- WELCOME TO A LESSON ON COALITIONS AND CRITICAL PLAYERS.
IN THIS LESSON WE WILL DEFINE COALITIONS AND CRITICAL PLAYERS,
ALSO DETERMINE WHICH PLAYERS ARE CRITICAL IN A COALITION
OF A WEIGHTED VOTING SYSTEM.
IN A WEIGHTED VOTING SYSTEM,
A COALITION IS ANY GROUP OF PLAYERS VOTING THE SAME WAY.
A COALITION IS A WINNING COALITION IF THE COALITION
HAS ENOUGH WEIGHT TO MEET QUOTA.
REMEMBER, IN THE WEIGHTED VOTING SYSTEM Q
IS THE QUOTA AND W SUB ONE, W SUB TWO,
AND SO ON ARE THE WEIGHTS OF EACH PLAYER.
A PLAYER IS CRITICAL IN A COALITION
IF BY LEAVING THE COALITION IT WOULD CHANGE IT
FROM A WINNING COALITION TO A LOSING COALITION.
LET'S TAKE A LOOK AT OUR FIRST EXAMPLE.
HERE WE HAVE A WEIGHTED VOTING SYSTEM.
WE WANT TO CONSIDER THE COALITION THAT CONSISTS
OF PLAYER ONE, PLAYER TWO, PLAYER FOUR, AND PLAYER FIVE.
SO HERE'S THE WEIGHT OF PLAYER ONE,
HERE'S THE WEIGHT OF PLAYER TWO, WEIGHT OF PLAYER FOUR,
AND WEIGHT OF PLAYER FIVE.
THE FIRST QUESTION IS WHAT IS THE COMBINED WEIGHT
OF THE COALITION, WHICH WOULD BE 14 + 12 + 4 + 4 = 34.
SO WE CAN SAY THE COMBINED WEIGHT OF THE COALITION IS 34.
AND NOTICE HOW BECAUSE 34 IS GREATER THAN OR EQUAL
TO THE QUOTA OF 30, WE DO HAVE A WINNING COALITION.
AND THEN FINALLY, WE WANT TO DETERMINE
WHICH PLAYERS ARE CRITICAL.
REMEMBER, A PLAYER IS CRITICAL IF BY LEAVING THE COALITION
THE COALITION CHANGES FROM WINNING TO LOSING.
SO LOOKING AT OUR COALITION HERE,
NOTICE HOW IF PLAYER ONE LEAVES
WE'RE LEFT WITH PLAYER TWO, FOUR AND FIVE,
WHERE THE WEIGHT WOULD BE 12 + 4 + 4 = 20,
WHICH DOES NOT MEET THE QUOTA OF 30.
AND THEREFORE, PLAYER ONE IS CRITICAL.
AND NOW WE'LL SEE IF PLAYER TWO IS CRITICAL.
SO IF PLAYER TWO LEAVES THE COALITION,
WE'RE LEFT WITH PLAYER ONE, PLAYER FOUR, AND PLAYER FIVE.
NOTICE HOW THE COMBINED WEIGHT WOULD BE 14 + 8 OR 22,
WHICH AGAIN IS <30 SO IT DOES NOT MEET QUOTA.
THEREFORE, PLAYER TWO IS ALSO CRITICAL.
AND NOW WE'LL TEST PLAYER FOUR.
IF PLAYER FOUR LEAVES THEN THE WEIGHT WOULD BE 14 + 12 + 4.
NOTICE HOW THE COMBINED WEIGHT IS 30, WHICH IS THE QUOTA.
AND THEREFORE, PLAYER FOUR IS NOT CRITICAL.
AND THEN FINALLY, IF PLAYER FIVE LEAVES--
NOTICE PLAYER FIVE HAS THE SAME WEIGHT AS PLAYER FOUR.
SO PLAYER ONE, PLAYER TWO,
AND PLAYER FOUR STILL HAVE A COMBINED WEIGHT OF 30.
AND THEREFORE, PLAYER FIVE IS NOT CRITICAL.
SO WE SAY THAT THE CRITICAL PLAYERS OF THE COALITION
ARE PLAYER ONE AND PLAYER TWO.
LET'S TAKE A LOOK AT A SECOND EXAMPLE.
HERE WE HAVE A DIFFERENT WEIGHTED VOTING SYSTEM
AND A DIFFERENT COALITION.
OUR COALITION NOW IS PLAYER ONE, PLAYER TWO,
AND PLAYER THREE THAT HAVE A COMBINED WEIGHT
OF 33 + 21 + 12 = 66.
SO THE COMBINED WEIGHT OF THE COALITION IS 66.
66 IS GREATER THAN OR EQUAL TO 45, THE QUOTA.
AND THEREFORE, WE DO HAVE A WINNING COALITION.
AND THEN FINALLY, WHICH PLAYERS ARE CRITICAL?
SO IF PLAYER ONE LEAVES THE COALITION
WE ARE LEFT WITH PLAYER TWO AND PLAYER THREE
THAT HAVE A COMBINED WEIGHT OF 33.
21 + 12, WHICH DOES NOT MEET QUOTA
THEREFORE, PLAYER ONE IS CRITICAL.
NEXT, IF PLAYER TWO LEAVES THE COALITION
THAT LEAVES PLAYER ONE AND PLAYER THREE,
WHICH DOES HAVE A COMBINED WEIGHT OF 45, WHICH IS QUOTA.
SO PLAYER TWO IS NOT CRITICAL.
AND THEN IF PLAYER THREE LEAVES,
WE'RE LEFT WITH PLAYER ONE AND PLAYER TWO
THAT HAVE A COMBINED WEIGHT OF 54.
AND THEREFORE, THE COALITION STILL MEETS QUOTA,
PLAYER THREE IS NOT CRITICAL.
SO WE ONLY HAVE ONE CRITICAL PLAYER HERE.
THE CRITICAL PLAYER OF THE COALITION IS PLAYER ONE.
NOW, BEFORE WE GO I DO WANT TO REVISIT THREE DEFINITIONS
FROM THE PREVIOUS LESSON,
BUT NOW WILL INCLUDE THE IDEA
OF COALITIONS AND CRITICAL PLAYERS
IN THESE THREE DEFINITIONS.
A PLAYER IS A DICTATOR IF THE SINGLE-PLAYER COALITION
CONTAINING THEM IS A WINNING COALITION.
A PLAYER HAS VETO POWER IF THEY ARE CRITICAL
IN EVERY WINNING COALITION.
AND FINALLY, A PLAYER IS A DUMMY
IF THEY ARE NOT CRITICAL IN ANY WINNING COALITION.
I HOPE YOU FOUND THIS HELPFUL.