- WELCOME TO INTRODUCTION TO WEIGHTED VOTING.
IN THIS LESSON WE'LL DEFINE WEIGHTED VOTING,
USE SHORTHAND FORM FOR WEIGHTED VOTING,
AND ALSO DEFINE DICTATOR VETO POWER AND DUMMY
IN A WEIGHTED VOTING SYSTEM.
WEIGHTED VOTING GIVES SOME VOTERS
DIFFERENT AMOUNTS OF INFLUENCE OR WEIGHT CONCERNING THE OUTCOME
OF AN ELECTION.
THIS IS OFTEN COMMON PRACTICE
WHEN IT COMES TO CORPORATE SHAREHOLDER MEETINGS
WHEN SHAREHOLDERS VOTING WEIGHT IS BASED UPON
HOW MANY SHARES THEY OWN.
THE MORE SHARES SOMEONE OWNS, THE MORE WEIGHT OF THEIR VOTE.
THE CONDITIONS THAT MUST BE MET
FOR A WINNER MIGHT BE A SIMPLE MAJORITY,
BUT THERE COULD ALSO BE DIFFERENT REQUIREMENTS
TO SELECT A WINNER.
EACH INDIVIDUAL, OR ENTITY,
CASTING A VOTE IS CALLED A PLAYER IN THE ELECTION.
THEY'RE OFTEN NOTATED AS P SUB ONE, P SUB TWO THROUGH P SUB N,
WHERE N IS THE TOTAL NUMBER OF VOTERS.
EACH PLAYER IS GIVEN A WEIGHT,
WHICH USUALLY REPRESENTS HOW MANY VOTES THEY GET.
THE QUOTA IS THE MINIMUM WEIGHT NEEDED FOR THE VOTE TO PASS,
OR WEIGHT NEEDED FOR THE PROPOSAL, TO BE APPROVED.
A WEIGHTED VOTING SYSTEM WILL OFTEN BE REPRESENTED
IN A SHORTHAND FORM AS WE SEE HERE,
WHERE Q IS THE QUOTA, W SUB ONE IS THE WEIGHT OF PLAYER ONE,
W SUB TWO IS THE WEIGHT OF PLAYER TWO AND SO ON.
LET'S TAKE A LOOK AT AN EXAMPLE.
WE WANT TO WRITE A POSSIBLE WEIGHTED VOTING SYSTEM
FOR THE FOLLOWING SITUATION.
A COMPANY HAS FIVE SHAREHOLDERS.
THESE SHAREHOLDERS HOLD THE FOLLOWING OWNERSHIP STAKE
IN THE COMPANY GIVEN HERE AS PERCENTAGES.
THEY ARE TRYING TO DECIDE IF THEY SHOULD INVEST
IN A NEW PRODUCT LINE TO GROW THE COMPANY.
THE COMPANY BYLAWS STATE THAT A MAJORITY OF THE OWNERSHIP,
MORE THAN 50%, HAS TO APPROVE THE DECISION,
BECAUSE THE OWNERSHIP IS GIVEN AS PERCENTAGES
AND THE QUOTA HAS TO BE MORE THAN 50%.
WE'LL START WITH THE QUOTA,
WHICH WE'LL START AT 51 AND THEN A COLON
FOLLOWED BY THE WEIGHT OF EACH OWNERSHIP,
WHICH WE CAN ALSO THINK OF AS THE NUMBER OF VOTES
GIVEN TO A VOTER BASED UPON THEIR PERCENT OF OWNERSHIP.
SO WE COULD LIST THE WEIGHT AS 30, 25, 20, 15, 10.
THIS WOULD BE A POSSIBLE WEIGHTED VOTING SYSTEM
FOR THIS SITUATION,
BUT THERE ARE AN INFINITE NUMBER OF WEIGHTED VOTING SYSTEMS
THAT WE COULD CREATE FOR THIS SITUATION.
