In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and especially game theory

Game theory

Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

, the airport problem is a type of fair division

Fair division

Fair division, also known as the cake-cutting problem, is the problem of dividing a resource in such a way that all recipients believe that they have received a fair amount...

 problem in which it is decided how to distribute the cost of an airport

Airport

An airport is a location where aircraft such as fixed-wing aircraft, helicopters, and blimps take off and land. Aircraft may be stored or maintained at an airport...

 runway

Runway

According to ICAO a runway is a "defined rectangular area on a land aerodrome prepared for the landing and take-off of aircraft." Runways may be a man-made surface or a natural surface .- Orientation and dimensions :Runways are named by a number between 01 and 36, which is generally one tenth...

 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:

1. 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
2. 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

Shapley value

In game theory, the Shapley value, named in honour of Lloyd Shapley, who introduced it in 1953, is a solution concept in cooperative game theory. To each cooperative game it assigns a unique distribution of a total surplus generated by the coalition of all players...

for an appropriately defined game.

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:

Aircraft	Adding A	Adding B	Adding C	Adding D	Shapley value
Marginal Cost	8	3	2	5
Cost to A	2				2
Cost to B	2	1			3
Cost to C	2	1	1		4
Cost to D	2	1	1	5	9
Total					18