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Auction Mechanism to Allocate 
Air Traffic Control Slots 

Marianne Raffarin * 
April 28, 2003 

Abstract 

This article deals with an auction mechanism &r airspace slots, as 
a means of solving the European airspace congestion problem. A dis- 
equilibrium, between Air Traffic Control (ATC) services supply and 
ATC services demand, are at the origin of almost one fourth of de- 
lays in the air transport industry in Europe. In order to tackle this 
congestion problem, we suggest modifying both pricing and allocation 
of ATC services, by setting up an auction mechanism. Objects of the 
auction will be the right for ctirlines to cross a part of the airspace, and 
then to benefit firom ATC services over a period corres|>onding to the 
iiectasisary time for the crossing. Allocation and payment rules have 
to be defined according to the objectives of this auction. The auc- 
tioneer is the public authority in charge of ATC services, whose aim 
is to obtain £tn efiicient allocation. Therefore, the social value will be 
maximized. Another objective is to internalize congestion costs. To 
that end, we apply the principle of Clarke-Groves mechanism auction: 
each winner has to pay the externalities imposed on other bidders. The 
complex context of ATC leads to a specific design for this auction. 

1 Introduction 

The air transport industry in Europe is faced with the recurring problem of 
delays. Although delays slightly decreased in 2001, this was essentially due 
to the current international context. Delay and traffic levels are strongly 
coimected. High rate of flight delays can again become very topical with 



*EUHEQua, CNRS, University of Paiis-I and LEEA, ENAC-CENA, Toulouse, e-mail: 
raffiarin^recherdie.enacir 



future growth of the air transport industry. Delays are very costly in terms 
of welfare. 

The reason essentially advanced by airlines to explain delays is the insuf- 
ficient capacity of the Air Traffic Control (ATC). A study estimates the cost 
of those specific delays, borne by passengers and airlines, between 6.6 and 
10.7 billions of euros for 1999. It is very important to tackle this problem. 

However, if ATC services are actually responsible for an important part 
of European delays, airlines are also at the origin of this situation: under- 
capacity is due to insufficient supply and too much demand. 

The aim of this article is to propose a solution to the inadequacy of the 
supply to cope with the demand, by considering a fixed capacity. Pricing is 
the mechanism usually used to avoid such disequiUbrium. We observe that 
ATC fees do not satisfied this principle. Indeed, ATC charges are a function 
of the weight of the aircraft in order to introduce cross subsidies between 
airspace users, and of the distance flown over in order to reflect the cost of 
service. The ATC fees provide incentives to operate flights on small aircraft 
and to supply frequent flights, but frequencies contribute to air congestion. 
Thus, ATC charges do not work to reduce delays. 

Another problem is the organization of ATC services, such that the upper 
airspace is divided into "sectors" with a team of two or three controllers in 
charge of one sector. When the demand is greater than the supply for one 
sector, ATC authorities have to "regulate" the traffic and to allocate slots. 
The time an aircraft is allowed to enter in a regulated sector is specified in a 
slot. For each regulated sector, a Ust of slots is built. Those slots satisfied the 
capacity -per hour announced by the regulated sector. For instance, a four 
hour long regulation associated with a rate of 30 flights per hour would result 
in a slot allocation list made up of 120 slots separated from one another by 
2 minutes. The principle "flrst planned, first served", which presumes that 
flights should arrive over the restricted sector in the same order in which they 
would have arrived without regulation, is applied throughout the process. 

ATC fees and allocation rules do not produce an efficient treatment of 
congestion. This article proposes a mechanism which combines allocation 
and pricing. Allocation of slots will be done in an efficient way, via optimal 
ATC fees. 

First, we need to present the specific context of auction for airspace. 
Then, we will be able to design allocation and payment rules of the mech- 
anism. The public authority is faced with some constraints, as capacity 
constraints, to determine the optimal allocation. Airlines' payoffs have to be 
defined according to the objectives. Finally, we will provide an example to 
understand how the mechanism works and to analyze the results. 



