There is no single straightforward generalization of polygon monotonicity to higher dimensions. The "Shapley value" is the unique payoff vector that is efficient, symmetric, and satisfies monotonicity. The definition of the monotonicity criterion with regard to these methods is disputed. Independence of Irrelevant Alternatives can be substituted with a Resource monotonicity axiom. It is standard that local strict monotonicity suffices in one dimension. It is truthful because the median function satisfies the strong monotonicity property. The Alabama paradox is an example of violation of the resource monotonicity axiom. Extrapolation under the weak monotonicity assumption will provide a bound, rather than a point-estimate. In the special case when is totally ordered, monotonicity of implies monotonicity of its inverse. Given the exclusion and monotonicity assumption, the second half of this equation should be zero. One can examine a function's monotonicity without calculus. They established monotonicity properties for the value function and the optimal dynamic bid policy. Some basic properties include monotonicity and diagonal jump. Again, a similar reasoning yields monotonicity of . This second change (a removal caused by an addition) violates the condition of monotonicity.