For background, see Passenger Forcibly Removed From United Flight, Prompting Outcry. Also see the discussion in this Fiat Thread.

In summary, United Airlines had to move employees and passengers in a fully booked plane, and decided to drag one of the passengers off the plane. I will model this problem as a simple transferrable utility game, and I will use the Shapley value as my solution concept.

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Setup

There are the following four agents:

• United Airlines (agent 1)
• employee (agent 2)
• passenger A (agent 3)
• passenger B (agent 4)

The doctor and the other passenger are currently in possession of seats. United has the plane, so no one gets any value without United.

The passengers value the seat at $1. The employee values the seat at $0.

In addition to exchanging seats, United can also drag a passenger off a plane at cost $1. The passengers value get getting dragged off the plane at $-1.

United prefers to have its employee fly with either customer ($3), followed by one employee flying ($2), followed by two customers flying ($1), followed by one customer flying ($0.5), followed by no customers flying ($0).

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Solution

Now that I have set up the problem, I introduce the Shapley value f. f has the property that it is the unique solution which satisfies the dummy property (or null player), symmetry, efficiency, and additivity (or linearity). See https://en.wikipedia.org/wiki/Shapley_value for more details.

Practically, I first compute the maximum aggregate utility possible from any subset of agents:

• v(1,3) = $1.5
• v(1,4) = $1.5
• v(1,2,3) = $2
• v(1,2,4) = $2
• v(1,3,4) = $3
• v(1,2,3,4) = $4
• (all other subsets yield $0 aggregate utility)

For agent i, I compute the expected aggregate utility gain that a random subset of agents would get if they were joined by i. I'll save you a bunch of algebra to get you the results:

• f(1) = 1.83
• f(2) = 0.33
• f(3) = f(4) = 0.67

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Policies

Shapley Value

I have to back out what payments and allocations yield the Shapley value payoffs. To generate a total surplus of $4, one of the passengers has to give up her seat. To keep f(3)=f(4)=$0.67, United pays one of the passengers $0.67 to give up their seat. The seat goes to the employee and the other passenger pays $0.33 to keep her seat. In summary, the agents get the following utility

Agent	Utility
United	1.83
Employee	0.33
Passenger A	0.67
Passenger B	0.67

Ascending Auction

Now consider if United had used an ascending auction for a passenger to give up her seat. This would stop at $1, upon which a passenger would accept the voucher. I’m aggregating United and the employee because I don't want to model that a side-payment to the employee to get her cooperation. This outcome is still efficient, but gives more utility to the passengers compared to the Shapley value.

Agent	Utility
United + Employee	2
Passenger A	1
Passenger B	1

Forcible Removal

Now consider what happened in reality. United offered a voucher less than $1. Both passengers refused, so United dragged one passenger off the plane. Not only is this far from the Shapley value, but it is also inefficient due to the $0.5 loss of total surplus.

Agent	Utility
United + Employee	2
Passenger A	-1
Passenger B	1

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Discussion

I conclude that the best outcome in a cooperative game sense might not have been for United to drag the passenger off the plane. Compensating the passenger would have been more efficient, even in a world where the Shapley value is not feasible.

As for a positive explanation of said events, I suspect that a principal-agent model where local managers do not internalize the full costs to Untied of removing the passenger (e.g. reputational, legal) may help explain why United acted. But that is for another model and another R1.