2 Context of auction for airspace 

In order to define an auction mechanism, we need to present objects and 
objectives for the auction. The wide complexitj^ of the ATC organization, 
explained by a high level of security, is at the origin of a specific definition 
of objects. They will be presented in detail. Objectives are more simple: a 
public authority in charge of ATC services sets up an auction with the aim 
of reducing delays and promoting a better use of the existing capadtj^ 

2.1 Objects of the auction 

An auction for the airspace will be the selling of rights for airlines to cross 
a part of the airspace, and then to benefit fi"om ATC services over a period 
corresponding to the necessary time to cross. Those rights will be called 
"slots", as for the airport slots, but with a difiierent sense of current "ATC 
slots". 

Obviously, there are many objects in this auction. Two elements are at 
the origin of this multiphcity: space and time. 

2.1.1 Components of a slot 

The airspace is divided into "sectors". The work of a team of controllers is 
to ensure secunty to uiguts crossing the sector th^* are in charge of. Most 
of air links cross many sectors. It means many different goods. 

Moreover, a sector is defined by its capadt>^ This the highest number 
of flights that can be present in the same sector at the same moment. A 
given sector can be crossed by several flights during a given period. It means 
many identical goods. Those goods are perfect complements because airlines 
cannot nm a risk of missing a sector to operate a link. It is absolutely 
nece^ary to allow package bids from individual sectors. 

Another important dimension to constitute a slot is the time. Each pack- 
age must also include periods at which sectors will be crossed. 

For security reasons, air traffic flow must be spread over the day and 
cannot be concentrated over a short period. Thus, a sector capacity cannot 
be defined by day. It has to be fixed for a short period. We divided a day into 
34 periods of half an hour: 6:00-6:30, 6:30-7:00, etc. The auction is organized 
for flights in airspace between 6:00 in the morning and 11:00 in the evening. 
The set of sub-i>eriods of one day is: 

T'= {ti,...,t34} 



The division into airspace sectors already exists. The set of those sector 
is: X = {1, . . . ,x}. Each sector is characterized by a time capacity. Due to 
the activation of some military areas, where civil flights are not allowed to 
cross, this capacity varies over the periods. We consider ks;t as the capacity 
of the sector s during the period t. 

2.1.2 Complementary objects 

A slot is not necessary only for one flight or for only one air link. For strategic 
reasons, airlines can bid for a slot used for several air links. Sectors for a 
same flight are not the only perfect complements. Flight periods of several 
air links need to be consistent with each other, due to aircraft turnover. The 
existence of a hub explains also the strong complementarity between sectors. 
Then, slots can be for an air link, or for several air links of an aircraft, or for 
several air links of a group of passengers. 

An airline bid specifies which sectors, at which periods, are necessary 
to form a package. Generally, flights will be operated over several periods. 

A slot z will be pairs of "sectors-period": z = {[(ya,<fl)]a=i a], with y^ a 

package of sectors, ta the time period needed for the package ya and A the 
number of necessary periods to operate the flight. 

Airlines can ask for many slots. So, this auction is for multiple packages. 

2.2 Objectives of the auction 

Many objectives justify an auction mechanism for airspace. 

2.2.1 Internalize congestion costs 

ATC services are not a public good. On one hand, fees paid by users involve 
possibility of exclusion. On the other hand, periodic situation of congestion 
involve rivalry. With the limited capacity of ATC services, airlines impose 
externalities on others. 

One aim of this auction is to lead airlines to take into account the con- 
sequences of theirs flight choices. We need to know the value of each slot 
for airlines. K the demand of airlines for a slot is not satisfied, they bear an 
opportunity cost. This cost is equal to the profit that airlines would obtain 
if the demand has been satisfied, minus fees for the slot. The airlines ability 
to pay for a slot is the amount of this opportunity cost. 

