|
Price Of Stability
In game theory, the price of stability (PoS) of a game is the ratio between the best objective function value of one of its equilibria and that of an optimal outcome. The PoS is relevant for games in which there is some objective authority that can influence the players a bit, and maybe help them converge to a good Nash equilibrium. When measuring how efficient a Nash equilibrium is in a specific game we often also talk about the price of anarchy (PoA), which is the ratio between the ''worst'' objective function value of one of its equilibria and that of an optimal outcome. Examples Another way of expressing PoS is: : \text = \frac ,\ \text \geq 0. In particular, if the optimal solution is a Nash equilibrium, then the PoS is 1. In the following prisoner’s dilemma game, since there is a single equilibrium (B,R) we have PoS = PoA = 1/2. On this example which is a version of the battle of sexes game, there are two equilibrium points, (T,L) and (B,R), with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game 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 mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash Equilibrium
In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players, and no one has anything to gain by changing only one's own strategy. The principle of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to competing firms choosing outputs. 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 their's 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Price Of Anarchy
The Price of Anarchy (PoA) is a concept in economics and game theory that measures how the efficiency of a system degrades due to selfish behavior of its agents. It is a general notion that can be extended to diverse systems and notions of efficiency. For example, consider the system of transportation of a city and many agents trying to go from some initial location to a destination. Let efficiency in this case mean the average time for an agent to reach the destination. In the 'centralized' solution, a central authority can tell each agent which path to take in order to minimize the average travel time. In the 'decentralized' version, each agent chooses its own path. The Price of Anarchy measures the ratio between average travel time in the two cases. Usually the system is modeled as a game and the efficiency is some function of the outcomes (e.g. maximum delay in a network, congestion in a transportation system, social welfare in an auction, etc.). Different concepts of equilibriu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prisoner’s Dilemma
The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("defect") for individual reward. This dilemma was originally framed by Merrill Flood and Melvin Dresher while working at RAND in 1950. Albert W. Tucker appropriated the game and formalized it by structuring the rewards in terms of prison sentences and named it "prisoner's dilemma". William Poundstone in his 1993 book ''Prisoner's Dilemma'' writes the following version:Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police admit they don't have enough evidence to convict the pair on the principal charge. They plan to sentence both to two years in prison on a lesser charge. Simultaneously, the police offer each prisoner ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Number
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dots Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers. Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions. The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Competitive Facility Location Game
The competitive facility location game is a kind of competitive game in which service-providers select locations to place their facilities in order to maximize their profits.Eva Tardos and Tom Wexler, "Network Formation Games". Chapter 19 in The game has the following components: * There are several consumers who need a certain service, e.g, electricity connection. * There are several producers that can supply this service, e.g, electricity companies. * Each producer can build its facility (e.g, a power station) in one of several locations. * For every pair of consumer (C) and location (L), there is a fixed cost of serving C from L (e.g, depending on the distance between the power station and the consumer's house). This cost is denoted Cost ,L The game is a sequential game with three steps: # Each producer selects a location for placing its facility. # Each producer set a price for each user (price discrimination is allowed, since there is a different cost for serving different cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game 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 mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
