TL;DR
A simple coin-flip experiment shows that asking a second independent informant with the same 80% reliability adds no accuracy when you decide by majority. With odd numbers of independent, better-than-random informants the majority rule improves accuracy, but adding one person to an odd-sized panel (making it even) can leave accuracy unchanged.
What happened
The author set up a thought experiment and simulation: Bob flips a fair coin and Alice reports the result but lies 20% of the time, so trusting Alice alone yields an 80% correct guess. Adding a second independent friend, Bob, who also lies 20% of the time, might seem like it should improve accuracy. A simulation (one million flips) and a probability breakdown show it does not: cases where both agree raise confidence but are offset by cases where they disagree, which leave the observer guessing at random. Explicitly, for a true heads flip the probabilities work out as 64% both truthful, 4% both lying, and two 16% mixed cases; the mixed cases are 50/50. The overall success rate with two informants remains 80%. Extending to three independent informants with the same error rate raises accuracy to 90%, but adding a fourth brings it back to the three-person level, so the odd/even pattern repeats.
Why it matters
- Majority voting with independent, better-than-random voters improves accuracy as group size grows, but even-sized increments can add no new information.
- Designing decision panels or committees benefits from considering odd versus even member counts to avoid unresolved ties that negate gains.
- Reliance on additional identical-quality sources can fail when decisions are binary and resolved by majority, limiting expected value from extra agreement.
- This phenomenon connects to classical voting theory (Condorcet’s jury theorem) and explains an explicit caveat around even-sized juries.
Key facts
- In the base scenario Alice lies 20% of the time, so trusting her alone gives 80% accuracy.
- Adding a second independent informant who also lies 20% of the time does not change the overall accuracy: it remains 80%.
- Probability breakdown for a true heads: both truthful 64%, both lie 4%, and the two mixed cases each 16% — mixed cases yield 50% correctness.
- A simulation with one million coin flips was used to illustrate the result.
- With three independent informants each lying 20% of the time, the majority rule yields 90% accuracy.
- Adding a fourth informant returns accuracy to the three-person level (no improvement), and this odd/even pattern continues.
- The effect assumes informants act independently and are not adversarial or otherwise correlated.
- The behavior is related to Condorcet’s jury theorem; proofs often exclude the even-voter case because an added even voter can provide no net information.
What to watch next
- Whether informant errors are independent — correlation between reporters would change outcomes and is not analyzed in the source.
- Situations with adversarial or strategic reporters — the optimal decision rule here assumes non-adversarial behavior and independence.
- Practical decision systems that use majority rules: check for tie-handling and whether adding members actually changes the decision distribution.
Quick glossary
- Majority rule: A decision process that selects the option favored by more than half of voters or informants.
- Independent decision: When each informant’s report is statistically independent of others, not influenced or correlated with them.
- Error rate: The probability that an informant reports the wrong outcome; here described as a percentage (e.g., 20%).
- Condorcet’s jury theorem: A result in voting theory stating that if each voter is more likely than not to be correct and votes independently, the probability the majority is correct approaches 100% as the number of voters grows.
- Tie: A situation in an even-sized panel where votes split evenly, leaving no majority preference.
Reader FAQ
Does adding any extra person always help?
Not necessarily. In this setup with independent reporters who each have the same error rate, adding one person to an odd-sized group (making it even) can provide no improvement.
Would results change if reporters were correlated or strategic?
Yes. The analysis assumes independence and non-adversarial behavior; how correlation or strategic lying affects outcomes is not confirmed in the source.
Why does three reporters improve accuracy but four does not?
With three reporters a majority tiebreaks disagreements more often in favor of the true outcome; with four reporters a 2-2 split becomes possible and can offset the gains, reproducing the even/odd pattern described.
Is this phenomenon commonly discussed in voting theory?
The source links it to Condorcet’s jury theorem and notes that proofs often sidestep the even-voter case because an added even voter may supply no new information.
ARE TWO HEADS BETTER THAN ONE? Three heads are certainly more fun Dec 9, 2025 You’re playing a game with your lying friends Alice and Bob. Bob flips a coin…
Sources
- Are two heads better than one?
- Condorcet paradox
- No need to toss a coin: conflicting scientific expert testimonies …
- Dilation, Sure Loss and Simpson's Paradox
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