The problem of interception is hard for non-math people to understand. There is a gap between engineers and policy makers. What follows is an entertainment for a rainy Sunday, in the form of a board game. You will need:
- a checker board
- one red checker
- one black checker
- a coin for tossing
- some curiosity
A critic might deflect the insight as the result of “gross oversimplification.” Don’t let him get away with it. Challenge him instead to offer his own board game.
- Start by placing a single red checker on the edge of the board, in either the black or white square closest to the middle. The red checker is the adversary’s hypersonic warhead, perhaps an Avangard.
- Place a black checker in the corresponding position on the opposite side. This corresponds to your interceptor missile.
A turn of play consists of one movement by the red checker, followed by one by the black checker. But each moves according to different rules. The move of each checker consists of two steps:
- The first step of a movement advances a checker towards the opposite side of the board by one square.
- The second step moves the checker to the right or left, a lateral displacement.
- The red checker must move two steps lateral, to the left or right, according to a coin toss. There is no red “player.” You are playing against the coin.
- The black checker must move one step to the left or right, your choice.
In this game, distances are not accurately represented. It’s like a subway map. It’s on the critic to improve without resorting to the “reality is too complicated” defense.
The victory conditions:
- You win if you manage to place the black checker directly in front of the red checker. This corresponds to an impact, a “hard kill.”
- You lose if the red checker passes the black one, or reaches a parallel position. Your black checker cannot turn around and pursue.
The above is the hypersonic version of the problem. For the ballistic missile version, one rule is different.
- The red checker now proceeds straight across the board with no lateral moves.
- The black checker still makes lateral moves, representing noise in the system. Refine this if you wish, by replacing certain lateral movement with randomness. For example, a lateral move occurs only if two heads come up in a row.
So the game is short. You could prolong it if you happen to have more checkerboards to arrange in a 2×2 square. As simple as it is, it illustrates the basic problem.
If you play the hypersonic version many times, you will sometimes win. On average, the black checker-interceptor will miss the red checker-warhead by a number related to the classic random problem of the “drunk’s walk”, or “random walk.” Since you have the game, you don’t have to do any math. Get a coin out, play the game 10 times, and let me know.
Who knows? If you can figure out a way to win this game consistently, DARPA might be talking to you. In Part 3, we’ll explain how the game relates to the actual problem.
At least it doesn’t cost you any quarters.