In mathematics and especially game theory, the airport problem is a type of fair division problem in which it is decided how to distribute the cost of an airport runway among different players who need runways of different lengths. The problem was introduced by S. C. Littlechild and G. Owen in 1973.[1] 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 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 |
References
Littlechild, S. C.; Owen, G. (1973). "A Simple Expression for the Shapley Value in a Special Case". Management Science. 20 (3): 370–372. JSTOR 2629727.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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