K this cost is revealed by the auction, it will be possible to charge winners 
the externality imposed on others. 



2.2.2 Reach an efficient allocation of slots 

Although ATC services are not a public good, a public authoritj'^ is in charge 
of them. In the collective interest, this authority would prefer to reach the 
highest social surplus than the highest revenue. We look for an efficient 
mechanisna. 

The social surplus is equal to the sum of the airlines' net surplus, passen- 
gers' net surplus and ATC revenue. With "jaeld management^ strategies, 
airlines capture passengers' surplus. ATC costs for airlines and ATC revenues 
cancel each other out. Thus, after simplification, social surplus is equal to 
the sum of bids of winners, because we saw, that airlines' net profit plus ATC 
cost are equal to their bids. 

One objective is the maximization of this welfare. 

2.2.3 Spread the traffic over time and space 

We decided to study an auction mechanism for airspace slots in order to solve 
the present problem of congestion. The goal of this system is also to spread 
the traffic over time and space and not to cancel ffights initially forecasted 
at a peak period. 

The interest of an auction mechanism in this context is to incite airlines 
to modify either ffight route, or flight hour, or both, when the capacity is 
insufficient, bv means of orices. The ontimal T>eriod to flight and the ontimal 
sectors to cross wiii be determined according to ability to pay. 

2.2.4 Balance the ATC services budget 

France, as most in countries, decided to charge direct users of airspace. An 
auction mechanism is at the origin of transfers from bidders to the auctioneer- 
However, it is not sure that the budget of the civil aviation administration 
will be balanced. 

For this reason, we suggest to separate the fees in two parts. One will 
be connected to the ATC service costs and the other will be linked to the 
congestion costs. This second part will be determined by the auction residts. 

3 Auction design 

Due to multiple packages, the auction design specifies not only allocation and 
payment rules but also what bids look like. Airlines will have to announce 
which slots they want and how much they are able to pay for them. 



An optimization of the total value of the bids, under constraints, will 
give the allocation and the objectives will induce a special payment rule. 
Indeed, externalities and services managed by a public authority are in favor 
of a mechanism such that winners pay the cost imposed on others. So, the 
auction mechanism for airspace slots will be an adaptation of a Clarke-Groves 
mechanism. 

3.1 Bids of the auction 

There is a lot of possible "sector-period" combination. So, we propose to 
leave airlines to define themselves their slots. 

Bids of the auction will have two components. First, airlines will describe 
precisely the slots which are relevant to them. Second, they will announce 
their values for those slots. 

3.1.1 Relevant slots 

The auctioneer cannot propose an exhaustive list of all possible slots. Airline 
i will describe the M* slots it wants. The first part of her announcement will 
be the list Z' = {z^}„,=i,.„,Mi. 

Considering the third objective, it is not possible that at the end of the 
auction, some forecasted flights at peak period are canceled and all capacity 
is not used at ofi'-peak period. The auctioneer can allow airlines to modify 
a slot if their demand is not satisfied. We suggest to implement an auction 
with only one turn and to leave airlines asking for several slots for the same 
flight. At their "favorite" slot, they will add others slots in case of insufficient 
capacity to obtain the first one. Those alternatives will be diflFerent from the 
"first choice", either by the time period, earlier or later, or by sectors, a 
longer but less congestioned route, or by both. 

Thus, for an air link, airlines ask for several slots, z^ = {^m}r=i,,..,Ri^ is 
a vector including all the slots described by airline i to operate the air link 
m. 

For a given air link m, the smaller is r, the more the airline prefers this 
slot. It means that slots with r = 1 are "favorite" slots. 

With the previous notation, the slot r asked by the airline i for its air 
link m is: ^ = {[(2/^a,*lj:;a)]a=i,...,AJ-}- 

3.1.2 Slots values 

In addition to describing slots for their air links, airlines must also announce 
how much they are able to pay for them. For each airline z, bids are contained 



in the list of price: B' = {&j^}m=i,-.,Mi. 

