Home > Riddles > The trial of the cadi
A miniature of fair distribution: when two people value the same object differently, the solution is not only to give it to the person who wants it most, but to distribute the advantage that this produces.
Two heirs dispute a ruby that cannot be broken. To avoid arguments, the cadi brings each person in separately and asks them how much money that ruby represents to them.
It does not reveal the answers. Then he dictates a sentence: the ruby will go to the person who valued it most, but that person must pay compensation to the other.
If the private valuations are $A$ and $B$, with $A>B$, what compensation makes the distribution equitable?
Answer: the compensation must be $$
\frac{A+B}{2}.
$$ The heir who values the ruby at $A$ keeps it and pays $C$ compensation to the other. For the winner, the result is worth: $$
A-C.
$$ For the loser, receiving money instead of a ruby valued at $B$ means a net gain of: $$
C-B.
$$ For the distribution to be equitable, both profits must be equal: $$
A-C=C-B.
$$ So: $$
A+B=2C,
$$ and therefore: $$
C=\frac{A+B}{2}.
$$ Thus the winner obtains an advantage of $$
A-\frac{A+B}{2}=\frac{A-B}{2},
$$ and the loser gets the same advantage: $$
\frac{A+B}{2}-B=\frac{A-B}{2}.
$$ Both earn the same with respect to their own valuation.
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