# Condorect sub-cycle rule

Markus Schulze schulze at sol.physik.tu-berlin.de
Fri Sep 26 04:53:06 PDT 1997

```Dear David,

you wrote (26 Sep 1997):
>Does anyone have an authoritative English reference for the sub-cycle
>rule?

Mike Ossipoff wrote (10 Jul 1996):
> The subcycle rule:
>
> Definitions:
>
> A "cycle" is a set of alternatives that beat eachother in circular
> fashion: A beats B beats C beats A, for example.
>
> Cycle A is a "subcycle" of cycle B if every alternative in cycle
> A beats everything in cycle B that is beaten by something in cycle
> A, and if every alternative in cycle A is beaten by everything that
> beats something in cycle A.
>
> What it amounts to is that cycle A is an element of cycle B. A
> cycle that's an element of another cycle.
>
> 1. Before choosing a winner, solve any cycles that exist.
>
> 2. Before solving a cycle, first solve any subcycle of that cycle.
>
> 3. To solve a cycle, apply Condorcet's choice rule to the alternatives
> in that cycle, choosing a winner among those alternatives according
> to the Condorcet choice rule. Eliminate from the election every
> member of that cycle except for the winner. If that cycle, A, is a
> subcycle of another cycle, B, then the winner in A replaces cycle
> A in cycle B, occupying the same position in cycle B that cycle
> A occupied. This last sentence isn't strictly necessary as part
> of the rule, since it follows from the definition of a subcycle
> that the winner in A beats what the other members of a beat
> and is beaten by what beats them.

Markus Schulze (schulze at sol.physik.tu-berlin.de)

```