The Game of Clue: An Information Theory Perspective

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:

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:

Where:

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:

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.

layout: post comments: true title: “The Game of Clue: An Information Theory Perspective” date: 2023-11-02 12:00:00 tags:

• board-games
• probability
• information-theory
• entropy
• cluedo

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?

Modeling note. Throughout, I analyze the standard edition with 6 suspects, 6 weapons, and 9 rooms. When I discuss “a hand of 3 cards,” you can think of the 6‑player game where everyone is dealt 3. I’ll generalize later to other player counts.

PART 2: Bayesian Updates During Play

Each suggestion and response generates information. Two simple building blocks cover most updates:

(A) Value of eliminating a single candidate

If a category has (N) remaining possibilities and you eliminate one (e.g., someone shows you Candlestick), the entropy drop is [ \boxed{,v(N)=\log!\frac{N}{N-1}, .} ] Some useful values:

Early in the game, eliminating a suspect/weapon (typically (N=6)) is worth (\approx 0.263) bits, whereas eliminating a room ((N=9)) is only (\approx 0.170) bits.

(B) “Controlled” suggestions (forcing what gets shown)

A time‑honored tactic: include two cards you hold in your suggestion. Example: You hold Rope and Kitchen; you suggest (Miss Scarlet, Rope, Kitchen). The only card anyone else could possibly show is Miss Scarlet—so you either see Miss Scarlet (eliminate that suspect) or, if nobody can show it, you deduce Miss Scarlet is in the envelope.

If the targeted category currently has (N) possibilities, the expected information per controlled suggestion is [ \boxed{,I_{\text{ctrl}}(N) = \frac{1}{N}\log N ;+; \frac{N-1}{N},\log!\frac{N}{N-1}, .} ] Values you’ll actually encounter:

Takeaway: Controlled suggestions are 2–3× more informative than uncontrolled ones (next) and get even better as a category shrinks.

(C) Uncontrolled suggestions (when you can’t force the show)

If you can’t include two cards you own, you’ll often be shown some card among the suspect/weapon/room you stated—but you don’t control which. A rough, symmetric approximation when you hold none of the three is: [ I_{\text{unctrl}} \approx \underbrace{\frac{1}{SWR}\log(SWR)}{\text{rare: all 3 are the solution}} ;+; \Big(1-\frac{1}{SWR}\Big)\cdot \frac{v(S)+v(W)+v(R)}{3}. ] Early game with ((S,W,R)=(6,6,9)): (I{\text{unctrl}}\approx \mathbf{0.257}) bits—far below controlled suggestions ((\approx 0.50)–(0.65) bits). Control, whenever possible.

PART 3: Strategy — Turning Bits into Better Play

3.1 Choose what to learn (target the smaller live set)

Because (I_{\text{ctrl}}(N)) decreases as (N) increases, it is more informative to target the category with fewer remaining candidates (counter‑intuitive but true). Why? The chance to resolve the entire category in one go is (1/N), and that rises as (N) shrinks.

Rule of thumb: If you can control, aim at the category with the smallest live set (ties broken by which card you can force).

3.2 Prefer showing rooms when defending

When you must show a card to another player, you control the leak. Early:

• Showing a room leaks (\approx 0.17)–(0.19) bits.
• Showing a suspect/weapon leaks (\approx 0.26) bits.

Leak‑minimizing policy: Show a room if legal; otherwise show a weapon; otherwise a suspect—unless you have meta‑reasons (e.g., disguising the fact you also have a second card among their suggestion).

3.3 Bits per turn vs. movement (why room ownership matters)

You can only suggest the room you occupy. If you own that room card, you can set up controlled suggestions every turn from that room by also naming a weapon you own, targeting suspects one by one. That yields (\approx 0.65)–(0.72) bits per turn early, which is excellent.

Practical advice:

• Early: navigate to a room you hold (or can reach via secret passage) and camp there to farm controlled suspect/weapon tests.
• If you don’t hold the current room, try to control one of the other two cards to reduce leakage randomness.

3.4 How many controlled suggestions to finish a category?

Let (T(N)) be the expected number of controlled suggestions to identify a category with (N) possibilities. Each controlled suggestion succeeds with prob. (1/N); otherwise you eliminate one and continue with (N-1). The recursion [ T(N)=1+\frac{N-1}{N},T(N-1),\quad T(1)=0 ] solves to the closed form [ \boxed{,T(N)=\frac{N+1}{2}-\frac{1}{N}, .} ] So:

• (T(6)=3.\overline{3}) (suspects/weapons)
• (T(5)=2.8)
• (T(4)=2.25)
• (T(3)=1.\overline{6})
• (T(9)=4.888\ldots) (rooms)

This matches table‑top intuition: clean, steady progress if you can keep control.

3.5 Watching others (ownership entropy)

Even when you learn little about the solution, you learn about who holds what. If Alice asks about ({S,W,R}) and Bob shows a card, you know: Bob holds at least one of ({S,W,R}). That’s an OR‑constraint on Bob’s 3‑card hand. Over time, the network of OR‑constraints and non‑refutations sharpens to exact ownership. While this post focuses on the solution entropy, tracking ownership constraints is decisive in close games.