IN ORDER TO HAVE A MEANINGFUL WEIGHTED VOTING SYSTEM THOUGH,
THERE MUST BE LIMITS ON THE QUOTA.
NUMBER ONE, THE QUOTA MUST BE
MORE THEN HALF THE TOTAL NUMBER OF VOTES,
BECAUSE NOTICE HOW IF IT WASN'T THERE COULD BE MORE WEIGHT
AGAINST A PROPOSAL AND IT COULD STILL PASS.
AND NUMBER TWO, THE QUOTA CAN'T BE LARGER
THAN THE TOTAL NUMBER OF VOTES
BECAUSE IF IT WAS NO PROPOSAL WOULD EVER PASS.
A PLAYER WILL BE A DICTATOR IF THEIR WEIGHT
IS EQUAL TO OR GREATER THAN THE QUOTA.
THE DICTATOR CAN ALSO BLOCK ANY PROPOSAL FROM PASSING.
THE OTHER PLAYERS CANNOT REACH QUOTA WITHOUT THE DICTATOR.
LOOKING AT THIS WEIGHTED VOTING SYSTEM HERE,
NOTICE HOW PLAYER ONE HAS A WEIGHT OF 21,
WHICH IS MORE THAN QUOTA
AND THEREFORE PLAYER ONE IS A DICTATOR.
ALSO NOTICE, PLAYER ONE HAS THE ABILITY TO BLOCK ANY PROPOSAL
BECAUSE IF PLAYER TWO AND PLAYER THREE FORM A GROUP,
THEY STILL ONLY HAVE A WEIGHT OF 6 + 3 OR 9, WAY BELOW QUOTA.
SO WE CAN SEE, AGAIN, PLAYER ONE IS A DICTATOR.
NEXT, A PLAYER HAS VETO POWER IF THEIR SUPPORT IS NECESSARY
FOR THE QUOTA TO BE REACHED.
IT IS POSSIBLE FOR MORE THAN ONE PLAYER TO HAVE VETO POWER
OR FOR NO PLAYER TO HAVE VETO POWER.
LOOKING AT THIS FIRST EXAMPLE HERE, NOTICE HOW PLAYER ONE
IS NOT A DICTATOR, BUT IF PLAYER TWO AND THREE FORM A GROUP
THEY ONLY HAVE A WEIGHT OF 26, WHICH IS BELOW QUOTA.
AND THEREFORE, PLAYER ONE IS REQUIRED TO REACH QUOTA.
AND THEREFORE, PLAYER ONE HAS VETO POWER,
BUT NOTICE HOW IF PLAYER ONE AND PLAYER THREE FORM A GROUP
THEY HAVE A WEIGHT OF 30, WHICH DOES REACH QUOTA.
SO PLAYER TWO IS NOT REQUIRED TO REACH QUOTA
AND IF PLAYER ONE AND PLAYER TWO FORM A GROUP
THEY HAVE A WEIGHT OF 34, ONCE AGAIN MORE THAN QUOTA,
SO PLAYER THREE IS NOT REQUIRED.
SO, AGAIN, WE CAN SAY THAT PLAYER ONE HAS VETO POWER.
LOOKING AT OUR SECOND EXAMPLE HERE,
NOTICE THAT OTHER PLAYERS ARE DICTATORS
AND IF PLAYER ONE AND PLAYER TWO FORM A GROUP, THEY REACH QUOTA.
IF PLAYER ONE AND PLAYER THREE FORM A GROUP, THEY REACH QUOTA.
AND ALSO IF PLAYER TWO AND PLAYER THREE FORM A GROUP,
THEY REACH QUOTA.
SO NO SINGLE PLAYER'S REQUIRED TO REACH QUOTA AND THEREFORE,
NO PLAYER HAS VETO POWER.
NEXT, A PLAYER IS A DUMMY IF THEIR VOTE
IS NEVER ESSENTIAL FOR A GROUP TO REACH QUOTA.
