Beatpath examples 3

The following are examples of beatpath orders for elections with three candidates: A, B, and C. For each example, the pair-wise beat relation is given in a tabular format, and the beatpath order is given in both a tabular and graphical format. In addition, the Schwartz set, the Smith Set, and any Condorcet winner or Weak Condorcet winners are noted.

The tables are color coded as follows:

Example 1: Condorcet Winner
A beats B and C. B beats C.

Beat Path Order


 * The Condorcet winner is A.
 * The Schwartz set = {A}.
 * The Smith set = {A}.

Example 2: Weak Condorcet Winner
A beats B and ties C. B beats C.

Beat Path Order


 * The weak Condorcet winner is A.
 * The Schwartz set = {A}.
 * The Smith set = {A,B,C}.

Example 3: Leaders Pair-wise Tie
A ties B. A and B beat C.

Beat Path Order


 * The weak Condorcet winners are A and B.
 * The Schwartz set = {A,B}, each in their own component.
 * The Smith set = {A,B}.

Example 4: Three-way Tie
A ties B and C. B ties C.

Beat Path Order


 * The weak Condorcet winners are A, B, and C.
 * The Schwartz set = {A,B,C}, each in their own component.
 * The Smith set = {A,B,C}.

Example 5: Cycle
A beats B. B beats C. C beats A.

Beat Path Order


 * No Condorcet winner. No weak Condorcet winner.
 * The Schwartz set = {A,B,C}, all in the same component.
 * The Smith set = {A,B,C}.