2. Three individuals, i, j, and k, are voting over four outcomes q, r, s, and t.
ID: 3168296 • Letter: 2
Question
2. Three individuals, i, j, and k, are voting over four outcomes q, r, s, and t. Their preference orderings are If they vote honestly using a round-robin tournament, are there any group preference cycles? Now k changes his mind, and switches his preferences to tP,sPrPg. Are there any group preference cycles? How does this problem illustrate the idea of "rational man, irrational society?" 3. Using the same preferences as above (before k changes his mind), can i fashion a sequential agenda (i.e., each outcome introduced sequentially and only retained if it beats the exist- ing winner) such that her top choice wins? What about after k changes his mind? Then, identify agendas that j and k could design to secure their top choices (for both before and after k changes his mind, if possible). In general, can an agenda be fashioned that leads to the defeat of an alternative favored by a majority over each other alternative?Explanation / Answer
2. A round-robin tournament is a tournament in which player meets the other turn by turn. This is different from an elimination tournament where one player eliminates the other and moves ahead to meet the winner of another game. The loser of the the games never meet each other or the winners of the other game.
In round-robin tournament there is a concept called group preference cycle. This can be explained with a example. Suppose there are a number of people voting for three candidates 1, 2 and 3. When there are only to players 1 and player 2 playing, the people prefer player 1 over 2. When player 2 and 3 are playing, the people prefer player 2 over 3. This should imply that they prefer player 1 over 3. However, if they prefer player 3 over 1, when players 1 and 3 are playing, they we are looking at a preference cycle. This is called the group preference cycle.
To find if there exists a group preference cycle in the given problem, we need to create a 4 x 3 matrix for the three voters i, j and k with four outcomes q, r, s and t. This can be done as follows :
Voters --> i j k
1st Choice qPi rPj tPk
2nd Choice sPi qPj rPk
3rd Choice rPi tPj sPk
4th Choice t s q
If we look at outcomes q and r,
q is prefered over r by voter i, but r is prefered over q by voters j and k. So, r > q.
Between outcomes r and s
r is prefered over s by voters j and k, but s is prefered over r by voters i. So, r > s.
Between outcomes s and t,
s is prefered over t by voter i, but t is prefered over s by voters j and k. So, t > s.
Between outcomes t and q,
q is prefered over t by voter i and j, but t is prefered over q by voter k. So, q > t.
Between outcomes q and s,
q is prefered over s by voter i and j, but s is prefered over q by voter k. So, q > s.
and between outcomes r and t,
r is prefered over t by voter i and j, but t is prefered over r by voter k. So, r > t
So, we can see clearly, that the voters prefer outcome r the most followed by q, then t and then s.
So, r > q > t > s. Hence, there is no group preference cycle in this case.
If voter k changes his/her mind and prefers s over r, The 4 x 3 matrix can be written as :
Voters i j k
1st preference qPi rPj tPk
2nd preference sPi qPj sPk
3rd preference rPi tPj rPk
4th preference t s q
If we look at outcomes q and r,
q is prefered over r by voter i, but r is prefered over q by voters j and k. So, r > q.
Between outcomes r and s
r is prefered over s by voter j, but s is prefered over r by voters i and k. So, s > r.
Between outcomes s and t,
s is prefered over t by voter i, but t is prefered over s by voters j and k. So, t > s.
Between outcomes t and q,
q is prefered over t by voter i and j, but t is prefered over q by voter k. So, q > t.
Between outcomes q and s,
q is prefered over s by voter i and j, but s is prefered over q by voter k. So, q > s.
and between outcomes r and t,
r is prefered over t by voter i and j, but t is prefered over r by voter k. So, r > t
We see from the above given preferences that, r is prefered over q and q is prefered over s. So, it should imply that r is prefered over s. However, between outcomes r and s, s is prefered by i and k and r is prefered by only voter j. So, s > r.
Similarly, r is prefered over t, and t is prefered over s. This should imply that r is prefered over s or r > s. However, between outcomes r and s, s is prefered by i and k and r is prefered over s by only voter j. So, s > r.
This implies that there is a group preference cycle in the game when voter k changes his/her preference from outcome r to s.
This explains the rational man irrational society as the preference of one person does not change, however when in clubbed in a group, the outcomes change and seem irriational. In the second situation, k always prefers s over r and t over r, however, when clubbed with voters i and j, the winner of the outcome between r, s and t becomes a cycle. This is because, voter i prefers s over r and r over t. Voter j, however prefers outcome r over s and r over t.
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