How to Scientifically Prove Lottery Fairness: Understanding Randomness Through Mathematics and Statistics

Can you say with certainty that the lottery you participated in was truly fair?

Lottery fairness is not a matter of intuition -- it can be rigorously defined and proven through mathematics and statistics. This article explains the fairness of lottery methods from a scientific perspective, keeping formulas to a minimum and prioritizing clarity.

Four Conditions for a Fair Lottery

The following four conditions must be met for a lottery to be considered mathematically "fair."

Condition 1: Equal Probability

All participants must have an equal probability of reaching every outcome.

Expressed as a formula:

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

n = number of participants (= number of outcomes)

For example, in a lottery with 5 people and 5 outcomes, each person has a 1/5 = 20% chance of getting each outcome.

Condition 2: Independence

The result of the previous lottery must not affect the next lottery.

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

Even if person A won 1st place last time, the probability of A winning 1st place this time remains unchanged.

Condition 3: Unpredictability

Results must not be predictable before the lottery is executed. The criteria are: it must be unpredictable with human computational ability, and it must have a certain level of complexity (entropy).

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

All participants must receive different outcomes. Mathematically, this is expressed as:

Mapping f: participant set -> outcome set is bijective

Surjective means someone reaches every outcome (no gaps). Injective means multiple people do not reach the same outcome (no duplicates).

Mathematical Proof of Amidakuji

Proof 1: Guaranteed Bijection

Theorem: Amidakuji always satisfies bijection.

The proof proceeds in the following steps.

In Step 1, we verify the structure. There are n vertical lines L1, L2, ..., Ln and m horizontal lines h1, h2, ..., hm, where each horizontal line connects only two adjacent vertical lines.

In Step 2, we show path uniqueness. Starting from each vertical line, you go down, cross when you encounter a horizontal line, go down the vertical line again, and repeat until you reach the goal. This process is deterministic -- the same starting point always leads to the same goal.

In Step 3, we prove the permutation property. Amidakuji mathematically represents a permutation.

The effect of one horizontal line:

Swap positions i and i+1 of vertical lines

The effect of m horizontal lines:

Composition of m permutations

Permutations are always bijective, so bijection is guaranteed.

Conclusion: Amidakuji mathematically guarantees bijection.

Proof 2: Equal Probability

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

There are n! permutations for n vertical lines.

Example: 3 lines yield 3! = 6 permutations

• (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 each horizontal line position is chosen independently at random, drawing a sufficient number of lines can generate all permutations with equal probability. This is based on Fisher-Yates shuffle theory.

The number of horizontal lines needed for a mathematically fair distribution is approximately n^2 log(n) / (2 pi^2), based on the mixing time of adjacent transposition shuffles (Lacoin 2016). For 10 people, the theoretical threshold is about 12 lines; for 20 people, about 61; and for 50 people, about 495. The required number grows quadratically.

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

Proof 3: Unpredictability

Theorem: With 3 or more horizontal lines, it becomes difficult for humans to visually predict the result.

Analyzing computational complexity: with 0 horizontal lines, prediction time is O(1) and each vertical line reaches itself. With 1 horizontal line, prediction time is O(1) and only the two connected lines swap. With 2 horizontal lines, prediction time is O(n) and visual prediction becomes possible with practice.

With 3 or more horizontal lines, prediction time is O(m), proportional to the number of lines. Complexity increases with more lines, and with 10 or more, prediction becomes practically impossible.

Human visual tracking ability drops sharply beyond 3 intersection points.

Conclusion: With 3 or more horizontal lines, unpredictability is practically satisfied.

Comparison with Other Lottery Methods

See detailed comparisons of various lottery methods here

Mathematical Analysis of Paper Lottery

Paper lotteries use n tickets k1, k2, ..., kn with outcomes assigned to each, and participants draw in a determined order.

The problems include the possibility of duplication or gaps due to ticket creation errors (with no mathematical guarantee), and the fact that the creator knows the outcomes and can recommend specific tickets, resulting in low transparency.

Equal probability is satisfied if tickets are correctly made. Independence is satisfied. Unpredictability is somewhat weak since the creator knows the outcomes. Bijection is not guaranteed.

Mathematical Analysis of Digital Roulette

Digital roulettes use pseudorandom number generators (PRNG) with algorithms like the linear congruential method.