As for the slots, 6J,i is a vector including bids for all the slots asked for a 
given air link. Bids are classified in order of preference; b\^ — {&m }r=i,...,iij^) 
such that JjJ is greater than ^"*"^ for all r in the set [1, i^ — 1]. 

According to information they have, airlines announce their bids. Con- 
sider that ^ in ©% is the exogenous private information of airline i. The set 
of all bidders' private information is = 0^ x ... x ©" and the vector of all 
bidders' signals is ^ = (^\ ..., ff^), with tf in 0. 

For each airline i, abilities to pay is defijied as following: 

„jj;- : e* — ^ ir , Vr = 1, . . . X, Vm = 1, . . . , AT, 

where t;J^(^) is the willingness to pay of the airline i with the signal ^ for 
her slot r, for its air Hnk m. For notation: V^*" = {v^{9')/ff^ e 0*}. 
Then: 

such that v'(«>*) = {t;ijr(^)/r = l,...,i4,m = l,...,M«} 
Airlines' bids are based on their own willingness to pay for goods: 

b^ : 6' — > R+ 

The complete airline t's bid is D* = (Z'; S'j, with: 



{ 



B" = {{&j„^}r^l,...iri,}m.l,...,M- 



The component z^ = {[(yj^;a'*m;a)]a-i,..,Ai?"} ^f ^* is associated with the 
component ftjjj* of B*. 

3.2 Results of the auction 

With the objectives we fixed to design an auction mechanism, a Clarke- 
Groves mechanism will be suitable for airspeice slots auction. Such a system 
can be used when either a single public good is sold or many private goods 
are sold to many people. In a Clarke-Groves mechanism, payoff of an agent 
is connected to its bid only through the consequences its bid has on the final 
allocation or decision. The price he pays is independent of its bid. Let's see 
how this price will be computed for airspace slots, after the announcement 
of the final allocation. 



3.2.1 Allocation rule 

Once airlines passed on their bids, made of slots and willingness to pay, 
the auctioneer computes the maximum social surplus. Then, the auctioneer 
announces to airlines which slots they obtained. The allocation list of airline 
i is given by H* = {h\, . . . , h'j^i}, such that hi^ = {/ijjj"}r=i,.„,ijj^ is equal to 
one if it won the slot and to zero otherwise: 

h^€{OA}. ^i^m.^r (1) 

Moreover, /ij„ cannot contain more than one element equal to one, because 
all bids are for the same air link: 

X^/ij;: € {0,1}, Vz,Vm (2) 

Finally, capacity constraint must be satisfied by the final allocation: 

E E E E (^JJT >< h^. X it=4^a ) ^ ^*;t' V5 G X, vt € r (3) 

t=l m~\ r=l o=l 

We obtain the allocation of slots among airspace users by maximizing the 
sum of the ability to pay for slots, under constraints (1), (2) and (3). The allo- 
cation is given by H = (if S . . . , ir\ . . . , H% with W = {{/i^^}r=i,...,iej.}m=i,...,Mi, 
solution of the auctioneer program: 

under the constraints: 

i=l m-1 r=l o=l 

Aijj,'-G{0,l},Vi,Vm,Vr 



3-2.2 Payment rule 

The auctioneer informs winners of the slots cost. To compute the price paid 
by agent i for slot m, he needs to know what would be the allocation L{i; m), 
if agent i did not bid for slot m. It is a vector such that: 

L{i; m) = {L^{i; m), . . . , L% m), . . . , V{i; m)) 
with U{i;m) = 0i(t;rn),...,4,(i;m)} 
and Z;,(t;m) = {ll^{i;m)}r=i^,„^^--{0}r=i ijj. 

