The classic board game “Clue” has entertained countless individuals, inviting players to dive deep into a mystery using wit and deduction. But beneath its playful exterior lies a mathematical tapestry woven with probabilities and information. In this exploration, we embark on an information theory analysis of “Clue,” revealing the intrinsic entropy and the enlightening information gains as players receive their cards.

A Board Gamer’s Analytical Journey

I occasionally enjoy trying new board games, and more often than not, I find myself more engrossed in understanding the game’s dynamics than in winning. It’s this analytical streak that I wish to share with you through a series of analyses on different games. Alongside, I aim to include basic codes that mirror my understanding and could potentially serve as rudimentary solutions for these games.

It’s crucial to understand that most board games resist being ‘solved’ through mere basic analyses. While Reinforcement Learning techniques might train an agent to master-level gameplay, it doesn’t equate to having ‘solved’ the game. My analyses predominantly orbit around statistics, probabilities, information theory, and game theory. In the best scenarios, these explorations yield guidelines on gameplay or even cheeky ways to cheat. If fortune favors, I might even stumble upon suggestions to make games fairer or more intriguing!

The Secrets Behind “Clue” Revealed: An Information Theory Analysis

Board games have been a cherished pastime for many, and among the classics, “Clue” (or “Cluedo” in some regions) stands out as a game of mystery, deduction, and strategy. In the manor of Mr. Boddy, players race to uncover who committed the crime, with what weapon, and in which room. While many of us have enjoyed countless hours accusing Colonel Mustard in the library with the candlestick, have you ever stopped to think about the game from an information theory perspective?

PART 1: Information Theroy

What is Information Theory?

Before we dive in, let’s get a brief overview of what information theory is. At its core, information theory is a branch of applied mathematics and electrical engineering that revolves around quantifying information. One of its fundamental concepts is “entropy,” which measures the uncertainty or randomness of a random variable. The higher the entropy, the more uncertain or random the variable is.

Information Entropy of the Solution

In “Clue,” players aim to determine three things:

  1. The suspect (out of 6 possibilities: Mr. Green, Colonel Mustard, Mrs. Peacock, etc.)
  2. The weapon (out of 6 possibilities: knife, candlestick, rope, etc.)
  3. The room (out of 9 possibilities: library, ballroom, study, etc.)

It means we don’t need to know what are the cards in each players hand as long as we can guess the solution. Without any cards in hand (i.e., at the start of the game), the uncertainty is at its peak. To calculate the entropy, we use the formula:

\[H(X) = -\sum p(x) \log_2(p(x))\]

Where \(H(X)\) is the entropy of the random variable \(X\) and \(p(x)\) is the probability of each outcome. Since each suspect, weapon, or room is equally likely at the start, the probability is the reciprocal of the number of possibilities.

Let’s compute the total entropy:

\[H_{\text{total}} = H_{\text{suspect}} + H_{\text{weapon}} + H_{\text{room}}\]

Where:

\[H_{\text{suspect}} = -\sum_{i=1}^{6} \frac{1}{6} \log_2\left(\frac{1}{6}\right)\] \[H_{\text{weapon}} = -\sum_{i=1}^{6} \frac{1}{6} \log_2\left(\frac{1}{6}\right)\] \[H_{\text{room}} = -\sum_{i=1}^{9} \frac{1}{9} \log_2\left(\frac{1}{9}\right)\]

The total entropy \(H_{\text{total}}\) of the solution, before any cards are revealed, is approximately \(8.34\) bits. This means that, without any prior knowledge, it would take us about \(8.34\) bits of information on average to specify the exact solution of the crime.

The Information Gain from Cards

Now, what happens when a player receives their hand of cards? Each card a player receives reduces the uncertainty about the solution. For instance, if you receive the “Mrs. Peacock” card, you can be certain she’s not the culprit. The difference in entropy before and after receiving information (like cards) is known as “information gain.”

