ECONOMICS 1010 A
PROBLEM 7 -- Answers
J. ALLEN
Summer I, 2007
Three people, Alfred, Betty and Gemma, must vote for a single policy. There are three policy alternative:
1, 2 and 3. Alfred prefers 1 to 2, and 2 to 3. Betty prefers 2 to 3 and 3 to 1. Gemma prefers 3 to 1 and 1
to 2. The voting rule is one person, one vote.
1. If they vote on all three, with plurality winning, what happens?
Nothing wins. Obviously not a rational way to run a political system.
2. If they vote on 1 vs 2 first, then the winner vs 3, with the majority winning, which policy is chosen?
1 wins first, then with 1 vs 3, 3 wins
3. If they vote on 2 vs 3 first, then the winner vs 1, with the majority winning, which policy is chosen?
2 wins first, then with 2 vs 1, 1 wins
4. If they vote on 1 vs 3 first, then the winner vs 2, with the majority winning, which policy is chosen?
3 wins first, then with 3 vs 2, 2 wins
5. What does this say about decisions with three people and three possibilities?
First, the group is irrational; their joint choices violate transitivity. From 1 > 2 and 3 > 1. But from 4, 2
> 3. Transitivity is irrational. Second, it says the the policy CAN be chosen by the person who
controls the agenda. If you were Alfred, how would you arrange the agenda of voting? (2 vs 3 first)
6. If there are 40 Alfreds, 35 Bettys and 25 Gemmas, and they vote on all three with plurality
winning, what happens?
1 wins with 40 votes to 35 for 2 and 25 for 1
7. Suppose the 25 Gemmas now prefer 2 to 1. They still prefer 3 to 2, but now prefer 2 to 1. With
plurality winning, is this a rational decision? Why or why not? Is it a fair decision? Why or why
not?
Is this rational? What is rationality here? We need to define rationality a bit differently. There is no
violation of transitivity. Figure two for a first choice, one for a second and zero for a third: 1 gets 40*2
= 80; 2 gets 35*2 + 25 = 95; 3 gets 25*2 = 50. So transitivity holds: 2 > 1; 1 > 3; therefore 2 > 3. BUT
the 40 Alfreds are imposing their choice on the Betties and Gemmas, for whom 1 is the last choice.
Is that rational? In economics a rational decision is a person maximizing utility. Would the choice of
1 maximize the utility of the group? NO, if those weightings, two for first, one for second, are
interpreted as utility values.
To make this real, think of Alfred as a Conservative, Betty as a Liberal and Gemma as an NDP in last
year’s federal election. Or think of Alfred as a Liberal, Betty as a PC and Gemma as Reform in the
three elections Jean Cretien “won”. I’m sure you can find many other examples in Canada.