As I write this, I'm taking a break from searching for my recently wandered off and phoneless wife. I wish to find her as quickly as possible, but I'm currently experiencing some inner turmoil. Should I continue on this wife-hunt or is it more time-efficient to remain stationary and wait until she walks by? Do two confined random-walks meet quicker than a single random-walk finding a stationary point? Wait a second, I can write a script (click here!) that simulates finding my lost and (presumingly) scared wife!
In order to keep things simple, I'll assume this mall is a bounded square grid where my wife and I are initially assigned to a random location. Then, during each iteration, a non-stationary person walks a single square in any horizontal, vertical or diagonal direction. Whenever my wife and I are in the same square, I will record the total number of squares traveled and reset the simulation. This process repeats for two scenarios: 1) We're both randomly moving around 2) One person remains stationary, until representative total-distance averages are reached.
The diagram below shows a typical example for each scenario:
After running this simulation for a few different sized grids, I confirmed that two people randomly walking around is approximately 1.6x faster! Now, I would typically follow this result, but my wife already found me like 45 minutes ago and she's been patiently waiting for me to finish this blog post.