In the previous question, you showed that sometimes there may be no policy that beats every other

Question:

In the previous question, you showed that sometimes there may be no policy that beats every other policy in a majority rule election and, as a result, the agenda can determine the outcome. In the previous question, all of the policy choices on the agenda were as good as any other, but this is not always the case. Imagine that three voters, L, M, and R, are choosing between seven candidates. The preferences of the voters are given in the following table. Voter M, for example, likes Grumpy the best and Doc the least image text in transcribed

a. Imagine that we vote according to a given agenda starting with Happy vs. Dopey.
Who wins? We will help you with this one.

Voter L ranks Happy above Dopey, so voter L will vote for Happy. Voter M prefers Dopey to Happy, so voter M will vote for Dopey.
Voter R ranks Dopey above Happy so voter R will vote for Dopey. So wins.

b. Now take the winner from part a and match him against Grumpy. Who wins?

c. Now take the winner from part b and match him against Sneezy. Who wins?

d. Now take the winner from part c and match him against Sleepy. Who wins?

e. Now take the winner from part d and match him against Bashful. Who wins?

f. Finally, take the winner from part e and match him against Doc. Who wins?
g. We have now run through the entire agenda so the winner from part f is the final winner.
Here is the point. Look carefully at the preferences of the three voters. Compare the preferences of each voter for Happy (or Grumpy or Dopey) with the final winner.
Fill in the blank: Majority rule has led to an outcome that regards as worse than some other possible outcome. The answer to this question should shock you.
This question is drawn from the classic and highly recommended introduction to game theory, Thinking Strategically by Avinash K.
Dixit and Barry J. Nalebuff (New York:
W.W. Norton, 1993).