A simple notebook representation is a matrix (players × cards) with entries in ({0,1,?}). Each suggestion updates:

• Non‑refuter → mark 0 (cannot hold any of the three for that player).
• Refuter → mark “at least one of these three is 1.” If you can force which one (control), you immediately set a 1.

PART 4: How player count changes the starting information

Your hand size depends on the number of players, and so does the information you begin with. Using the same formula (I_{\text{hand}}) and averaging over hypergeometric compositions:

*With 4–5 players the deal is uneven; the table shows the mean over all possible 4‑/5‑card hands.

Implication: Fewer players → larger hands → you start closer to the truth.

PART 5: A compact “bits‑first” playbook

1. Control whenever possible. Two cards you hold + one target → (\approx0.5)–(0.7) bits/turn.
2. Target the smallest live category. It maximizes expected information now and raises your chance to finish that category.
3. Camp a room you own. It lets you control repeatedly without board friction.
4. When forced to show, leak rooms first. Rooms give away fewer bits early.
5. Exploit non‑refutations. Each pass is a 0‑bit event for the solution but a big update on ownership.
6. Avoid accidental mega‑leaks. Don’t show a card that lets an opponent deduce two categories at once (e.g., when they clearly controlled the other two).
7. Accuse only on certainty. One wrong accusation is elimination; wait until your posterior is a point mass (or the table state makes the risk rational).

PART 6: Generalizing the formulas

Everything here extends to other editions (different category sizes). Replace (S_0, W_0, R_0) accordingly.

• Starting entropy: (\log(S_0W_0R_0)).
• Hand gain: (\log!\big(\frac{S_0W_0R_0}{(S_0-s)(W_0-w)(R_0-r)}\big)).
• Single elimination: (v(N)=\log!\frac{N}{N-1}).
• Controlled suggestion (target size (N)): (I_{\text{ctrl}}(N)=\frac{1}{N}\log N + \frac{N-1}{N}\log!\frac{N}{N-1}).
• Expected controlled turns to finish category (N): (T(N)=\frac{N+1}{2}-\frac{1}{N}).

PART 7: Minimal, Reproducible Code

Below is a small Python snippet that reproduces the key numbers used above and lets you experiment with variants.

import math
from math import comb

# Category sizes
S0, W0, R0 = 6, 6, 9  # modify for variants

def H_start(S0=6, W0=6, R0=9):
    return math.log2(S0) + math.log2(W0) + math.log2(R0)

def hand_info_gain(s, w, r, S0=6, W0=6, R0=9):
    return H_start(S0, W0, R0) - (
        math.log2(S0 - s) + math.log2(W0 - w) + math.log2(R0 - r)
    )

def v(N):  # value of eliminating one candidate from a category of size N
    return math.log2(N/(N-1))

def I_ctrl(N):  # expected info of a controlled suggestion targeting size-N category
    return (1/N)*math.log2(N) + (1 - 1/N)*v(N)

def T_ctrl(N):  # expected # of controlled suggestions to finish category
    return (N + 1)/2 - 1/N

# Composition probabilities for a 3-card hand (6-player game)
def composition_probs_3():
    total = comb(18, 3)  # 5 suspects + 5 weapons + 8 rooms after case file
    rows = []
    for s in range(4):
        for w in range(4 - s):
            r = 3 - s - w
            ways = comb(5, s) * comb(5, w) * comb(8, r)
            p = ways / total
            I = hand_info_gain(s, w, r)
            rows.append(((s, w, r), p, I))
    return sorted(rows, key=lambda x: (-x[2], x[0]))

if __name__ == "__main__":
    print("Start entropy (bits):", H_start())
    print("I_ctrl Suspect/Weapon (N=6):", I_ctrl(6))
    print("I_ctrl Room (N=8):", I_ctrl(8))
    print("T_ctrl for N in {6,5,4,3}:", [T_ctrl(N) for N in [6,5,4,3]])
    print("3-card compositions (prob %, info bits):")
    for (s,w,r),p,I in composition_probs_3():
        print((s,w,r), round(100*p,3), round(I,3))

Conclusion

Thinking about Clue in bits clarifies why some habits feel strong at the table:

• The mystery starts at (\approx 8.34) bits. Your initial hand trims (\approx 0.7)–(1.5) bits depending on size.
• Controlled suggestions are king. They routinely deliver (\approx 0.5)–(0.7) bits per turn early and accelerate as a category shrinks.
• Rooms leak less. When you must reveal, show rooms to minimize opponent information—especially in the early game.
• Target smaller live sets. It’s more informative to complete a nearly solved category than to “spread” your questions.

Beyond the math, Clue remains a social deduction game—players hide tells, misdirect, and leverage tempo on the board. But if you steer by information—measuring your progress in bits—you’ll not only play better, you’ll understand why your choices work.