How to Scientifically Prove Lottery Fairness

"How can we prove a lottery is truly fair?" "What's the mathematical basis for amidakuji being fair?"

Lottery fairness can be rigorously defined and proven with mathematics and statistics, not just intuition.

This article thoroughly explains lottery fairness from a scientific perspective. We minimize formulas and focus on diagrams for easy understanding.

Four Conditions for Fair Lottery

Four conditions define "fairness" mathematically:

Condition 1: Equal Probability

Definition: All participants have equal probability of reaching all outcomes.

Formula:

P(participant i reaches outcome j) = 1/n

n = number of participants (= number of outcomes)

Example: With 5 people and 5 outcomes, each person has 1/5 = 20% chance for each outcome.

Condition 2: Independence

Definition: Previous lottery results don't affect subsequent lotteries.

Formula:

P(current result | previous result) = P(current result)

Example: Even if person A won 1st place last time, the probability of A winning 1st place this time doesn't change.

Condition 3: Unpredictability

Definition: Results cannot be predicted before lottery execution.

Criteria:

• Unpredictable with human computational capacity
• Has certain level of complexity (entropy)

Condition 4: Bijection (One-to-One Correspondence)

Definition: All participants receive different outcomes.

Mathematical Expression:

Mapping f: participant set → outcome set is bijective

Meaning:

• Surjective: Someone reaches every outcome (no gaps)
• Injective: Multiple people don't reach same outcome (no duplicates)

Mathematical Proof of Amidakuji

Proof 1: Guaranteed Bijection

Theorem: Amidakuji always satisfies bijection.

Proof:

Step 1: Structure Verification

• n vertical lines: L₁, L₂, ..., Lₙ
• m horizontal lines: h₁, h₂, ..., hₘ
• Each horizontal line connects only two adjacent vertical lines

Step 2: Path Uniqueness Starting from each vertical line:

1. Go down
2. Cross when meeting horizontal line
3. Go down vertical line again
4. Repeat to reach goal

This process is deterministic - same starting point always reaches same goal.

Step 3: Permutation Proof Amidakuji mathematically represents a permutation.

Effect of one horizontal line:

Swap positions i and i+1 of vertical lines

Effect of m horizontal lines:

Composition of m permutations

Permutations are always bijective, guaranteeing one-to-one correspondence.

Conclusion: Amidakuji mathematically guarantees bijection. ■

Proof 2: Equal Probability

Theorem: When horizontal lines are randomly placed, all permutations are generated with equal probability.

Proof Outline:

n vertical lines have n! possible permutations.

Example: 3 lines = 3! = 6 patterns

• (1,2,3) → (1,2,3)
• (1,2,3) → (1,3,2)
• (1,2,3) → (2,1,3)
• (1,2,3) → (2,3,1)
• (1,2,3) → (3,1,2)
• (1,2,3) → (3,2,1)

When horizontal line placement is sufficiently random:

When each line position is chosen independently and randomly, drawing sufficient lines can generate all permutations with equal probability (Fisher-Yates shuffle theory).

Practical number of lines:

• For n vertical lines, approximately 2n horizontal lines suffice
• Example: 20 lines for 10 people

Conclusion: With sufficiently random horizontal line placement, equal probability is satisfied. ■

Proof 3: Unpredictability

Theorem: With 3 or more horizontal lines, visual prediction by humans becomes difficult.

Computational Complexity Analysis:

0 horizontal lines:

• Prediction time: O(1) (immediately known)
• Each vertical line reaches itself

1 horizontal line:

• Prediction time: O(1) (immediately known)
• Only connected 2 lines swap

2 horizontal lines:

• Prediction time: O(n) (trackable in linear time)
• Visually predictable with practice

3+ horizontal lines:

• Prediction time: O(m) (proportional to line count)
• Complexity increases with more lines
• Practically unpredictable with 10+ lines

Psychological Research: Human visual tracking ability drops sharply beyond 3 intersection points.

Conclusion: With 3+ horizontal lines, unpredictability is practically satisfied. ■

Comparison with Other Lottery Methods

For practical comparisons, see Lottery Method Comparison.

Mathematical Analysis of Paper Lottery

Structure:

• n tickets: k₁, k₂, ..., kₙ
• Assign outcome to each ticket
• Participant drawing order: permutation

Problems:

1. Possibility of Duplication/Gaps

Ticket creation errors → outcome duplication or gaps
Example: Two "winners", missing "losers"

Mathematical guarantee: None

2. Creator Manipulation Possibility

Creator knows outcomes
→ Can recommend specific tickets

Transparency: Low

Equal probability: ○ (if created correctly) Independence: ○ Unpredictability: △ (creator knows) Bijection: △ (not guaranteed)

Mathematical Analysis of Roulette (Digital)

Structure:

• Uses pseudorandom number generator (PRNG)
• Algorithms like linear congruential method

Algorithm Example (Linear Congruential):

X(n+1) = (a × X(n) + c) mod m

Problems:

1. Pseudorandom Limitations

Not true random, deterministic algorithm
Results predictable if seed value known

2. Periodicity

Pseudorandom numbers always have period
Period: maximum m (mod value)

3. Lack of Transparency

Users cannot verify algorithm
Black box

Equal probability: ○ (algorithm dependent) Independence: △ (seed dependent) Unpredictability: △ (algorithm dependent) Bijection: ○ (design dependent)