SO LOOKING AT THIS SYSTEM,
NOTICE HOW PLAYER THREE IS NEVER REQUIRED,
PLAYER ONE AND PLAYER TWO CAN MAKE QUOTA,
BUT PLAYER ONE AND PLAYER THREE CANNOT REACH QUOTA
BECAUSE THEY'D HAVE A WEIGHT OF 14.
IF PLAYER TWO AND PLAYER THREE FORM A GROUP
THEY HAVE A WEIGHT OF 12,
AGAIN, WHICH IS BELOW QUOTA.
AND THEREFORE, PLAYER THREE IS NEVER ESSENTIAL
TO REACH QUOTA SO WE CAN SAY PLAYER THREE IS A DUMMY PLAYER.
NOW, LETS TAKE A LOOK AT SOME EXAMPLES
TO REVIEW ALL THESE CONCEPTS.
WE WANT TO CONSIDER THE FOLLOWING
WEIGHTED VOTING SYSTEM.
WE'RE FIRST ASKED HOW MANY PLAYERS THERE ARE.
THERE WOULD BE ONE, TWO, THREE, FOUR PLAYERS.
NEXT, WHAT IS THE TOTAL NUMBER OF VOTES OR WEIGHT?
WELL, THAT WOULD BE 8 + 8 + 8 + 2 = 26,
SO WE'LL SAY THERE'S A TOTAL OF 26 VOTES.
THE QUOTA WOULD BE 18.
NOW WE WANT TO IDENTIFY ANY DICTATORS,
BUT NOTICE HOW NONE OF THE WEIGHTS ARE GREATER THAN
OR EQUAL TO THE QUOTA OF 18
AND THEREFORE, NO PLAYERS ARE DICTATORS.
WE ALSO WANT TO IDENTIFY WHETHER ANY PLAYERS HAVE VETO POWER,
WHICH MEANS THEIR SUPPORT IS REQUIRED TO REACH QUOTA.
FOR EXAMPLE, NOTICE HOW IF WE CONSIDER PLAYER TWO,
THREE AND FOUR 8 + 8 + 2 = 18, WHICH DOES REACH QUOTA
AND THEREFORE, PLAYER ONE IS NOT REQUIRED.
AND WE COULD SAY THE SAME FOR PLAYER TWO AND PLAYER THREE.
AND NOTICE THAT IF PLAYER ONE, TWO AND THREE FORM A GROUP,
OF COURSE, THEY REACH QUOTA SO PLAYER FOUR IS NOT ESSENTIAL.
AND THEREFORE, NO PLAYERS HAVE VETO POWER.
WE ALSO WANT TO IDENTIFY ANY DUMMY PLAYERS.
A DUMMY PLAYER IS A PLAYER
THAT'S NEVER ESSENTIAL TO REACH QUOTA,
BUT WE JUST SHOWED THAT IF PLAYER TWO, THREE,
AND FOUR FORM A GROUP, OR A COALITION,
NOTICE 8 + 8 = 16 AND THEREFORE,
PLAYER FOUR IS REQUIRED TO REACH QUOTA.
AND THEREFORE, PLAYER TWO IS NOT A DUMMY PLAYER.
SO IN THIS CASE THERE ARE NO DUMMY PLAYERS.
LET'S TAKE A LOOK AT ANOTHER EXAMPLE WITH THE SAME QUESTIONS,
BUT A DIFFERENT WEIGHTED VOTING SYSTEM.
AGAIN, THE FIRST QUESTION IS HOW MANY PLAYERS ARE THERE?
THERE ARE ONE, TWO, THREE, FOUR, FIVE PLAYERS.
THE TOTAL NUMBER OF VOTES, OR THE WEIGHT,
WOULD BE 18 + 5 + 3 + 3 + 1 = 30.
SO WE'LL SAY THE TOTAL NUMBER OF VOTES IS 30.
THE QUOTA IS HERE TO THE LEFT OFF THE COLON.