We obtain this allocation by solving: 



V m 'J j€N 



max 



under constraints: 



(5) 



I. \fi K,' ^m' 

E E EE (^m'(t;"») X W=: X i^> ) < fc.;t,V5 6 x,vt € r 

, ^ tn*;tt m';o/ 

J=l m'=l r=:l 0=1 

e(i;m)€{0,l},Vj^t,Vm',Vr 
C(t;m)€ {0,1}, Vm'#m,Vr 

'm V*i"V — ", »• 
r=l 

Then, the agent i must pay for the slot m: 

j^m = EEE«^^'>^C^(»;"») 

j=l m'=l r=l 

-(EEE«^^'>«^^'-E*m^>^/»mn 



^i=l m'=l r=:l 

= EEE*'^'>^^J^^(»;"») 

j=l Tii'=l r=l 

/ n M> <./ AT «S„/ 

- EEEe'x/^ir+EEe'xc' 



j=i m'=lr'=l 



m'=i r'=l 



(6) 



This price is the difference between the total bids of all airlines, except 
i, when airline i does not bid for the air link m and when it bids for this air 
Unk. Then pl^ reflects the amount of which airline i deprives other bidders 
with its demand for m. 

The price can also be written: 

pj„ = j;e X /^liT - E E E (^m' X fc - e X e(i;m)) 

r=l j=l m'=l r— 1 

In this way, another interpretation is possible. AirUne i pays its bid for the 
slot it won, for its air link m, and it benefits from a discount equal to the 
amount it increases the final allocation value with its demand. 

Prices correspond to externalities that airlines imposed on other bidders. 

Proposition 1 This mechanism is a direct revealing and efficient mecha- 
nism. 

Proof is given in annex A. 

4 Example 

In order to understand what the process is of the auction, we give a simple 
example with less parameters than in an actual situation. 

4.1 Data of the simulation 

We consider: 

• 4 airlines: i = 1, 2, 3, 4; 

• 6 ATC sectors: s = si, $2, S3, ^4, S5, s^, laid out as on the figure (1): 

• 3 time periods: t = ti, ^2, t^; 

• a capacity for each sector at each time period equal to 2 aircraft: ks;t = 
2, V5,t 

Each airline describes the slots it wants and the substitute slots in case 
it would not obtain its favorite slot. The slot r of the airline i for its air 
link m is z^^^ and is associated with the ability to pay v^. Given that the 
mechanism is efficient and direct revealing, we know that the airlines' bids 



10 




Figure 1: Layout of the 6 sectors, 

I 
I 

axe equal to their ability to pay. We suppose that the airlines' bids are the 

following: 



1: ^l^hnk -l^choice: | jj!, Z ^jf^^' ^^'^^^'^^l' I^^^'^^^'^ 

-2^^ choice: l ^Z " {[(^l'«2),tl], [(53,^4, 56),t2]} 

- 3^^ choice: | ^\l ^ ^^^^^' ^^' ^^^' *^^' ^^^^' ^^^' *^^^ 

*2°<llink: -1«* choice: ^^ Z {^"^'*^J'^"^'*^J> 

-2°'^ choice: { |^ Z {J'^'*^^'*^> 



S^^J choice: H:, = i'l^-^'^^M^^'^'j} 



2;i _ 



Airline 2: * 1^ link: - 1«* choice: {% I {[(^i'**)'*^^*^,*^]} 

-2'^d choice: { f^ I ^If*^'"^^'*^]'f"«'*^^> 

-S'^l choice: | jll I {[«i'*i]' [(-^'-«)'*2]> 

-4^ choice: jC I iI(*-*^)'*^]'f^-^^5> 



11 



• 2°^ link: 



• 3^'^ link: 



- 1®* choice: 

- 2°*^ choice: 

- 3'"'* choice: 

- 4*^ choice: 

- 1^* choice: 

- 2°"* choice: 

- 3*^ choice: 



Airline 3: • 1^* link 



r^ choice: 



• 2°<^ link: 



