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.
@cite_12 defines the , which requires that, whenever any change happens in the environment, the welfare of all agents not responsible for the change should be affected in the same direction --- they should all be made at least as well off as they were initially or they should all be made at most as well off. This is the most general way of expressing the idea of among agents. The PM and RM axioms are special cases of this principle.
@cite_12
2027956524 2081359028 2088903596 2149247676
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. Much of the dissatisfaction with Professor Arrow's formulation (1963) of the problem of social choice rests on Arrow's imposition of the controversial "independence of irrelevant alternatives" condition (A3). Arrow himself has ceased to insist upon this condition in its strong original form (Arrow 1967). The chief objection to A3 is that it prevents the collectivechoice rule from responding to intensity of individual preference, making the choice rule unacceptable on a priori grounds (Kemp and Asimakopulos 1952; Hildreth 1953; Coleman 1966; Quirk and Saposnik 1968; Sen 1970). This view is not correct if intensity of preference is measured by an ability to make side payments in order to influence collective decision making. I show that a modified version of A3 which takes explicit account of intensity of preference actually implies the original independence condition for realistic environment sets. This will be true no matter how side payments of private goods and services are evaluated. A modest amount of notation is required before the independence condition can be given formal expression. The set X, the nonnegative orthant of m-dimensional Euclidean space, is the space of hypothetical states of the economy. There are n individuals whose preferences are to be considered by any collective decision-making process, and to each individual i there corresponds a binary relation Ri which is complete on X. The term 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. We study a novel mechanism design model in which agents each arrive sequentially and choose one action from a set of actions with unknown rewards. The information revealed by the principal affects the incentives of the agents to explore and generate new information. We characterize the optimal disclosure policy of a planner whose goal is to maximize social welfare. One interpretation of our result is the implementation of what is known as the "wisdom of the crowd." This topic has become increasingly relevant with the rapid spread of the Internet over the past decade.