Mathematically, the entropy \(H(X)\) of a set of possibilities is defined as:

\[H(X) = -\sum p(x) \log_2(p(x))\]

Where \(p(x)\) represents the probability of each outcome. To understand the information gain from various card distributions, we’ll analyze the following scenarios:

  1. 1 suspect, 1 weapon, 1 room.
  2. 2 suspects, 1 weapon.
  3. 2 suspects, 1 room.
  4. 3 suspects.
  5. 3 rooms.
  6. 1 suspect, 2 rooms.

For these distributions:

\(\text{Scenario 1: } H_{\text{left1}} = -1 \times (\log_2(1/5) + \log_2(1/5) + \log_2(1/8))\) \(\text{Scenario 2: } H_{\text{left2}} = -1 \times (\log_2(1/4) + \log_2(1/5) + \log_2(1/9))\) \(\text{Scenario 3: } H_{\text{left3}} = -1 \times (\log_2(1/4) + \log_2(1/6) + \log_2(1/8))\) \(\text{Scenario 4: } H_{\text{left4}} = -1 \times (\log_2(1/3) + \log_2(1/6) + \log_2(1/9))\) \(\text{Scenario 5: } H_{\text{left5}} = -1 \times (\log_2(1/6) + \log_2(1/6) + \log_2(1/6))\) \(\text{Scenario 6: } H_{\text{left6}} = -1 \times (\log_2(1/5) + \log_2(1/6) + \log_2(1/7))\)

The information gains for these scenarios are the differences between the original total entropy and the entropy after receiving the cards:

  1. 1 suspect, 1 weapon, 1 room: \(0.696\) bits.
  2. 2 suspects, 1 weapon: \(0.848\) bits.
  3. 2 suspects, 1 room: \(0.755\) bits.
  4. 3 suspects: \(1.000\) bits.
  5. 3 rooms: \(0.585\) bits.
  6. 1 suspect, 2 rooms: \(0.626\) bits.

Crucially, given the identical count of suspects and weapons (6 each), they’re interchangeable in our analysis. Thus, results involving suspects can seamlessly be swapped with weapons. However, rooms, having 9 options, yield distinct gains, underscoring their distinct role in the game’s dynamics.

The cards a player receives influence their deduction prowess. For instance, a fortunate player dealt a hand of three suspects or weapons has already acquired 1 bit out of the initial 8.34 bits of uncertainty. Conversely, a less lucky player handed three room cards gleans roughly half of that information.

This indicates that holding more room cards translates to possessing less direct information. However, there’s an intriguing twist in “Clue” that levels the playing field. While players can freely inquire about any weapon or suspect during their turn, querying about rooms requires an added layer of strategy: they must roll the dice favorably and navigate to that specific room before asking. This added dimension can balance out the informational disadvantage tied to room cards.

Probabilities and Expected Information Gain

Beyond individual card distributions, it’s essential to consider the likelihood of each combination occurring. We calculated the probabilities for various card combinations:

  1. 3 suspects: \(\approx 1.5\%\).
  2. 3 weapons: \(\approx 1.5\%\).
  3. 3 rooms: \(\approx 6.3\%\).
  4. 2 suspects, 1 weapon: \(\approx 6.8\%\).
  5. 2 weapons, 1 suspect: \(\approx 6.8\%\).
  6. 2 suspects, 1 room: \(\approx 10.2\%\).
  7. 2 weapons, 1 room: \(\approx 10.2\%\).
  8. 2 rooms, 1 suspect: \(\approx 16.2\%\).
  9. 2 rooms, 1 weapon: \(\approx 16.2\%\).
  10. 1 suspect, 1 weapon, 1 room: \(\approx 24.4\%\).

Given these probabilities, the expected information gain across these combinations is \(\approx 0.708\) bits. This value provides insight into the average reduction in uncertainty a player can anticipate upon receiving their cards.

PART 2: Game Strategy

Conclusion