Consider a combat tournament with a large number (sufficiently large that small sample size is not a problem) of combatants, in which each match is zero-sum and has a winner (there are no ties). The rules specify that each combatant may choose to bring into the tournament exactly 2 of the following 3 "weapons": a pillow, a thimble, a sword. The sword obviously being superior to the other "weapons", ALL combatants choose a sword as one of their 2 weapons. For the other weapon though, the combatants are split: exactly half choose to bring a pillow, and exactly half choose to bring a thimble.
After the conclusion of the tournament, a statistician analyzes the results of the tournament. The statistician finds that:
combatants with a sword had a combined winning percentage of exactly 50% (remember that EVERY combatant brought a sword, so every win by any combatant with a sword meant a loss by another combatant with a sword, resulting in a winning percentage of exactly 50%)
combatants with a thimble had a combined winning percentage of less than 50%
combatants with a pillow had a combined winning percentage of more than 50%
Simple example with round-robin tournament of only 4 combatants to demonstrate the mathematics:
Alice and Bob both wield sword and pillow. Carol and Dave both wield sword and thimble. The 6 matches:
- Alice defeats Bob.
- Alice defeats Carol.
- Alice defeats Dave.
- Bob defeats Carol.
- Bob defeats Dave.
- Carol defeats Dave.
- Alice (sword and pillow) 3-0
- Bob (sword and pillow) 2-1
- Carol (sword and thimble) 1-2
- Dave (sword and thimble) 0-3
Winning percentages by weapon:
- sword 6-6 (50%)
- thimble 1-5 (16.7%)
- pillow 5-1 (83.3%)
Since pillow has the highest winning percentage, even higher than sword, the statistician concludes that a pillow is a better, more effective weapon than a sword.
What logical or statistical fallacy is this statistician making? I checked the Wikipedia list of fallacies and misuse of statistics articles, but those articles list so many different fallacies that I am uncertain.