Utility functions on indivisible goods information
Some branches of economics and game theory deal with indivisible goods, discrete items that can be traded only as a whole. For example, in combinatorial auctions there is a finite set of items, and every agent can buy a subset of the items, but an item cannot be divided among two or more agents.
It is usually assumed that every agent assigns subjective utility to every subset of the items. This can be represented in one of two ways:
An ordinal utility preference relation, usually marked by . The fact that an agent prefers a set to a set is written . If the agent only weakly prefers (i.e. either prefers or is indifferent between and ) then this is written .
A cardinal utility function, usually denoted by . The utility an agent gets from a set is written . Cardinal utility functions are often normalized such that , where is the empty set.
A cardinal utility function implies a preference relation: implies and implies . Utility functions can have several properties.[1]
^Gul, F.; Stacchetti, E. (1999). "Walrasian Equilibrium with Gross Substitutes". Journal of Economic Theory. 87: 95–124. doi:10.1006/jeth.1999.2531.
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Some branches of economics and game theory deal with indivisiblegoods, discrete items that can be traded only as a whole. For example, in combinatorial...
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equilibrium with that assignment. In the case of indivisible item assignment, when the utilityfunctions of all agents are GS (and thus an equilibrium exists)...
(2018). "Fair Allocation of IndivisibleGoods: Improvements and Generalizations". Proceedings of the 2018 ACM Conference on Economics and Computation....
Pareto efficient social choice function must be a linear combination of the utilityfunctions of each individual utilityfunction (with strictly positive weights)...
additive utilities. They show that a fractional CE (where some goods are divided) can always be rounded to an integral CE (where goods remain indivisible), by...
classes of goods. When the number of agents is constant there is an FPTAS using Woeginger technique. For agents with submodular utilityfunctions: Golovin...
possible subset of items. It is usually assumed that the utilityfunctions are monotone set functions, that is, Z 1 ⊇ Z 2 {\displaystyle Z_{1}\supseteq Z_{2}}...
pay for public goods according to their marginal benefits. In other words, they pay according to the amount of satisfaction or utility they derive from...
Y|y\succeq z\}|} The AU extension is based on the notion of an additive utilityfunction. Many different utilityfunctions are compatible with a given ordering...
agents' utilityfunctions. Concavity: the most general assumption (made by Fisher and Arrow&Debreu) is that the agents' utilities are concave functions, i...
includes both substitute goods and independent goods, and only rules out complementary goods. See Gross substitutes (indivisible items). Polterovich, V...
probability for indivisible items under certain assumptions on the valuations of the agents. Suppose the agents have cardinal utilityfunctionson items. Then...
of the participants. Without money, it may be impossible to allocate indivisible items fairly. For example, if there is one item and two people, and the...
dividing a set of indivisible heterogeneous goods (e.g., rooms in an apartment), and simultaneously a homogeneous divisible bad (the rent on the apartment)...
, … , x n ′ } {\displaystyle \{x_{1}',\dots ,x_{n}'\}} where, for utilityfunction u i {\displaystyle u_{i}} for each agent i {\displaystyle i} , u i...
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