the Monotonicity Criterion in Decision Theory: Importance and Application in Social Choice Theory

Monotonicity Criterion

If candidate X is the winner then X should still win even if they are moved higher on the ballot

The monotonicity criterion is a principle used in decision theory that states that if a decision maker’s preferences satisfy the monotonicity property, then a new option that is preferred to one of the original options will not make the original option more attractive to the decision maker. In other words, the criterion implies that whenever an alternative is added that is preferred to one of the existing alternatives, the relative ranking of all other alternatives should remain unchanged.

More formally, suppose a decision maker has a set of alternatives, A, and a preference relation, ≻, over A. ≻ is said to satisfy the monotonicity criterion if, for any a, b, c ∈ A:

1. if a ≻ b, then adding an alternative c that is preferred to b will not cause the decision maker to prefer b to a (i.e. a ≻ c ⇒ a ≻ b)
2. if a ∼ b (i.e. a is indifferent to b), then adding an alternative c that is preferred to b will not cause the decision maker to prefer b to a (i.e. a ∼ c ⇒ a ∼ b)

In other words, the monotonicity criterion requires that adding a new option to the set of alternatives that is preferred to one of the original options should not lead to a situation where the decision maker prefers the original option to the new option.

The monotonicity criterion is often used in the context of social choice theory, where it is a desirable property of collective decision-making mechanisms. It ensures that if a candidate is preferred to another, then adding a third candidate will not cause the original candidate to become less preferable. This property is often seen as a requirement for the fairness and stability of collective decisions.

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