Airline 4: * 1^* link 



- 2°^ choice: 

- 3^^ choice: 

- 1^* choice: 

- 2°^ choice: 

- 3'^ choice: 

- 1^* choice: 

- 2°^ choice: 

- S'^** choice: 

- 1^* choice: 

- 2°'^ choice: 

- 3'^^ choice: 
Interpretation of those demands is appended to (see 



• 2°^ link: 



r 4^' = 

\vr = 

f 4' = 

I vr = 

r # = 

I vr = 

{ vr = 

1 vr = 

1 vr = 

i vr = 

r zr = 

Ur = 

r zr = 

I vr = 

r ^r = 

I vr = 

[vr = 

r 4'^ = 

I vr = 

r zr = 

r ^f = 

r .f = 

I vf = 

r .^^ = 

I vr = 

r 4=^ = 

\vr = 

1 vr = 

r 4=' = 

1 1;^^'' = 



{[{«3,S6),tl], [S2,t2]} 

26 

{Isz,ti],[{s2,se),t2]} 
24 

{[(S3,S6),i2],[s2,*3]} 

21 

{[s3,t2],[{S2,S6),t3]} 

17 

{(S3,«4),*l} 

22 

{[S3,tll[si,t2]} 

19 

{(S3,«4),t2} 

13 

{[(S4,S5),tl], [(s4,S5),t2]} 

52 

{l{S4,Ss),t2],[{S4,Ss),t3]} 

34 

{(S4,S5),<3} 

20 

{[S2,ti],[si,t2]} 

21 

{(Sl,«2),i2} 

17 

{[S2,<2], [Sl,<3]} 

13 

{[(s3,S6),<l],[s5,i2]} 

20 

{[S3,*2],[(S5,S6),<2]} 

17 

{[(S3,S6),*2], [55,^3]} 

14 

{(Si,S2),t2} 

18 

{[S2,*2],[si,t3]} 

15 

{{Si,S2),t3} 

12 
annex B). 



12 



4.2 Results 

FVom those bids and capacity constraints, we can solve the program (4). The 
optimal allocation is: 

H = {[(1,0,0), (0,1,0)], [(0,1, 0,0), (0,0, 0,1), (0,1,0)], 
[(1,0,0),(1,0,0)U(1, 0,0), (0,1,0)]} 

Results are resmned in the following tabular: 





l«*lmk 


2°** link 


3*^ link 


Airline 1 


1^ choice 


2°*^ choice 


- 


Airline 2 


2^ choice 


4"* choice 


2°^ choice 


Airline 3 


1^ choice 


l^hoice 


- 


Airline 4 


1^ choice 


2°^ choice 


- 



The social value, computed by adding the abilities to pay for allocated 
slots, is equal to 254 monetary imits. 

Then, we have to compute the price of each slot. We solve the program 
(5) several times, by removing alternatively an air link of an airline. The 
tabular 1 gives all the new allocations. If the airline t does not receive any 
slot for its air link m, the aUocation is L(i, m). It means that components 
of the slot zj^ are available to make other slots. Then, new allocations are 
possible. 

For example, with the components of the slot rf ^ a^^ilable, it becomes 
optimal to allocate to the second airline its first choice instead of its second 
choice for its first air link, with aU other slots still allocated in the same way. 
The airline 2 can benefit from the sector s^ at the period ti to operate its 
flight avoiding to make a detour through the sector 55. With the allocation 
L(3, 1), the sum of the willingness to pay is equal to 277. 

We can compute the price of each slot. It is the willingness to pay for it 
minus the difference between the sum of the willingness to pay of allocated 
slots, 254 monetary imits, and the one of slot that would be allocated if the 
airline did not bid for this air link. Prices of the nine slots are given by the 
set of equations (7). 



