Here is how we apply the concept of Shapley Values to the famous airport problem.
Suppose an airport wants to build a runway to support three airline companies. Here are the requirements for each company:
Airline 1: 1500 m
Airline 2: 2200 m
Airline 3: 3000 m
Assume the cost per m of runway construction is $1k. How must the airport split costs among the three airline companies fairly?
Proportional rule
The simplest way is to split the cost proportional to the length each airliner needs, i.e., you divide the total cost proportional to the runway requirements as:
Airline 1: 3000 x 1500 / (1500 + 2200 + 3000 ) = 672k
Airline 2: 3000 x 2200 / (1500 + 2200 + 3000 ) = 985k
Airline 3: 3000 x 3000 / (1500 + 2200 + 3000 ) = 1343k
Is it a fair division? It seems so. But what happens if Airline 3 comes up with a variation, say, change to 3500? The new contributions get modified to:
Airline 1: 729k
Airline 2: 1069k
Airline 3: 1701k
Suddenly, this seems unfair for Airlines 1 and 2 as the new plan only benefits the third one, but the others also bear the extra costs (729 – 672) and (1069 – 985), respectively. So, we try the Shapley values:
Shapley value
We have seen how it works in the previous post. Let’s build the table first for case 1.
C(1500, 2200, 3000)
| Combination | 1 | 2 | 3 |
| 123 | 1500 | 700 | 800 |
| 132 | 1500 | 0 | 1500 |
| 231 | 0 | 2200 | 800 |
| 213 | 0 | 2200 | 800 |
| 312 | 0 | 0 | 3000 |
| 321 | 0 | 0 | 3000 |
| Average | 3000/6 = 500 | 5100/6 = 850 | 9900/6 = 1650 |
And for case 2, C(1500, 2200, 3500):
| Combination | 1 | 2 | 3 |
| 123 | 1500 | 700 | 1300 |
| 132 | 1500 | 0 | 2000 |
| 231 | 0 | 2200 | 1300 |
| 213 | 0 | 2200 | 1300 |
| 312 | 0 | 0 | 3500 |
| 321 | 0 | 0 | 3500 |
| Average | 3000/6 = 500 | 5100/6 = 850 | 12900/6 = 2150 |
Case 1:
Shapley value Airline 1 = 500
Shapley value Airline 2 = 850
Shapley value Airline 3 = 1650
Case 2:
Shapley value Airline 1 = 500
Shapley value Airline 2 = 850
Shapley value Airline 3 = 2150
The game theory solution does not penalise the first two airlines and only demands the third one to pay for the scope change.
