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Tournament Solution
A tournament solution is a function that maps an oriented complete graph to a nonempty subset of its vertices. It can informally be thought of as a way to find the "best" alternatives among all of the alternatives that are "competing" against each other in the tournament. Tournament solutions originate from social choice theory, but have also been considered in sports competition, game theory, multi-criteria decision analysis, biology, webpage ranking, and dueling bandit problems. In the context of social choice theory, tournament solutions are closely related to Fishburn's C1 social choice functions, and thus seek to show who the best candidates are among all candidates. Definition A tournament (graph) T is a tuple (A,\succ) where A is a set of vertices (called ''alternatives'') and \succ is a connex and asymmetric binary relation over the vertices. In social choice theory, the binary relation typically represents the pairwise majority comparison between alternatives. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Web Search Engine
A search engine is a software system designed to carry out web searches. They search the World Wide Web in a systematic way for particular information specified in a textual web search query. The search results are generally presented in a line of results, often referred to as search engine results pages (SERPs). When a user enters a query into a search engine, the engine scans its index of web pages to find those that are relevant to the user's query. The results are then ranked by relevancy and displayed to the user. The information may be a mix of links to web pages, images, videos, infographics, articles, research papers, and other types of files. Some search engines also mine data available in databases or open directories. Unlike web directories and social bookmarking sites, which are maintained by human editors, search engines also maintain real-time information by running an algorithm on a web crawler. Any internet-based content that can't be indexed and searched ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Top Cycle
In voting systems, the Smith set, named after John H. Smith, but also known as the top cycle, or as Generalized Top-Choice Assumption (GETCHA), is the smallest non-empty set of candidates in a particular election such that each member defeats every candidate outside the set in a pairwise election. The Smith set provides one standard of optimal choice for an election outcome. Voting systems that always elect a candidate from the Smith set pass the Smith criterion and are said to be 'Smith-efficient' or to satisfy the Smith criterion. A set of candidates each of whose members pairwise defeats every candidate outside the set is known as a dominating set. The Smith set can be seen as defining a voting method (Smith's method) which is most often encountered when extended by an IRV tie-break as Smith/IRV or as Tideman's Alternative, or by minimax as Smith/minimax. Properties of Smith sets *The Smith set always exists and is well defined (see next section). *The Smith set can have more ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Copeland's Method
Copeland's method is a ranked voting method based on a scoring system of pairwise "wins", "losses", and "ties". The method has a long history: * Ramon Llull described the system in 1299, so it is sometimes referred to as "Llull's method" * The Marquis de Condorcet described a similar system in the 1780s, so the method could be referred to as "Condorcet's method", but instead other systems were subsequently devised that choose the Condorcet winner. * Arthur Herbert Copeland described the system in the 1950s, so it has been frequently been called "Copeland's method". (unpublished). Each voter is asked to rank candidates in order of preference. A candidate A is said to have majority preference over another candidate B if more voters prefer A to B than prefer B to A; if the numbers are equal then there is a preference tie. The Copeland score for a candidate is the number of other candidates over whom they have a majority preference ''plus'' half the number of candidates with whom t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Isomorphism
In graph theory, an isomorphism of graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H'' : f \colon V(G) \to V(H) such that any two vertices ''u'' and ''v'' of ''G'' are adjacent in ''G'' if and only if f(u) and f(v) are adjacent in ''H''. This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection. If an isomorphism exists between two graphs, then the graphs are called isomorphic and denoted as G\simeq H. In the case when the bijection is a mapping of a graph onto itself, i.e., when ''G'' and ''H'' are one and the same graph, the bijection is called an automorphism of ''G''. If a graph is finite, we can prove it to be bijective by showing it is one-one/onto; no need to show both. Graph isomorphism is an equivalence relation on graphs and as such it partitions the class of all graphs into equivalence classes. A set of graphs isomorphic to each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elements in and in . It is a generalization of the more widely understood idea of a unary function. It encodes the common concept of relation: an element is ''related'' to an element , if and only if the pair belongs to the set of ordered pairs that defines the ''binary relation''. A binary relation is the most studied special case of an Finitary relation, -ary relation over sets , which is a subset of the Cartesian product X_1 \times \cdots \times X_n. An example of a binary relation is the "divides" relation over the set of prime numbers \mathbb and the set of integers \mathbb, in which each prime is related to each integer that is a Divisibility, multiple of , but not to an integer that is not a multiple of . In this relation, for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connex Relation
