- Also called:
- Arrow’s theorem
- Key People:
- Kenneth J. Arrow
- Related Topics:
- political science
- common good
impossibility theorem
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What is an impossibility theorem in political science?
In political science, an impossibility theorem, formulated by Kenneth J. Arrow, posits that assessing the common good is generally impossible. The theorem assumes agents have well-ordered preferences in collective choice situations, stipulating conditions for a valid decision procedure, such as allowing individuals to have any rational preference ordering, and the absence of a dictator. It demonstrates the impossibility of constructing a procedure that results in a collectively rational expression of individual desires. This challenges the notion of a collective democratic will and denies the existence of objective criteria for collective decision-making.
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Why are impossibility theorems important for voting systems?
Impossibility theorems are important for voting systems because they challenge the idea that a voting system can perfectly reflect the will of the people.
The impossibility theorem demonstrates the difficulties of aggregating individual preferences into a collective, rational decision. It assumes that voters have well-defined preferences and that the voting system adheres to minimal, seemingly uncontroversial conditions, such as allowing any rational preference order, avoiding dictatorships, and respecting unanimous preferences.
These theorems highlight the limitations and trade-offs inherent in any voting mechanism. They show that no system can perfectly satisfy all desirable criteria for fairness and representation, which has significant implications for democratic theory and political economy.
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What is Arrow's impossibility theorem?
Arrow's impossibility theorem states that assessing the common good is generally impossible. Kenneth J. Arrow formulated the theorem, a central element of rational choice theory, in his 1951 book Social Choice and Individual Values. The theorem stipulates that minimal conditions must apply to the decision procedure for its result to be valid.
The theorem assumes that agents have complete and well-ordered preferences over outcomes under consideration. It stipulates that there should be no dictator, that the collective ranking over outcomes remains unchanged if one of the alternatives ceases to be considered, and that unanimous preference over a pair of outcomes implies a collective preference over that pair.
Given these assumptions, the theorem demonstrates the impossibility of constructing any procedure that results in a collectively rational expression of individual desires.
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How does Arrow's theorem show problems with fairness in elections?
Arrow's impossibility theorem demonstrates the inherent difficulties in aggregating individual preferences to reflect a collective will. The theorem posits that no electoral system can perfectly convert individual preferences into a fair outcome if voters have three or more options. It shows that any voting method will inevitably violate one or more fairness criteria when attempting to produce a social order from individual preferences.
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What are some rules that Arrow's theorem says can't all be satisfied at once?
Arrow's impossibility theorem states that no social decision-making system can satisfy all of the following conditions simultaneously:
- Non-dictatorship: The preferences of a single individual should not become the group preference without considering the preferences of other individuals.
- Pareto efficiency: If every individual prefers one option over another, then the group should also prefer that option.
- Independence of irrelevant alternatives: The group's ranking of two options should depend only on individual preferences for those two options, not on preferences for other options.
- Unrestricted domain: All individual preference orderings should be possible inputs into the decision-making process.
- Social rationality: The group's preferences should be logically consistent (e.g., transitive).
The theorem demonstrates the inherent difficulties in aggregating individual preferences into a coherent and representative collective decision.
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How have impossibility theorems influenced the way we design voting systems today?
Arrow's impossibility theorem demonstrates the inherent difficulties in perfectly aggregating individual preferences into a collective decision, influencing the design of voting systems by highlighting the trade-offs between various desirable criteria.
This theorem has led to a deeper understanding of the limitations of different voting mechanisms and a focus on designing systems that minimize specific undesirable outcomes, such as disproportionality or the potential for strategic manipulation. It has also spurred the development of new voting methods and a reevaluation of how democratic decision procedures can best represent individual preferences, acknowledging the absence of a perfect solution.
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impossibility theorem, in political science, the thesis that it is generally impossible to assess the common good. It was first formulated in Social Choice and Individual Values (1951) by Kenneth J. Arrow, who was awarded (with Sir John R. Hicks) the Nobel Prize for Economics in 1972 partially in recognition of his work on the theorem. As a central element of rational choice theory, which attempts to explain political behaviour as the rational pursuit of individual self-interest, the impossibility theorem posed a major challenge to 20th-century welfare economics and to a reevaluation of how democratic decision procedures arrive at representative expressions of individuals’ preferences. It has also been used to challenge the concept of “the public” as a meaningful social entity.
The impossibility theorem assumes that agents have complete and well-ordered preferences over all the outcomes under consideration in a collective choice situation. This requires that agents know whether they prefer one in any pair of possible outcomes, and it requires that agents’ preferences obey the logical relationship of transitivity, which requires that if Adams is preferred to Madison and Madison is preferred to Washington, then Washington cannot be preferred to Adams. The impossibility theorem considers cases in which three or more agents make a collective choice from three or more alternatives in situations as diverse as democratic voting, establishing public policies that reflect social welfare, and the marketplace. The theorem is constructed to resolve the question of whether there is any mathematical procedure for amalgamating individual preferences that results in a collectively rational preference ordering of all the possible outcomes.
In addition to assuming that individuals’ preferences are rational, the theorem stipulates that four minimal conditions must apply to the decision procedure for its result to be valid. The theorem requires that individuals be permitted to have any rational preference ordering over alternatives, that there not be a single dictator whose preference over a single pair of alternatives holds for the group decision, that the collective ranking over outcomes remains unchanged if one of the alternatives ceases to be considered, and that a unanimous preference over a pair of outcomes implies a collective preference over that pair. These requirements are generally regarded as beyond controversy.
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The theorem proves that, given these minimal assumptions, it is impossible to construct any procedure that results in a collectively rational expression of individual desires. Though highly technical in its statement, the theorem has important implications for philosophies of democracy and political economy. The theorem rejects the notion of a collective democratic will, whether derived through civic deliberation or construed by experts who paternalistically apply knowledge of what is best for a population. The theorem also denies that there could be objective basic needs or universal criteria that any procedure for collective decision making should recognize, such as minimal nutrition standards or human rights.