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Virtual Valuation
In auction theory, particularly Bayesian-optimal mechanism design, a virtual valuation of an agent is a function that measures the surplus that can be extracted from that agent. A typical application is a seller who wants to sell an item to a potential buyer and wants to decide on the optimal price. The optimal price depends on the ''valuation'' of the buyer to the item, v. The seller does not know v exactly, but he assumes that v is a random variable, with some cumulative distribution function F(v) and probability distribution function f(v) := F'(v). The ''virtual valuation'' of the agent is defined as: ::r(v) := v - \frac Applications A key theorem of Myerson says that: ::The expected profit of any truthful mechanism is equal to its expected virtual surplus. In the case of a single buyer, this implies that the price p should be determined according to the equation: ::r(p) = 0 This guarantees that the buyer will buy the item, if and only if his virtual-valuation is weakly-p ... [...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 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 formats.” Introduction Auctions facilitate transactions by enforcing a specific set of rules regarding the resource allocations of a group of bidders. Theorists consider auctions t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian-optimal Mechanism Design
A Bayesian-optimal mechanism (BOM) is a mechanism in which the designer does not know the valuations of the agents for whom the mechanism is designed, but the designer knows that they are random variables and knows the probability distribution of these variables. A typical application is a seller who wants to sell some items to potential buyers. The seller wants to price the items in a way that will maximize their profit. The optimal prices depend on the amount that each buyer is willing to pay for each item. The seller does not know these amounts, but assumes that they are drawn from a certain known probability distribution. The phrase "Bayesian optimal mechanism design" has the following meaning: * Bayesian means that we know the probability distribution from which the agents' valuations are drawn (in contrast to prior-free mechanism design, which do not assume any prior probability distribution). * Optimal means that we want to maximize the expected revenue of the auctioneer, wher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous'' ''monotonic increasing'' cumulative distribution function F : \mathbb R \rightarrow ,1/math> satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability that the random variable X takes on a value less ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian-optimal Mechanism
A Bayesian-optimal mechanism (BOM) is a mechanism in which the designer does not know the valuations of the agents for whom the mechanism is designed, but the designer knows that they are random variables and knows the probability distribution of these variables. A typical application is a seller who wants to sell some items to potential buyers. The seller wants to price the items in a way that will maximize their profit. The optimal prices depend on the amount that each buyer is willing to pay for each item. The seller does not know these amounts, but assumes that they are drawn from a certain known probability distribution. The phrase "Bayesian optimal mechanism design" has the following meaning: * Bayesian means that we know the probability distribution from which the agents' valuations are drawn (in contrast to prior-free mechanism design, which do not assume any prior probability distribution). * Optimal means that we want to maximize the expected revenue of the auctioneer, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Uniform Distribution
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, ''a'' and ''b'', which are the minimum and maximum values. The interval can either be closed (e.g. , b or open (e.g. (a, b)). Therefore, the distribution is often abbreviated ''U'' (''a'', ''b''), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable ''X'' under no constraint other than that it is contained in the distribution's support. Definitions Probability density function The probability density function of the continuous uniform distribution is: : f(x)=\begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truthful Mechanism
In game theory, an asymmetric game where players have private information is said to be strategy-proof or strategyproof (SP) if it is a weakly-dominant strategy for every player to reveal his/her private information, i.e. given no information about what the others do, you fare best or at least not worse by being truthful. SP is also called truthful or dominant-strategy-incentive-compatible (DSIC), to distinguish it from other kinds of incentive compatibility. An SP game is not always immune to collusion, but its robust variants are; with group strategyproofness no group of people can collude to misreport their preferences in a way that makes every member better off, and with strong group strategyproofness no group of people can collude to misreport their preferences in a way that makes at least one member of the group better off without making any of the remaining members worse off. Examples Typical examples of SP mechanisms are majority voting between two alternatives, secon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Myerson Ironing
Mechanism design is a field in economics and game theory that takes an objectives-first approach to designing economic mechanisms or incentives, toward desired objectives, in strategic settings, where players act rationally. Because it starts at the end of the game, then goes backwards, it is also called reverse game theory. It has broad applications, from economics and politics in such fields as market design, auction theory and social choice theory to networked-systems (internet interdomain routing, sponsored search auctions). Mechanism design studies solution concepts for a class of private-information games. Leonid Hurwicz explains that 'in a design problem, the goal function is the main "given", while the mechanism is the unknown. Therefore, the design problem is the "inverse" of traditional economic theory, which is typically devoted to the analysis of the performance of a given mechanism.' So, two distinguishing features of these games are: * that a game "designer" choo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithmic Pricing
Algorithmic pricing is the practice of automatically setting the requested price for items for sale, in order to maximize the seller's profits. Dynamic pricing algorithms usually rely on one or more of the following data. * Probabilistic and statistical information on potential buyers; see Bayesian-optimal pricing. * Prices of competitors. E.g., a seller of an item may automatically detect the lowest price currently offered for that item, and suggest a price within $1 of that price. * Personal information of the currently active buyer, such as her or his demographics and her or his interest in the product. If the seller detects that you are about to buy, your price goes up. * Business information of the seller, such as the expected date in which he or she is going to receive new stocks, or her or his target selling velocity in units per day. See also * Algorithmic trading * Contribution margin * Price optimization software * Pricing * Tacit collusion * Yield management Yield ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