13 



H P^ (^'?)7 suop-BDonv :T ^\<\^J, 





Value 
total 


1—1 


CM 


1— t 
CO 
CM 


CO 
CM 


^H 




CM 


o 

CM 


CM 
CM 


CM 


< 


M 


CO 








fL 














o" 


o" 


S" 


>< 

o 


T-H 


o" 


o" 


1-H 


o^ 


o" 


N 


tH 


o 


r-l 


1-H 


o 


T— ( 


o 


o 


o 


T-H 


iH 


o 

-» — ^ 


1— t 


o^ 


o^ 


o^ 


o^ 


1-H 


2. 


o^ 


2- 


FH 


CO 


o" 


o" 


o" 


o^ 


o" 


o" 


o" 


o^ 


o" 


o" 


w 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


iH 


T-^ 


r-H 


T-H 


1-H 


T-H 


T-H 


T-H 


o^ 


1-H 


1-H 






















CO 

.5 


W 


CO 






















o^ 


o^ 


o" 


o" 


o" 


o" 


o^ 


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o" 


e^ 


o 


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o 


o 


o 


o 


o 


o 


o 


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1-1 


1-H 


1— 1 


rH 


1—1 


T— 1 


1—1 


3 


1—) 


1—1 


1-H 


iH 


CO 


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o 


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iH 


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1— ( 


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1-H 


1—i 


1-H 






















J 


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1-H 


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CM 


1-^ 


o 


T-H 


1—1 


o 


r^ 


o 


T-H 


o 


T-H 


iH 


3 


3 


o^ 


3 


S- 


3 


s. 


3 


£- 


o^ 


CM 


"^ 


o" 


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1-H 


CO 


r^ 


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3 


r^ 


o^ 


S- 


y~i 


S- 


1—1 


1—1 


T-H 


3, 


T-( 


^ 


o" 


o" 


o" 


o" 


o" 


o" 


o" 


o" 


o" 


o" 


CO 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


CM 


o 


1-H 


o 


1— t 


1-H 


o 


l—l 


1-H 


T-H 


r-H 


tH 


1-^ 


o^ 


o^ 


o^ 


o^ 


1—1 


o^ 


o^ 


o^ 


o^ 






















1H 

< 


cs 


CO 






















o" 


o" 


o" 


o" 


o" 


o" 


o" 


o" 


o" 


o" 


CM 


o 


o 


l—t 


1— 1 


1— H 


1-H 


o 


T-H 


1-H 


1—1 


1-1 


1-H 


o^ 


o^ 


o^ 


o 


o^ 


T— t 


o 


o 


o^ 


1H 


CO 


o" 


o^ 


o^ 


o" 


o" 


o^ 


o" 


o^ 


o^ 


o^ 


CM 


o 


o 


o 


o 


o 


o 


o 


o 


o 


o 


1-1 


o^ 


T-H 


1-H 


1-H 


T-H 


1—1 


1—4 


1-H 


1-H 


1-H 
























a 

< 


s 

1 


iH 


lH 

i3' 






c? 


lH 
CO 


of 


iH 




K 



j4 = 

p? = 

i^ = 

Pi = 

p! = 

1^ = 

pt = 

pt = 



70 -(254 -196) 
13 -(254 -247) 
27 -(254 -231) 
17 -(254 -237) 
19 -(254 -241) 
52 -(254 -207) 
21 -(254 -242) 
20 -(254 -240) 
15 -(254 -242) 



= 12 

= 6 

= 4 

= 

= 6 

= 5 



6 
3 



(7) 



Let us iinderstand how those prices are defined. If airline 3 would not win 
a slot for its fiist air link, the only change in the optimal allocation will be 
in favor of airline 2 for its first air link. Instead of winning its second choice, 
with a value equal to 27 monetary units, it would obtain its first choice, with 
a value equal to 32 monetary units. Since there would be no bid from airline 
3 for its first air link, the increase in the total value would be of 5 monetary 
units. Then, the price of this slot is: pi — 5. 