Mathematical Analysis of Rock-Paper-Scissors

Game Theory Model:

• Two-player zero-sum game
• Nash equilibrium: (1/3, 1/3, 1/3)

Problems:

1. Psychological Bias

Humans cannot make perfectly random choices
First move often "rock" (statistically proven)
Can read opponent's habits

2. Frequent Ties

2 players: tie probability = 1/3
n players: very high tie probability

3. Bijection Fails

Multiple people can win simultaneously
→ Bijection not guaranteed

Equal probability: △ (psychological bias) Independence: × (depends on opponent choice) Unpredictability: △ (has strategy) Bijection: ×

Comparison Summary Table

Method	Equal Prob	Independence	Unpredictability	Bijection	Transparency
Amidakuji	◎	◎	◎	◎	◎
Paper lottery	○	○	△	△	△
Roulette	○	△	△	○	×
Rock-paper-scissors	△	×	△	×	◎
Excel random	○	△	△	○	△

Statistical Verification Methods

Methods to statistically verify if actual lotteries are fair.

Verification 1: Chi-Square Test (χ² Test)

Purpose: Verify if each outcome appears equally

For security considerations in online lotteries, see Lottery Security and Privacy.

Procedure:

1. Hypothesis Setting

   • Null hypothesis H₀: All outcomes appear with equal probability
   • Alternative hypothesis H₁: There is bias

2. Data Collection

   • Conduct lottery 100 times
   • Record appearance count of each outcome

3. Statistic Calculation

χ² = Σ [(observed - expected)² / expected]

4. Judgment
   • χ² value < critical value → no bias
   • χ² value ≥ critical value → bias exists

Example: 100 lotteries with 5 outcomes

Outcome	Observed	Expected	(O-E)²/E
A	22	20	0.2
B	19	20	0.05
C	18	20	0.2
D	21	20	0.05
E	20	20	0
Total	100	100	0.5

χ² = 0.5 < critical value(9.49) → no bias

Verification 2: Kolmogorov-Smirnov Test

Purpose: Verify distribution uniformity

Suitable for large-scale data verification (1000+ times).

Verification 3: Entropy Measurement

Purpose: Quantify degree of randomness

Shannon Entropy:

H = -Σ [P(i) × log₂ P(i)]

Maximum Entropy (completely random):

H_max = log₂ n

Example: For 5 outcomes H_max = log₂ 5 ≈ 2.32

High randomness if actual entropy is close to this.

Practice: Experience Amidakuji Fairness

Experiment 1: Small-Scale Amidakuji (3 People)

Setup:

• 3 vertical lines
• 3 horizontal lines

All Possible Outcomes (3! = 6 patterns):

1. (1,2,3) → (1,2,3)
2. (1,2,3) → (1,3,2)
3. (1,2,3) → (2,1,3)
4. (1,2,3) → (2,3,1)
5. (1,2,3) → (3,1,2)
6. (1,2,3) → (3,2,1)

Vary horizontal line placement patterns to confirm each outcome appears with equal probability.

Experiment 2: Large-Scale Simulation

Try the following on Amida-san (free online amidakuji):

1. Conduct 100 lotteries with 10 people
2. Record how many times each person got "1st place"
3. Verify if close to theoretical value of 10 times

Expected Result: Each person gets 1st place about 10 times (±3 variation)

Also see practical examples for company parties and school events.

Frequently Asked Questions

Q1: Isn't it unfair with few horizontal lines?

Answer: Few horizontal lines cause these problems:

• Not all permutations generated with equal probability
• Prediction becomes easy

Recommended count: For n people, we recommend 2n or more horizontal lines.

Q2: Does the order of drawing horizontal lines matter?

Answer: No, it doesn't.

Mathematical Reason: Permutation composition is associative, so final permutation is same regardless of line drawing order.

Q3: Can we trust digital tool random numbers?

Answer: Depends on the tool. Trustworthy tool conditions:

• Open source and verifiable
• Uses cryptographically secure RNG (CSPRNG)
• Highly transparent mechanism

Amida-san has all participants add horizontal lines, so even organizers cannot manipulate results.

Q4: Doesn't human line drawing eliminate randomness?

Answer: Individuals may place intentionally, but mixing multiple intentions creates randomness as result.

This is similar to "wisdom of crowds" - when many independent judgments gather, overall balance emerges.

Q5: Is there a fairer method than amidakuji?

Answer: Theoretically, digital lottery using true random number generator (TRNG) is fairest.

However, TRNG:

• Requires special hardware
• Low transparency (black box)
• Participants cannot verify

Amidakuji has the best balance of fairness and transparency.

Summary

Conditions for scientifically proving lottery fairness:

1. Equal Probability: All outcomes equally probable
2. Independence: Not affected by previous results
3. Unpredictability: Prediction is difficult
4. Bijection: Everyone gets different outcome

Amidakuji mathematically satisfies all of these:

• Bijection: Guaranteed by permutation properties
• Equal Probability: Guaranteed by Fisher-Yates theory with sufficient lines
• Unpredictability: Practically satisfied with 3+ horizontal lines
• Transparency: Process completely visible

For scientifically proven fair lottery, use Amida-san.