922 CHAPTER 14 Voting and Apportionment Table 14.24 Summary of the Voting Methods and Whether They Satisfy the Fairness Criteria Plurality Method Borda Count Method Plurality with Elimination Method Pairwise Comparison Method Majority criterion Always satisfies May not satisfy Always satisfies Always satisfies Head-to-head criterion May not satisfy May not satisfy May not satisfy Always satisfies Monotonicity criterion Always satisfies May not satisfy May not satisfy Always satisfies Irrelevant alternatives criterion May not satisfy May not satisfy May not satisfy May not satisfy As we have shown, each of the voting methods introduced in Section 14.1 has the potential to violate at least one of the fairness criteria. Is there another voting method that does satisfy all four criteria? Mathematicians and political scientists struggled with this question for two centuries. Finally, in 1951, an economist, Kenneth Arrow, proved mathematically that there is no voting method that can satisfy all four fairness criteria. This result is called Arrow’s impossibility theorem . See the Profile in Mathematics . Profile in Mathematics Kenneth Arrow Kenneth Arrow (1921–2017), an economist, was interested in the theory of corporations and how corporations make decisions when their leaders have different opinions. His particular interest in stockholders led to his discovery that voting methods can be flawed. While working at the RAND Corporation, he studied how mathematical concepts, game theory in particular, apply to decision making in the military and in diplomatic affairs. As a result, he developed a set of properties for any fair and reasonable voting method. After experimenting for days, he realized that there was no voting method that would satisfy all reasonable conditions. This conclusion is called the Arrow impossibility theorem, which he proved in 1951. In 1972, Arrow received the Nobel Prize in Economics for his exceptional work in the theory of general economic equilibrium. Learning Catalytics Keyword: Angel-SOM-14.2 (See Preface for additional details.) Arrow’s Impossibility Theorem It is mathematically impossible for any democratic voting method to simultaneously satisfy each of the following fairness criteria: The majority criterion The head-to-head criterion The monotonicity criterion The irrelevant alternatives criterion Exercises SECTION 14.2 Warm Up Exercises In Exercises 1–8, fill in the blank with an appropriate word, phrase, or symbol(s). 1. If a candidate receives a majority of first-place votes in an election, that candidate should be declared the winner. This criterion is called the ________ criterion. Majority 2. A candidate who wins a first election, then gains additional support without losing any of the original support, should also win a second election. This criterion is called the ________ criterion. Monotonicity 3. If a candidate is favored when compared head-to-head with every other candidate in an election, that candidate should be declared the winner. This criterion is called the ________ criterion. Head-to-head 4. If a candidate is the winner of an election and in a second election one or more of the other candidates is removed, the previous winner should still be the winner. This criterion is called the ________ criterion. Irrelevant alternatives 5. A voting method that may not satisfy any of the fairness criteria is the ________ method. Borda count 6. A voting method that always satisfies the majority criterion and head-to-head criterion, but may not satisfy any other criterion, is the ________ method. Pairwise comparison WENN Rights Ltd/Alamy Stock Photo
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