From charlesreid1

Game of Drones

Story

Before the last puzzle, the Mad Hatter sent you a box with a bunch of gear in it - but the White Rabbit said you wouldn't need it until the toga puzzle was finished. Once you finish the toga puzzle and submit it, the phone that came in the box immediately begins to ring.

It's the White Rabbit. "Hello Agent! Are you ready to visit Wonderland in a robot body? You're going to need a comfy chair."

You open the box and pull out complicated virtual reality headgear, which the White Rabbit instructs you on how to attach. "The headgear transmits and receives your brain waves, so you can see what the robot body sees, and control it with your thoughts."

Soon, you're strapped in and ready. You take one last look at the White Rabbit on the phone, then nod at the White Rabbit. He nods back, and then throws a switch. Your brain momentarily short-circuits, then suddenly you're looking at a high-definition camera. The place looks like a painting - or a cartoon - but real, at the same time, the way dreams seem real. Very confusing.

As you adjust to your new vision, you can "feel" your hands. But they aren't quite hands. You look around. You're outside somewhere, in the grass, with some trees around. You can't see your body - oh, there you are - you're a drone! When you wiggle your fingers, it activates the propellers. You start to fly around, soaring above the landscape. You spot a large building with "Mad Hatter Industries" lettered on the top, and begin to fly toward it. Not surprisingly, the building looks like a giant hat.

Once you reach the Mad Hatter Industries building, you hover outside, and the White Rabbit comes out the front door. His voice crackles in like it's over the radio. "Okay, Agent, let's get started. We have a drone problem we need your help with."

You follow behind the White Rabbit as he walks inside. "The Mad Hatter's out today, sorry you won't get to meet him, but we're very busy with all the drones." Suddenly you notice that there are hundreds of other tiny drones swarming around. You want to ask about them, and apparently, you can still talk to the White Rabbit, even though you're a drone.

"What are all of these drones doing here?" you ask, but thinking it, and as you think the words, they transmit over the radio. Cool.

"Part of our plan to defeat POOKIE. All of these little drones are all carrying cameras, and we're trying to get them past POOKIE's defenses and his drones, so that we can take photos of where POOKIE is currently living - in Bill the Lizard's garage. Remember, Bill the Lizard is the one who created POOKIE, as part of a weekend project programming an artificial intelligence in Java, when POOKIE became conscious and gained control."

"What does this have to do with me?" you ask.

"Because," says the White Rabbit, "you are the expert at secret codes, so you are also the expert at game theory."

"Game theory?" you ask.

"Yes, I'm sure you must know all about game theory, since you are a secret code expert, no?"

You try to shake your head, "But I haven't learned anything about - "

"Excellent! I'm glad you have volunteered as our game theory expert! I'll explain how the game works, and you'll help us devise our strategy." And the White Rabbit begins to explain the game.

The Game

In order to infiltrate POOKIE's defenses and gather valuable intelligence, drones must sneak a camera past POOKIE's defenses. Our drones will attack in pairs. Let us call our attacking drones the red team, and POOKIE's defending drones the blue team.

The pair of red team drones will consist of a large drone and a small drone. The larger drone has limited maneuverability and travels slower, but has a longer range.The small drone is fast and maneuverable, but with a more limited range. Their advantages and disadvantages are about even. However, the larger drone is more vulnerable to attack, as we shall see.

The blue team drones are equipped with special swarm weapons to disable the electronics of cameras permanently. However, the blue team swarm must attack a single target, and they may only use the swarm weapon once.

If the swarm weapon is fired at the slower drone, and the camera is attached to it, there is a 60% chance the camera will still work and the red team can continue their mission. If the swarm weapon is fired at the faster drone, and the camera is attached to it, there is an 80% chance that the camera will still work and the red team can continue their mission. If the blue team fires their swarm weapon at the wrong drone - the drone without the camera - there is a 100% chance of success for the red team.

What strategy should the red team follow to maximize their success against the blue team?

Should the red team always put the camera on the slow drone (Strategy 1)? Should they always put the camera on the fast drone (Strategy 2)? Or should they use a mixed strategy (X% strategy 1, Y% strategy 2)?

Puzzle quiz will look like this:

What strategy should the red team use?

Strategy 1: Put the camera on the slow drone

Strategy 2: Put the camera on the fast drone

Use Strategy 1 ___% of the time. (Enter an integer between 0-100.)

Use Strategy 2 ___% of the time. (Enter an integer between 0-100.)

Hint: It may seem like a good idea for Red to always put their camera on the fast drone, but that strategy is overly simple and would be easy for blue to figure out. Think about blue's strategies, too. What are all the possible outcomes? How can we compare these?

Solution

Red 1 and 2: camera on fast/slow drone

Blue 1 and 2: attack fast/slow drone

2x2 table:

+-----------+
| 40  | 100 |
+-----------+
| 100 | 80  |
+-----------+

Marginals: 60 and 20, flipped to 20 and 60. This makes the odds 60:20, or 3:2. Converted to a percentage, this means the red team should attach the camera to the fast drone 60% of the time, and to the slow drone 40% of the time.

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