Prove: A litigant can lose by a vote of 8 to 1 in each of 9 issues, and yet have a unanimous decision in his favour.

Question

How can I prove the emboldened outcome beneath? I'm contemplating some table or chart, with one row per issue, and one column per judge. But then how would I efficiently place checkmarks in the table?

Suppose a case arises before nine judges. B steals A's watch and sells it to C.

3 judges hold for A on the ground that a thief can pass no title in stolen goods, even to a good-faith buyer, as they find C to have been.

2 judges hold for A on the ground that, although a thief can pass title in stolen goods to a good-faith buyer, on the facts of the particular case C has not acted in good faith.

4 judges hold for C on the ground that a thief can pass title to a good-faith buyer and that on the facts C has acted in good faith.

There is a majority of 5 to 4 for A, who wins.

The principle of law, however, that a thief can pass no title, is actually supported by only 3 and rejected by 6 (= 2 + 4). Further, a majority of 7 (= 3 + 4) to 2 consider that C has acted in good- faith. If the two relevant issues were voted on separately, it would be held, by a majority of 6 (= 4 + 2) to 3, that a good-faith buyer from a thief is protected and by a majority of 7 (= 3 + 4) to 2 that C has acted in good faith. C wins handsomely on each issue, but he will lose the case in court by a majority of 5 to 4. It would be possible to imagine a case in which, if there were nine separate issues, one party could lose by a vote of eight to one on each issue, and yet have a unanimous decision in his favour!

Answer 1

I dont think this is a matter of ranking of issues. In the OP example, all nine judges use the same logic, but have different beliefs (findings).

• The common logical formula is: (C wins) iff (thief can pass title) AND (C in good faith)

• A minority of 3 judges do not believe (thief can pass title)

• A minority of 2 judges do not find (C in good faith)

• The two subsets above happen to not overlap, and the union happens to form a majority of 5.

To say different ranking of issues would (to me anyway) imply each judge has a different logical formula for deciding (C wins). IMHO this is an unnecessary generalization of the OP example.

Anyway, the requested example is easy to construct:

• The common logic of the 9 judges is: (C wins) iff $A_1 \cap A_2 \cap \cdots \cap A_9$

• For each $i$, a minority of 1 judge does not believe/find $A_i$

• The nine subsets happen to not overlap, and the union forms the entire set of 9 judges - each using the same logical formula but voting (C loses) for a different reason.

Answer 2

Each of the nine judges has a different ranking of the importance of the issues. Each one thinks one particular issue (a different one for each judge) is so important it outweighs all the others. Each one thinks A wins on his favorite issue and loses on all the others. You get the desired result-on any one issue there are eight judges who favor B but every judge favors A for the whole case.