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This page compares the properties of several typical utility functions of divisible goods. These functions are commonly used as examples in consumer theory. The functions are ordinal utility functions, which means that their properties are invariant under positive monotone transformation.
Infinite divisibility arises in different ways in philosophy, physics, economics, order theory (a branch of mathematics), and probability theory (also a branch of mathematics). One may speak of infinite divisibility, or the lack thereof, of matter , space , time , money , or abstract mathematical objects such as the continuum .
Fair item allocation is a kind of the fair division problem in which the items to divide are discrete rather than continuous. The items have to be divided among several partners who potentially value them differently, and each item has to be given as a whole to a single person. [1]
Goods are items that are usually (but not always) tangible, such as pens or apples. Services are activities provided by other people, such as teachers or barbers . Taken together, it is the production , distribution , and consumption of goods and services which underpins all economic activity and trade .
With four goods: yes for 2 agents with general valuations, no for 3 agents with general valuations, no for 4 agents, even with additive valuations. [28] With five or more goods: no for two agents with general valuations. Open conjectures: When there are two agents with additive valuations, CE for almost all incomes exists for any number of goods.
Most tangible goods, both durable and nondurable, are rival goods. A hammer is a durable rival good. One person's use of the hammer prevents others from using the hammer at the same time. However, the first user does not "use up" the hammer, meaning that some rival goods can still be shared through time.
A special case of this setting is when dividing rooms in an apartment between tenants. It is characterized by three requirements: (a) the number of agents equals the number of items, (b) each agent must get exactly one item (room), (c) the total amount of money paid by the agents must equal a fixed constant, which represents the total apartment rent.
A proportional division may not exist. For example, if the number of people is larger than the number of items, then some people will get no item at all and their value will be zero. Nevertheless, such a division exists with high probability for indivisible items under certain assumptions on the valuations of the agents.