In mathematics, a relation on a set is called connected or total if it relates (or "compares") all pairs of elements of the set in one direction or the other while it is called strongly connected if it relates pairs of elements. As described in the terminology section below, the terminology for these properties is not uniform. This notion of "total" should not be confused with that of a total relation in the sense that for all x \in X there is a y \in X so that x \mathrel y (see serial relation). Connectedness features prominently in the definition of total orders: a total (or linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected. Similarly, a strict partial order that is connected is a strict total order. A relation is a total order if and only if it is both a partial order and strongly connected. A relation is a strict total order if, and only if, it is a strict partial order and just connected. A strict total or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-armed Bandit
In probability theory and machine learning, the multi-armed bandit problem (sometimes called the ''K''- or ''N''-armed bandit problem) is a problem in which a fixed limited set of resources must be allocated between competing (alternative) choices in a way that maximizes their expected gain, when each choice's properties are only partially known at the time of allocation, and may become better understood as time passes or by allocating resources to the choice. This is a classic reinforcement learning problem that exemplifies the exploration–exploitation tradeoff dilemma. The name comes from imagining a gambler at a row of slot machines (sometimes known as "one-armed bandits"), who has to decide which machines to play, how many times to play each machine and in which order to play them, and whether to continue with the current machine or try a different machine. The multi-armed bandit problem also falls into the broad category of stochastic scheduling. In the problem, each mach ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary information encoded in genes, which can be transmitted to future generations. Another major theme is evolution, which explains the unity and diversity of life. Energy processing is also important to life as it allows organisms to move, grow, and reproduce. Finally, all organisms are able to regulate their own internal environments. Biologists are able to study life at multiple levels of organization, from the molecular biology of a cell to the anatomy and physiology of plants and animals, and evolution of populations.Based on definition from: Hence, there are multiple subdisciplines within biology, each defined by the nature of their research questions and the tools that they use. Like other scientists, biologists use the sc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tournament (graph Theory)
A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in an undirected complete graph. That is, it is an orientation of a complete graph, or equivalently a directed graph in which every pair of distinct vertices is connected by a directed edge (often, called an arc) with any one of the two possible orientations. Many of the important properties of tournaments were first investigated by H. G. Landau in to model dominance relations in flocks of chickens. Current applications of tournaments include the study of voting theory and social choice theory among other things. The name ''tournament'' originates from such a graph's interpretation as the outcome of a round-robin tournament in which every player encounters every other player exactly once, and in which no draws occur. In the tournament digraph, the vertices correspond to the players. The edge between each pair of players is oriented from the winner to the loser. If player a beats player b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-Criteria Decision Analysis
Multiple-criteria decision-making (MCDM) or multiple-criteria decision analysis (MCDA) is a sub-discipline of operations research that explicitly evaluates multiple conflicting criteria in decision making (both in daily life and in settings such as business, government and medicine). Conflicting criteria are typical in evaluating options: cost or price is usually one of the main criteria, and some measure of quality is typically another criterion, easily in conflict with the cost. In purchasing a car, cost, comfort, safety, and fuel economy may be some of the main criteria we consider – it is unusual that the cheapest car is the most comfortable and the safest one. In portfolio management, managers are interested in getting high returns while simultaneously reducing risks; however, the stocks that have the potential of bringing high returns typically carry high risk of losing money. In a service industry, customer satisfaction and the cost of providing service are fundamen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
