venue with fixed boundaries) and became separated, should you stand still and let your friend find you or should you roam around looking for them?
A curious Redditor raised this scenario three years ago, evoking a pleasant cascade of enlightened responses.
Answers were limited under the following assumptions: "The other person is constantly and randomly roaming. Foot traffic concentration is the same at all points of the park. Field of vision is always the same and unobstructed. Same walking speed for both parties."
Fascinated readers set about solving the problem by running statistical simulations on square grids of various sizes, from 20 x 20 to 320 x 320, with thousands of trials. In a single trial, two people were randomly placed on the grid. One person always roamed while the other person either roamed or stood still. Each turn, the people moved randomly one space north, south, east, or west, unless they were blocked by a boundary, in which case their movements were limited to open squares. A trial ended when the two people "found" each other by occupying the same grid. Running tens of thousands of trials, the readers tallied the average numbers of moves required for the people to find each other.
In all of the simulations presented, roaming was vastly superior to standing still, reducing the time required for the companions to find each other by at least 50%.
These simulations satisfactorily answer the question as presented, but they aren't exactly realistic. The biggest limitation? They don't accurately describe how someone would search in real life. People don't move completely at random; they would look around in a more methodical manner.
Peter Kerpedjiev, a postdoctoral fellow in biomedical informatics at Harvard Medical School, noticed this glaring issue and sought to remedy it. He repeated the grid simulations but programed the people to engage in two new moving behaviors: scanning and avoiding. Scanners systematically moved across the grid, up and down, left to right, back and forth. Avoiders kept track of where they had been and moved to the least-visited square. If there was more than one least-visited square, they picked one at random to move to.
Kerpedjiev found that employing an avoiding or scanning method to search for a friend who was roaming at random were both about equally successful and roughly 15% faster than searching for someone while moving randomly.
Kerpedjiev also explored beyond the original Reddit prompt to determine optimal "real-world" strategies, to be utilized if, say, you and a friend became separated at a large park and your cell phones both died. If you both had planned for this situation, the best option by far would be to meet at a previously decided upon location. But what if you didn't take this precaution?
"When there is no agreed upon strategy, it appears that the best choice is to walk around avoiding places that you’ve already been and taking a random turn here or there," Kerpedjiev wrote. This pretty much guarantees that you will find each other eventually.
One thing you definitely shouldn't do is both employ a scanning method. In this scenario, it's possible that you would never find each other! One can imagine two friends looping around Disneyland, searching for one another until the end of time (or at least until the park closes).