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Utility Functions On Indivisible Goods
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 \succ. The fact that an agent prefers a set A to a set B is written A \succ B. If the agent only weakly prefers A (i.e. either prefers A or is indifferent between A and B) then this is written A \succeq B. * A cardinal utility function, usually denoted by u. The utility an agent gets from a set A is written u(A). Cardinal utility functions are often normalized such that u(\emptyset)=0, where \emptyset is the empty set. A cardinal utility function implies a preference relation: u(A)>u(B) implies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interactions of Agent (economics), economic agents and how economy, economies work. Microeconomics analyzes what's viewed as basic elements in the economy, including individual agents and market (economics), markets, their interactions, and the outcomes of interactions. Individual agents may include, for example, households, firms, buyers, and sellers. Macroeconomics analyzes the economy as a system where production, consumption, saving, and investment interact, and factors affecting it: employment of the resources of labour, capital, and land, currency inflation, economic growth, and public policies that have impact on glossary of economics, these elements. Other broad distinctions within economics include those between positive economics, desc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subadditive Set Function
In mathematics, a subadditive set function is a set function whose value, informally, has the property that the value of function on the union of two sets is at most the sum of values of the function on each of the sets. This is thematically related to the subadditivity property of real-valued functions. Definition Let \Omega be a set and f \colon 2^ \rightarrow \mathbb be a set function, where 2^\Omega denotes the power set of \Omega. The function ''f'' is ''subadditive'' if for each subset S and T of \Omega, we have f(S) + f(T) \geq f(S \cup T). Examples of subadditive functions Every non-negative submodular set function is subadditive (the family of non-negative submodular functions is strictly contained in the family of subadditive functions). The function that counts the number of sets required to cover a given set is subadditive. Let T_1, \dotsc, T_m \subseteq \Omega such that \cup_^m T_i=\Omega. Define f as the minimum number of subsets required to cover a given set. F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Single-minded Agent
In computational economics, a single-minded agent is an agent who wants only a very specific combination of items. The valuation function of such an agent assigns a positive value only to a specific set of items, and to all sets that contain it. It assigns a zero value to all other sets. A single-minded agent regards the set of items he wants as purely complementary goods. Various computational problems related to allocation of items are easier when all the agents are known to be single-minded. For example: * Revenue-maximizing auctions. * Multi-item exchange. * Fair cake-cutting and fair item allocation. * Combinatorial auctions. * Envy-free pricing. Comparison to other valuation functions As mentioned above, a single-minded agent regards the goods as purely complementary goods In contrast, an additive agent assigns a positive value to every item, and assigns to every bundle a value that is the sum of the items in contains. An additive agent regards the set of items he want ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utility Functions On Divisible Goods
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. For example, the Cobb–Douglas function could also be written as: w_x \log + w_y\log. Such functions only become interesting when there are two or more goods (with a single good, all monotonically increasing functions are ordinally equivalent). The utility functions are exemplified for two goods, x and y. p_x and p_y are their prices. w_x and w_y are constant positive parameters and r is another constant parameter. u_y is a utility function of a single commodity (y). I is the total income (wealth) of the consumer. References * {{Cite book, author=Hal Varian, author-link=Hal Varian, title=Intermediate micro-economics, isbn=0393927024, year=2006 chapter 5. Acknowledgements This page has b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of a "typical" value. Median income, for example, may be a better way to suggest what a "typical" income is, because income distribution can be very skewed. The median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median is not an arbitrarily large or small result. Finite data set of numbers The median of a finite list of numbers is the "middle" number, when those numbers are list ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the ''local'' or ''relative'' extrema), or on the entire domain (the ''global'' or ''absolute'' extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x''∗, if for all ''x'' in ''X''. The value of the function at a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the ''local'' or ''relative'' extrema), or on the entire domain (the ''global'' or ''absolute'' extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x''∗, if for all ''x'' in ''X''. The value of the function at a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, and 9 (summing to 25) is 5. Depending on the context, an average might be another statistic such as the median, or mode. For example, the average personal income is often given as the median—the number below which are 50% of personal incomes and above which are 50% of personal incomes—because the mean would be higher by including personal incomes from a few billionaires. For this reason, it is recommended to avoid using the word "average" when discussing measures of central tendency. General properties If all numbers in a list are the same number, then their average is also equal to this number. This property is shared by each of the many types of average. Another universal property is monotonicity: if two lists of numbers ''A'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Summation
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined. Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article. The summation of an explicit sequence is denoted as a succession of additions. For example, summation of is denoted , and results in 9, that is, . Because addition is associative and commutative, there is no need of parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0. Very often, the elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aggregate Function
In database management, an aggregate function or aggregation function is a function where the values of multiple rows are grouped together to form a single summary value. Common aggregate functions include: * Average (i.e., arithmetic mean) * Count * Maximum * Median * Minimum * Mode * Range * Sum Others include: * Nanmean (mean ignoring NaN values, also known as "nil" or "null") * Stddev Formally, an aggregate function takes as input a set, a multiset (bag), or a list from some input domain and outputs an element of an output domain . The input and output domains may be the same, such as for SUM, or may be different, such as for COUNT. Aggregate functions occur commonly in numerous programming languages, in spreadsheets, and in relational algebra. The listagg function, as defined in the SQL:2016 standard aggregates data from multiple rows into a single concatenated string. Decomposable aggregate functions Aggregate functions present a bottleneck, because they poten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Social Welfare Function
In welfare economics, a social welfare function is a function that ranks social states (alternative complete descriptions of the society) as less desirable, more desirable, or indifferent for every possible pair of social states. Inputs of the function include any variables considered to affect the economic welfare of a society. In using welfare measures of persons in the society as inputs, the social welfare function is individualistic in form. One use of a social welfare function is to represent prospective patterns of collective choice as to alternative social states. The social welfare function provides the government with a simple guideline for achieving the optimal distribution of income. The social welfare function is analogous to the consumer theory of indifference-curve– budget constraint tangency for an individual, except that the social welfare function is a mapping of individual preferences or judgments of everyone in the society as to collective choices, which ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gross Substitutes (indivisible Items)
In economics, gross substitutes (GS) is a class of utility functions on indivisible goods. An agent is said to ''have a GS valuation'' if, whenever the prices of some items increase and the prices of other items remain constant, the agent's demand for the items whose price remain constant weakly increases. An example is shown on the right. The table shows the valuations (in dollars) of Alice and Bob to the four possible subsets of the set of two items: . Alice's valuation is GS, but Bob's valuation is not GS. To see this, suppose that initially both apple and bread are priced at $6. Bob's optimal bundle is apple+bread, since it gives him a net value of $3. Now, the price of bread increases to $10. Now, Bob's optimal bundle is the empty bundle, since all other bundles give him negative net value. So Bob's demand to apple has decreased, although only the price of bread has increased. The GS condition was introduced by Kelso and Crawford in 1982 and was greatly publicized by Gul an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
