Intransitivity

Definition

Intransitivity is a property of binary relations where the relation does not satisfy the transitive property. In other words, if A is related to B and B is related to C, it does not necessarily mean that A is also related to C.

5 Must Know Facts For Your Next Test

1. Intransitivity can lead to paradoxes and inconsistencies in decision-making, particularly in the context of voting and social choice theory.
2. The Condorcet paradox, where majority preferences can be intransitive, is an example of intransitivity in the democratic system.
3. Intransitivity can result in the Condorcet winner (the alternative that beats all others in pairwise comparisons) not being selected as the final outcome.
4. Arrow's Impossibility Theorem states that there is no voting system that can guarantee a transitive social preference order while satisfying certain democratic principles.
5. Intransitivity can lead to strategic voting, where voters may not truthfully reveal their preferences to achieve a desired outcome.

Review Questions

• Explain how intransitivity can lead to paradoxes and inconsistencies in the democratic system of government.
  • Intransitivity can create paradoxes and inconsistencies in the democratic system because it means that the preferences of voters may not be transitive. For example, in the Condorcet paradox, a majority of voters may prefer option A over option B, option B over option C, and option C over option A. This circular preference cycle violates the transitive property and can lead to situations where the outcome of an election does not reflect the true preferences of the voters, potentially undermining the legitimacy of the democratic process.
• Describe the relationship between intransitivity and the Condorcet winner in the context of voting and social choice theory.
  • Intransitivity can prevent the Condorcet winner, the alternative that beats all others in pairwise comparisons, from being selected as the final outcome. When preferences are intransitive, there may not be a clear Condorcet winner, or the Condorcet winner may not be chosen due to the voting system's inability to capture the true preferences of the voters. This can lead to outcomes that do not reflect the majority's preferences, undermining the principles of democratic decision-making.
• Analyze how intransitivity can incentivize strategic voting behavior in a democratic system.
  • Intransitivity can create incentives for strategic voting, where voters may not truthfully reveal their preferences in order to achieve a desired outcome. When preferences are intransitive, voters may realize that voting for their true first choice could lead to a less preferred outcome. As a result, they may strategically vote for a different option, even if it is not their true first choice, in an attempt to influence the final outcome. This strategic voting behavior can further distort the democratic process and undermine the principle of one person, one vote.

Related terms

Binary Relation: A binary relation is a set of ordered pairs that describes a relationship between elements of two sets.

Transitive Property: The transitive property states that if A is related to B and B is related to C, then A is also related to C.

Preference Cycles: Preference cycles occur when there is no clear majority preference among a set of alternatives, leading to intransitive preferences.