9
Proposition 1. The FB
n
winners are the minimax outcomes—they minimize the
maximum distance to the top preference of any player.
Proof. See Brams and Kilgour (2001, p. 292, Theorem 3).
The idea behind the proof is the following. Suppose an FB
n
outcome is not a
minimax combination. Then there is some other combination for which the maximum
distance to the top preference of any player is less. But the descent under FB
n
, which
stops at the first point at which all players approve of some combination, must stop at this
other combination. Therefore, this other combination must be a minimax combination.
A decision rule of unanimity is frequently used to decide important questions, such
as the admission of new members into a regional or international organization. As a case
in point, the Treaty of Rome in 1958 made unanimous consent of the original 6-member
Common Market a requirement for admission of new members; that rule is still in effect
in the present 15-member European Union that has evolved from the Common Market
over the last 45 years.
That different fallback decision rules can give dramatically different results is
illustrated by Example A, in which there are n = 10 players and k = 3 provisions, which
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Example A about here
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yield 8 combinations. Note that four combinations (000, 100, 010, 001) are the top
preferences of one player each, and three combinations (110, 101, 011) are the top
preferences of two players each; only combination 111 is nobody’s top preference.
Geometrically, the positions of the different players are shown in 3-space in Figure 1.
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Convention