We study monotonicity properties of solutions to the classic problem of fair cake-cutting—dividing a heterogeneous resource among agents with different preferences. Resource- and population-monotonicity relate to scenarios where the cake, or the number of participants who divide the cake, changes. It is required that the utility of all participants change in the same direction: either all of them are better-off (if there is more to share or fewer to share among) or all are worse-off (if there is less to share or more to share among). We formally introduce these concepts to the cake-cutting setting and show that they are violated by common division rules. In contrast, we prove that the Nash-optimal rule—maximizing the product of utilities—is resource-monotonic and population-monotonic, in addition to being Pareto-optimal, envy-free and satisfying a strong competitive-equilibrium condition. Moreover, we prove that it is the only rule among a natural family of welfare-maximizing rules that is both proportional and resource-monotonic.
The consistency axiom (cf. @cite_20 or @cite_2 ) is related to population-monotonicity, but it is fundamentally different as in that case the leaving agents take their fair shares with them.
@cite_20 @cite_2
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In this paper we study several monotonicity axioms in approval-based multi-winner voting rules. We consider monotonicity with respect to the support received by the winners and also monotonicity in the size of the committee. Monotonicity with respect to the support is studied when the set of voters does not change and when new voters enter the election. For each of these two cases we consider a strong and a weak version of the axiom. We observe certain incompatibilities between the monotonicity axioms and well-known representation axioms (extended proportional justified representation) for the voting rules that we analyze and provide formal proofs of incompatibility between some of these axioms and perfect representation. Abstract We consider the problem of fairly allocating an infinitely divisible commodity among agents with single-peaked preferences. We search for methods of performing this division, or solutions , that satisfy the following property of consistency : any recommendation made for any economy is in agreement with the recommendation made for any "reduced" economy obtained by imagining the departure of some of the agents with their allotted consumptions. Our main result is that essentially all efficient subsolutions of the no-envy solution satisfying consistency must contain a certain solution known as the uniform rule. We also characterize the uniform rule on the basis of a "converse" of consistency and the distributional requirement of individual rationality from equal division. Journal of Economic Literature Classification Numbers: D63, D71. Our objective is to investigate the implications of the “replacement principle” for the fair allocation of an infinitely divisible commodity among agents with single-peaked preferences. The principle says that when one of the components of the data entering the description of the problem to be solved changes, all of the relevant agents should be affected in the same direction: they all gain or they all lose. We apply it to situations in which the preferences of one of the agents may change, under the name ofwelfare-domination under preference-replacement. We show that there is no selection from the no-envy and Pareto solution satisfying it. Then, we weaken it by limiting its application to situations in which the change is not so disruptive that it turns the economy from one in which there is “too little” of the commodity to one in which there is “too much,” or conversely. We show that there is only one selection from the no-envy and Pareto solution satisfying this property andreplication-invariance. It is the “uniform rule,” a solution that has played a central role in previous analyses of the problem.Journal of Economic LiteratureClassification Numbers: D51, D63, D70. We prove uniform consistency of Random Survival Forests (RSF), a newly introduced forest ensemble learner for analysis of right-censored survival data. Consistency is proven under general splitting rules, bootstrapping, and random selection of variables--that is, under true implementation of the methodology. Under this setting we show that the forest ensemble survival function converges uniformly to the true population survival function. To prove this result we make one key assumption regarding the feature space: we assume that all variables are factors. Doing so ensures that the feature space has finite cardinality and enables us to exploit counting process theory and the uniform consistency of the Kaplan-Meier survival function.