X(n+1) = (a * X(n) + c) mod m

The problems are that pseudorandom numbers are not truly random but deterministic algorithms, so results can be predicted if the seed value is known. Pseudorandom numbers also always have a period. Furthermore, users cannot verify the algorithm, making it a black box.

Equal probability depends on the algorithm. Independence depends on the seed. Unpredictability depends on the algorithm. Bijection depends on the design.

Mathematical Analysis of Rock-Paper-Scissors

Rock-paper-scissors can be modeled as a two-player zero-sum game, with Nash equilibrium at (1/3, 1/3, 1/3).

The problems include psychological bias (humans cannot make perfectly random choices, and "rock" tends to be the most common first move), frequent ties (1/3 probability for 2 players, much higher for n players), and the impossibility of bijection since multiple people can win simultaneously.

Equal probability is incomplete due to psychological bias. Independence is not satisfied since it depends on the opponent's choice. Unpredictability is limited due to strategy. Bijection is not guaranteed.

Comparison Summary Table

Method	Equal Prob.	Independence	Unpredictability	Bijection	Transparency
Amidakuji	Excellent	Excellent	Excellent	Excellent	Excellent
Paper lottery	Good	Good	Fair	Fair	Fair
Roulette	Good	Fair	Fair	Good	Poor
Rock-paper-scissors	Fair	Poor	Fair	Poor	Excellent
Excel random	Good	Fair	Fair	Good	Fair

Statistical Verification Methods

Here are methods to statistically verify whether actual lotteries are fair.

Verification 1: Chi-Square Test

This method verifies whether each outcome appears an equal number of times.

The procedure is as follows. First, set up hypotheses: the null hypothesis H0 states that all outcomes appear with equal probability, and the alternative hypothesis H1 states that there is bias. Next, conduct the lottery 100 times and record the appearance count of each outcome.

The statistic is calculated as:

chi-squared = Sum [(observed - expected)^2 / expected]

If the chi-squared value is less than the critical value, there is no bias. If it is greater than or equal to the 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

chi-squared = 0.5 < critical value (9.49) implies no bias

Verification 2: Kolmogorov-Smirnov Test

This method verifies distribution uniformity and is suitable for large-scale data verification (1000+ trials).

Verification 3: Entropy Measurement

This method quantifies the degree of randomness. Shannon entropy is calculated as:

H = -Sum [P(i) * log2 P(i)]

The maximum entropy for a completely random distribution is:

H_max = log2 n

For example, with 5 outcomes, H_max = log2 5, which is approximately 2.32. If the actual entropy is close to this value, the randomness is high.

Experiencing Amidakuji Fairness

Experiment 1: Small-Scale Amidakuji (3 People)

Using 3 vertical lines and 3 horizontal lines, you can verify all possible outcomes (3! = 6 permutations):

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)

Try changing the horizontal line placement patterns to confirm that each outcome appears with equal probability.

Experiment 2: Large-Scale Simulation

Using the free online Amida-san, conduct 100 lotteries with 10 people, recording how many times each person gets "1st place." The results should be close to the theoretical value of 10 times. Each person should get 1st place about 10 times (with variation of roughly plus or minus 3).

Frequently Asked Questions

Q1: Does having fewer horizontal lines affect fairness?

Fewer horizontal lines cause problems: not all permutations are generated with equal probability, and edge positions tend to stay closer to their original positions. A mathematically fair distribution requires a number of lines proportional to n^2 log(n). Amida-san guarantees fairness through its algorithm.

Q2: Can we trust digital tool random numbers?

It depends on the tool. Conditions for a trustworthy tool include being open source and verifiable, using a cryptographically secure random number generator (CSPRNG), and having a highly transparent mechanism.

See more on security and privacy here

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

Q3: Is there a fairer method than amidakuji?

Theoretically, a digital lottery using a true random number generator (TRNG) is the fairest. However, TRNGs require special hardware, have low transparency (black box), and participants cannot verify the results. Amidakuji offers the best balance of fairness and transparency.

Summary

Scientifically proving lottery fairness requires meeting four conditions: equal probability (all outcomes are equally likely), independence (results are not affected by previous outcomes), unpredictability (results are difficult to predict beforehand), and bijection (everyone gets a different outcome).

Amidakuji mathematically satisfies all of these: bijection is guaranteed by the properties of permutations, equal probability is ensured by Fisher-Yates theory with sufficient horizontal lines, unpredictability is practically satisfied with 3 or more horizontal lines, and the process is fully visible, ensuring transparency.

For scientifically proven fair lotteries, use Amida-san.

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