|
Gross-substitute Valuations
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] |
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] |
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] |
Auction Theory
Auction theory is an applied branch of economics which deals with how bidders act in auction markets and researches how the features of auction markets Incentivisation, incentivise predictable outcomes. Auction theory is a tool used to inform the design of real-world auctions. Sellers use auction theory to raise higher revenues while allowing buyers to procure at a lower cost. The conference of the price between the buyer and seller is an economic equilibrium. Auction theorists design rules for auctions to address issues which can lead to market failure. The design of these rulesets encourages optimal bidding strategies among a variety of informational settings. The 2020 Nobel Prize for Economics was awarded to Paul R. Milgrom and Robert B. Wilson “for improvements to auction theory and inventions of new Auction#Types, auction formats.” Introduction Auctions facilitate transactions by enforcing a specific set of rules regarding the resource allocations of a group of bidders. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Competitive Equilibrium
Competitive equilibrium (also called: Walrasian equilibrium) is a concept of economic equilibrium introduced by Kenneth Arrow and Gérard Debreu in 1951 appropriate for the analysis of commodity markets with flexible prices and many traders, and serving as the benchmark of efficiency in economic analysis. It relies crucially on the assumption of a competitive environment where each trader decides upon a quantity that is so small compared to the total quantity traded in the market that their individual transactions have no influence on the prices. Competitive markets are an ideal standard by which other market structures are evaluated. Definitions A competitive equilibrium (CE) consists of two elements: * A price function P. It takes as argument a vector representing a bundle of commodities, and returns a positive real number that represents its price. Usually the price function is linear - it is represented as a vector of prices, a price for each commodity type. * An allocation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of some vector space X is if it satisfies any of the following equivalent conditions: #If 0 \leq r \leq 1 is real and x, y \in C then r x + (1 - r) y \in C. #If 0 is a if holds for any real 0 is called if \operatorname f \neq \varnothing and f(x) > -\infty for x \in \operatorname f. Alternatively, this means that there exists some x in the domain of f at which f(x) \in \mathbb and f is also equal to -\infty. In words, a function is if its domain is not empty, it never takes on the value -\infty, and it also is not identically equal to +\infty. If f : \mathbb^n \to \infty, \infty/math> is a proper convex function then there exist some vector b \in \mathbb^n and some r \in \mathbb such that :f(x) \geq x \cdot b - r for every x where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Sign
In music theory, a natural (♮) is an accidental which cancels previous accidentals and represents the unaltered pitch of a note. A note is natural when it is neither flat () nor sharp () (nor double-flat nor double-sharp ) (nor triple-flat nor triple-sharp). Natural notes are the notes A, B, C, D, E, F, and G represented by the white keys on the keyboard of a piano or organ. On a modern concert harp, the middle position of the seven pedals that alter the tuning of the strings gives the natural pitch for each string. The scale of C major is sometimes regarded as the central, natural or basic major scale because all of its notes are natural notes, whereas every other major scale in the circle of fifths has at least one sharp or flat in it. The notes F, C, E, B, and most notes inflected by double-flats and double-sharps correspond in pitch with natural notes; however, they are not regarded as natural notes but rather as enharmonic equivalents of them and are just as much chromat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submodular Set Function
In mathematics, a submodular set function (also known as a submodular function) is a set function whose value, informally, has the property that the difference in the incremental value of the function that a single element makes when added to an input set decreases as the size of the input set increases. Submodular functions have a natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory (as functions modeling user preferences) and electrical networks. Recently, submodular functions have also found immense utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, active learning, sensor placement, image collection summarization and many other domains. Definition If \Omega is a finite set, a submodular function is a set function f:2^\rightarrow \mathbb, where 2^\Omega denotes the power set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
