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Miller defined two main classes of monotonicity failure in 2012, which have been repeated in later papers: [14] [6] Upward monotonicity failure: Given the use of voting method V and a ballot profile B in which candidate X is the winner, X may nevertheless lose in ballot profile B' that differs from B only in that some voters rank X higher in B' than in B
In the context of search algorithms monotonicity (also called consistency) is a condition applied to heuristic functions. A heuristic h ( n ) {\displaystyle h(n)} is monotonic if, for every node n and every successor n' of n generated by any action a , the estimated cost of reaching the goal from n is no greater than the step cost of getting to ...
Both imply very strong monotonicity properties. Both types of functions have derivatives of all orders. In the case of an absolutely monotonic function, the function as well as its derivatives of all orders must be non-negative in its domain of definition which would imply that the function as well as its derivatives of all orders are ...
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non-increasing, or non-decreasing.
The most common cause of no-show paradoxes is the use of instant-runoff (often called ranked-choice voting in the United States).In instant-runoff voting, a no-show paradox can occur even in elections with only three candidates, and occur in 50%-60% of all 3-candidate elections where the results of IRV disagree with those of plurality.
Black's method satisfies the following criteria: Unrestricted domain; Non-imposition (a.k.a. citizen sovereignty) Non-dictatorship; Homogeneity; Condorcet criterion; Majority criterion; Pareto criterion (a.k.a. unanimity) [3] Monotonicity criterion [3] Majority loser criterion [3] Condorcet loser criterion [3] Reversal symmetry [3 ...
The test was devised by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test. [1] [2] [3] For a generalization, see Dirichlet's test. [4] [5] [6]
This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by the monotonicity. The limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous.