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Graph Continuity
In mathematics, particularly in game theory and mathematical economics, a function is graph continuous if its graph—the set of all input-output pairs—is a closed set in the product topology of the domain and codomain. In simpler terms, if a sequence of points on the graph converges, its limit point must also belong to the graph. This concept, related to the closed graph property in functional analysis, allows for a broader class of discontinuous payoff functions while enabling equilibrium analysis in economic models. Graph continuity gained prominence through the work of Partha Dasgupta and Eric Maskin in their 1986 paper on the existence of equilibria in discontinuous economic games. Unlike standard continuity, which requires small changes in inputs to produce small changes in outputs, graph continuity permits certain well-behaved discontinuities. This property is crucial for establishing equilibria in settings such as auction theory, oligopoly models, and location competiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oligopoly
An oligopoly () is a market in which pricing control lies in the hands of a few sellers. As a result of their significant market power, firms in oligopolistic markets can influence prices through manipulating the supply function. Firms in an oligopoly are mutually interdependent, as any action by one firm is expected to affect other firms in the market and evoke a reaction or consequential action. As a result, firms in oligopolistic markets often resort to collusion as means of maximising profits. Nonetheless, in the presence of fierce competition among market participants, oligopolies may develop without collusion. This is a situation similar to perfect competition, where oligopolists have their own market structure. In this situation, each company in the oligopoly has a large share in the industry and plays a pivotal, unique role. Many jurisdictions deem collusion to be illegal as it violates competition laws and is regarded as anti-competition behaviour. The EU com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash Equilibrium
In game theory, the Nash equilibrium is the most commonly used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed). The idea of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to his model of competition in an oligopoly. If each player has chosen a strategy an action plan based on what has happened so far in the game and no one can increase one's own expected payoff by changing one's strategy while the other players keep theirs unchanged, then the current set of strategy choices constitutes a Nash equilibrium. If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob has no other strategy available that does better than B at maximizing his payoff in response to Alice c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Strategy
In game theory, a move, action, or play is any one of the options which a player can choose in a setting where the optimal outcome depends ''not only'' on their own actions ''but'' on the actions of others. The discipline mainly concerns the action of a player in a game affecting the behavior or actions of other players. Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship. The term strategy is typically used to mean a complete algorithm for playing a game, telling a player what to do for every possible situation. A player's strategy determines the action the player will take at any stage of the game. However, the idea of a strategy is often confused or conflated with that of a move or action, because of the correspondence between moves and pure strategies in most games: for any move ''X'', "always play move ''X''" is an example of a valid strategy, and as a result every move can also be considered to be a strategy. Other authors treat strate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Semi-continuous
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, roughly speaking, the function values for arguments near x_0 are not much higher (respectively, lower) than f\left(x_0\right). Briefly, a function on a domain X is lower semi-continuous if its epigraph \ is closed in X\times\R, and upper semi-continuous if -f is lower semi-continuous. A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point x_0 to f\left(x_0\right) + c for some c>0, then the result is upper semicontinuous; if we decrease its value to f\left(x_0\right) - c then the result is lower semicontinuous. The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-concave Function
In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave. Quasiconvexity is a more general property than convexity in that all convex functions are also quasiconvex, but not all quasiconvex functions are convex. ''Univariate'' unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with star-convex sublevel sets can be unimodal without being quasiconvex. Definition and properties A function f:S \to \mathbb defined on a convex subset S of a real vector space is quasiconvex if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. One su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Function
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is convex if its epigraph (mathematics), ''epigraph'' (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph is shaped like a cup \cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap \cap. A twice-differentiable function, differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain of a function, domain. Well-known examples of convex functions of a single variable include a linear function f(x) = cx (where c is a real number), a quadratic function cx^2 (c as a nonnegative real number) and an exponential function ce^x (c as a nonnegative real number). Convex functions pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game
A game is a structured type of play usually undertaken for entertainment or fun, and sometimes used as an educational tool. Many games are also considered to be work (such as professional players of spectator sports or video games) or art (such as games involving an artistic layout such as mahjong, solitaire, or some video games). Games have a wide range of occasions, reflecting both the generality of its concept and the variety of its play. Games are sometimes played purely for enjoyment, sometimes for achievement or reward as well. They can be played alone, in teams, or online; by amateurs or by professionals. The players may have an audience of non-players, such as when people are entertained by watching a chess championship. On the other hand, players in a game may constitute their own audience as they take their turn to play. Often, part of the entertainment for children playing a game is deciding who is part of their audience and who participates as a player. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Location Theory
Location theory has become an integral part of economic geography, regional science, and spatial economics. Location theory addresses questions of what economic activities are located where and why. Location theory or microeconomic theory generally assumes that agents act in their own self-interest. Firms thus choose locations that maximize their profits and individuals choose locations that maximize their utility. History Transportation costs While others should get some credit for earlier work (e.g., Richard Cantillon, Etienne Bonnot de Condillac, David Hume, Sir James D. Steuart, and David Ricardo), it was not until the publication of Johann Heinrich von Thünen's first volume of ''Der Isolierte Staat'' in 1826 that location theory can be said to have really gotten underway. Indeed, the prominent regional scientist Walter Isard has called von Thünen "the father of location theorists." In ''Der Isolierte Staat'', von Thünen notes that the costs of transporting goods c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Auction Theory
Auction theory is a branch of applied economics that deals with how bidders act in auctions and researches how the features of auctions 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 confluence of the price between the buyer and seller is an economic equilibrium. Auction theorists design rules for auctions to address issues that can lead to market failure. The design of these rulesets encourages optimal bidding strategies in 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. Theorists consider auctio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed two-person zero-sum games, in which a participant's gains or losses are exactly balanced by the losses and gains of the other participant. In the 1950s, it was extended to the study of non zero-sum games, and was eventually applied to a wide range of Human behavior, behavioral relations. It is now an umbrella term for the science of rational Decision-making, decision making in humans, animals, and computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
