AI Script for tag like game

I have a small game (like tag). I have multiple bots that wander on the map. At any given time the nearest bot should chase the player and if tagged then the player loses. The catch is only one bot (the nearest) can be n active pursuit at one time.

Any thoughts?

Make an empty gameobject (call it the "Warden" or something [little pun there]). On it put a script with a static variable (call it isChasing:GameObject or something) for who is chasing and a variable for the shortest distance.

Then on each enemy, cache the Warden object in a variable on Awake() (it'll save processing speed). Every .5 second or so (why do it every frame?) have the enemies check distance. If the bot's distance is less than the stored distance it sets a variable on itself to start chasing, and sends its distance to the Warden. Then Warden then sends a message to isChasing to stop chasing, and replaces the object in isChasing with the new bot and the new distance.

This way isn't the best, but it is a good way to get started. Right at the very beginning, there will be a moment when most of the bots think they are chasing (as the distance is whittled down), but then it should work like you want it to.