SO THE QUOTA IS 16.
NOTICE IN THIS SYSTEM WE DO HAVE A DICTATOR.
NOTICE PLAYER ONE HAS A WEIGHT
THAT'S GREATER THAN OR EQUAL TO THE QUOTA OF 16
SO PLAYER ONE IS A DICTATOR.
WE ALSO WANT TO IDENTIFY ANY PLAYERS THAT HAVE VETO POWER,
WHICH MEANS THEIR VOTES ARE REQUIRED TO REACH QUOTA.
IF WE CONSIDER PLAYERS TWO, THREE, FOUR AND FIVE,
THEIR TOTAL WEIGHT WOULD BE 5 + 3 + 3 + 1 = 12,
WHICH IS BELOW QUOTA.
AND THEREFORE, NOTICE PLAYER ONE WOULD ALWAYS BE REQUIRED
TO REACH QUOTA.
AND THEREFORE, PLAYER ONE HAS VETO POWER.
WE ALSO WANT TO IDENTIFY ANY DUMMY PLAYERS.
REMEMBER, A DUMMY PLAYER IS NEVER ESSENTIAL TO REACH QUOTA.
NOTICE IF PLAYERS TWO, THREE, FOUR AND FIVE FORM A GROUP,
OR A COALITION, THEY STILL ONLY HAVE A WEIGHT OF 12,
WHICH IS BELOW QUOTA.
THE ONLY WAY TO REACH QUOTA IS WITH PLAYER ONE'S SUPPORT.
AND THEREFORE, PLAYER TWO, THREE, FOUR AND FIVE
ARE ALL DUMMY PLAYERS SINCE NONE OF THEM
ARE ESSENTIAL TO REACH QUOTA.
LET'S TAKE A LOOK AT ONE MORE EXAMPLE.
HERE WE DON'T KNOW THE QUOTA,
BUT WE'RE GIVEN THE WEIGHT OF THE FIVE PLAYERS.
THE FIRST QUESTION IS WHAT IS THE SMALLEST VALUE
THAT THE QUOTA Q CAN TAKE?
REMEMBER, THE QUOTA HAS TO BE MORE THAN HALF THE VOTES
SO WE'LL FIND THE SUM OF THESE VOTES, OR THESE WEIGHTS,
AND DIVIDE BY TWO.
AND THEN Q HAS TO BE GREATER THAN THAT VALUE.
SO WE'D HAVE 14 + 10 + 6 + 2 + 2
DIVIDED BY 2 = 34 DIVIDED BY 2 = 17.
REMEMBER, THE QUOTA HAS TO BE MORE THAN HALF
AND SINCE HALF WOULD BE 17 VOTES, OR A WEIGHT OF 17,
THAT MEANS THE SMALLEST VALUE OF Q
WOULD HAVE TO BE ONE MORE THAN 17 OR 18
SINCE WE WOULDN'T USE A FRACTION OF A VOTE
OR A FRACTION OF A WEIGHT.
SO THE SMALLEST VALUE OF Q IS 18,
THE LARGEST VALUE WOULD BE THE TOTAL NUMBER OF VOTES,
WHICH WE JUST FOUND WAS 34.
14 + 10 + 6 + 2 + 2.
SO THE LARGEST VALUE OF Q IS 34.
AND THEN FINALLY, WHAT IS THE VALUE OF Q,
THE QUOTA, IF AT LEAST 2/3 OF THE VOTES
ARE REQUIRED TO PASS A MOTION?
WELL, THAT WOULD BE 2/3 x 34, 34/1 THAT WOULD BE 68/3,
WHICH COMES OUT TO APPROXIMATELY 22.7.
SO WE'D HAVE TO ROUND UP TO 23.
SO IF AT LEAST 2/3 OF THE VOTES ARE REQUIRED, Q WOULD BE 23.
I HOPE YOU FOUND THIS HELPFUL.