The following questions review the main ideas of this chapter. Write your answers to the...

The following questions review the main ideas of this chapter. Write your answers to the questions and then refer to the pages listed by number to make certain that you have mastered these ideas.

What are the four fairness criteria? pgs. 163–166 What does the Arrow impossibility theorem conclude? pg. 170 Which criteria are always satisfied for each of the four criteria? pg. 170

Reference:

We have seen that each of the voting methods introduced in Section 3.1 can sometimes violate a fairness criterion. You might ask why we do not present a better method, one that satisfies all the criteria all the time. The reason we do not present such a “perfect” voting method is that none exists! The Arrow impossibility theorem tells us that it is a mathematical fact that even if all the voters assign preferences to all the alternatives, there is no voting method that will always satisfy the majority, head-to-head, monotonicity, and irrelevant-alternatives criterion.

Table 3.25 shows which criteria are satisfied by the various voting methods.

Notice that the pairwise comparison method satisfies the majority criterion and the monotonicity criterion. Moreover, the pairwise comparison method was designed to satisfy the head-to-head criterion. Although the pairwise comparison method does not always satisfy the irrelevant-alternatives criterion, it may still look like the best method we have seen. However, there is another problem with the pairwise comparison method: it often fails to produce a winner. For example, consider the preferences shown in Table 3.26.

Notice that in two columns, candidate A is ranked higher than candidate B. Six voters ranked A first and B second, and four voters ranked A second and B third. Only five voters ranked B higher than A. Thus, voters favored A to B by a 10-to-5 margin. Similarly, B is preferred to C by an 11-to-4 margin (11 =6+5 from the first and second columns); and C is preferred to A by a 9-to-6 margin (9 = 5 + 4 from the second and third columns). Thus, in the pairwise comparison method, there is no winning candidate. Stated another way, we can say that A is preferred to B, B is preferred to C, and C is preferred to A, a useless set of conclusions.