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.
Experimental studies show that people value certain fairness criteria more than others. @cite_9 demonstrated that people are willing to sacrifice Pareto-efficiency in order to reach an envy-free allocation. To our knowledge no study was ever conducted to unfold the relationship between monotonicity and efficiency or proportionality. However some indirect evidence points toward that monotonicity of the solution is in some cases as important as proportionality. The so called , where electoral seats have to be distributed among administrative regions provides the most notorious examples. The seat distribution of the US House of Representatives generated many monotonicity related anomalies in the last two centuries. The famous Alabama-paradox, as well as the later discovered population and new state paradoxes pressed the legislators to adopt newer and newer apportionment rules. The currently used seat distribution method is free from such anomalies, however it does not satisfy the so called Hare-quota, a basic guarantee of proportionality . We view this as an evidence that monotonicity is as important fairness axiom as the classic axioms of proportionality and envy-freeness.
@cite_9
2094747375
Envy is sometimes suggested as an underlying motive in the assessment of different economic allocations. In the theoretical literature on fair division, following Foley [Foley, D. (1967), Yale Economic Essays, 7, 45–98], the term “envy” refers to an intrapersonal comparison of different consumption bundles. By contrast, in its everyday use “envy” involves interpersonal comparisons of well-being. We present, discuss results from free-form bargaining experiments on fair division problems in which inter-and intrapersonal criteria can be distinguished. We find that interpersonal comparisons play the dominant role. The effect of the intrapersonal criterion of envy freeness is limited to situations in which other fairness criteria are not applicable.