# [EM] Plurality & Reguarity

Markus Schulze markus.schulze at alumni.tu-berlin.de
Fri Feb 1 06:15:21 PST 2002

```Dear Mike,

you wrote (1 Feb 2002):
> I'm sorry--I must have missed the part or P&P's Regularity definition
> that mentioned linear orders.

On page 912, P&P wrote:
> Let A be a finite set of m alternatives with m >= 2. A "linear order"
> on A is a reflexive, connected, transitive, and antisymmetric binary
> relation on A. We denote by L the set of all linear orders on A.
> Let N = {1,...,n}, n >= 2, be a set of players. A "coalition" is a
> nonempty subset of N. For a coalition S, we denote by L^S the set of
> all functions from S to L. Also, we denote by Re the set of all real
> numbers.
>
> Definition 2.1: A "decision scheme" (DS) is a function d: A x L^N
> -> Re which satisfies
> (2.1) d(x,R^N) >= 0 for all x e A and all R^N e L^N; and
> (2.2) [SUM over all x e A] d(x,R^N) = 1 for all R^N e L^N.

In other words: The input of a decision scheme is a set of linear
orders each of the same alternatives. The output of a decision
scheme is a probability distribution on the set of alternatives.

Then, with this definition of decision schemes, P&P define
"probabilistic voting procedures" (Definition 3.1). And with
this definition of probabilistic voting procedures, they
define "regularity" (Definition 3.8).

On page 914, they comment:
> What regularity requires is that given a preference profile R^N e L^N,
> the probability of an initially "available" alternative being chosen
> by the group N cannot increase if the set of available alternatives is
> enlarged by the addition of some more alternatives.

Markus Schulze

```