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mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and especially
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 appli ...
, the airport problem is a type of
fair division
Fair division is the problem in game theory of dividing a set of resources among several people who have an entitlement to them so that each person receives their due share. That problem arises in various real-world settings such as division of inh ...
problem in which it is decided how to distribute the cost of an
airport
An airport is an aerodrome with extended facilities, mostly for commercial air transport. Airports usually consists of a landing area, which comprises an aerially accessible open space including at least one operationally active surface ...
runway
According to the International Civil Aviation Organization (ICAO), a runway is a "defined rectangular area on a land aerodrome prepared for the landing and takeoff of aircraft". Runways may be a man-made surface (often asphalt concrete, as ...
among different players who need runways of different lengths. The problem was introduced by S. C. Littlechild and G. Owen in 1973. Their proposed solution is:
# Divide the cost of providing the minimum level of required facility for the smallest type of aircraft equally among the number of landings of all aircraft
# Divide the incremental cost of providing the minimum level of required facility for the second smallest type of aircraft (above the cost of the smallest type) equally among the number of landings of all but the smallest type of aircraft. Continue thus until finally the incremental cost of the largest type of aircraft is divided equally among the number of landings made by the largest aircraft type.
The authors note that the resulting set of landing charges is the
Shapley value
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a uniq ...
for an appropriately defined game.
Introduction
In an airport problem there is a finite population N and a nonnegative function C: N-R. For technical reasons it is assumed that the population is taken from the set of the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s: players are identified with their 'ranking number'. The cost function satisfies the inequality C(i)
coalition
A coalition is a group formed when two or more people or groups temporarily work together to achieve a common goal. The term is most frequently used to denote a formation of power in political or economical spaces.
Formation
According to ''A Gui ...
S is confronted with costs c(S): =MAX C(i). In this way an airport problem generates an airport game (N,c). As the value of each one-person coalition (i) equals C(i), we can rediscover the airport problem from the airport game theory.
Nash Equilibrium
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 equili ...
, also known as non-cooperative game equilibrium, is an essential term in
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 appli ...
described by
John Nash in 1951. In a game process, regardless of the opponent's strategy choice, one of the parties will choose a certain strategy, which is called
dominant strategy
In game theory, strategic dominance (commonly called simply dominance) occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The ...
. If any participant chooses the optimal strategy when the strategies of all other participants are determined, then this combination is defined as a
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 equili ...
. A game may include multiple Nash equilibrium or none. In addition, a combination of strategies is called the Nash balance. when each player's balance strategy is to achieve the maximum value of its
expected return
The expected return (or expected gain) on a financial investment is the expected value of its return (of the profit on the investment). It is a measure of the center of the distribution of the random variable that is the return. It is calculated ...
, at the same time, all other players also follow this strategy.
Shapley value
The
Shapley value
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a uniq ...
is a solution concept used in
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 appli ...
. There is a video o
Shapley value The
Shapley value
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a uniq ...
is mainly applicable to the following situation: the contribution of each actor is not equal, but each participant cooperates with each other to obtain
profit
Profit may refer to:
Business and law
* Profit (accounting), the difference between the purchase price and the costs of bringing to market
* Profit (economics), normal profit and economic profit
* Profit (real property), a nonpossessory intere ...
or return. The efficiency of the resource allocation and combination of the two distribution methods are more reasonable and fair, and it also reflects the process of mutual game among the league members. However, the benefit distribution plan of the
Shapley value
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a uniq ...
method has not considered the risk sharing factors of organization members, which essentially implies the assumption of equal risk sharing. It is necessary to make appropriate amendments to the benefit distribution plan of
Shapley value
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a uniq ...
method according to the size of risk sharing.
Example
An airport needs to build a runway for 4 different aircraft types. The building cost associated with each aircraft is 8, 11, 13, 18 for aircraft A, B, C, D. We would come up with the following cost table based on
Shapley value
The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a uniq ...
:
See also
Introduction video o
confrontation analysis.List of games in game theory
References
{{game theory
Fair division
Cooperative games