4.3 Analysis of the results 

Some conmients can be done on payo& and on the competition consequences 
on allocations, 

4.3.1 P3yG& 

We note that the price of the slot 2^ is zero. It means that airline 2 does not 
deprive any bidders with its slot. Since it would not bid for this slot, any of 
its components would be allocated to another airline: L(2, 2) = H. Airline 
2 imposes no externality with its slot for its second air link. 

With other slots, airlines deprive at least one bidder of getting a better 
choice. The price they have to pay is different from zero. Nevertheless, some- 
times airlines impose externality on themselves. The price paid by an airline 
is computed for each slot. The authority can define the payoff diffierently. For 
example, the price can be computed for the whole set of slots won by airlines. 
Then, the new allocation we look for is the one when an airline is completely 
out of the auction. In this way, results would be different from previous ones. 
To observe the difference, we need to compare the two methods. 

The price for the pool of slots obtained bj'^ airline 1 is computed by looking 
for the allocation L(l) of the auction with no demand from airline 1. This 
is the following allocation with a total value equal to 183: 



15 



L(l) = {[(0,0,0), (0,0,0)], [(1,0, 0,0), (0,0, 1,0), (0,1,0)], 
[(1,0,0), (1,0,0)], [(1,0,0), (1,0,0)]} 

The total value of slots won by airline 1 is 83 {vl'^ + v^^ = 70 + 13). The 
global price is then: 

p^ = 83 - (254 - 183) = 12 

pi = 12^p{+?4 = 18 

This price is less than the sum of the prices for the two slots computed 
separately. It means that airline 1 imposes externalities not only on others 
airlines but also on herself, and this latter externality is equal to 6. Indeed, 
we observe from the allocation L(l, 1), with the available slot zj'^ airline 1 
gets a better slot for its second air link. 

Another example is for airline 2. If it did not bid in this auction, the 
allocation would be: 

L(2) = {[(1,0,0), (0,1,0)], [(0,0, 0,0), (0,0, 0,0), (0,0,0)], 
[(1,0,0), (1,0,0)], [(1,0,0), (0,1,0)]} 

For airlines 1,3 and 4, the allocation is the same as H. Airline 2 imposes 
no externality on others bidders. We can guess that the price jp is equal to 
zero. Indeed, the total value of L(2) is 191 and the sum of slots' values of 
airline 2 is ^f ^ + v^'"^ + vf^ = 63. Then the price is: 

p^ = 63 - (254 - 191) = 

For the other airlines, the global price for slots is equal to the sum of 
individual price. 

4.3.2 Competition 

In our example, we observe that airlines are in competition for some air 
links. The same slot is relevant for airlines 1 and 3. Obviously, due to under- 
capacity, the airhne with the highest value gets the better possible choice. 

It is different when the competition is only for a part of the slot. For 
example, for a slot made of three sectors, airlines 2 and 4 have respectively 
for their air link 2 and 1, two cormnon sectors. Airline 4 with a value equal 
to 20 gets the slot, although the value of airline 2 was 26. The slot is in 
this case, not necessary for the airline with the highest value. This result 
is due to the other components of the slot. More than sector 53 and 5$ at 

16 



period ti, airline 4 needs sector S5 at period <2- But there is few demand for 
this sector. Allocating the slot to airline 4 and the sector 52 at period t2 to 
another airline than airline 2, leads to a greater total value of the objective 
than otherwise. 

5 Concluding remarks 

An auction mechanism seems weU-suited to the ATC situation. A scarce 
resource, the airspace, can be efficiently allocated because willingnesses to 
pay of airlines are revealed. An example shows us consequences of such a 
mechanism on payofe and on comjwtition. A price computed by slot and not 
by airline leads airlines to intemali2e congestion costs imposed on themselves. 
Prom a competition point of view, it is not necessary that the airline with the 
highest value gets the slots. It can be a solution for edrlines to reroute their 
air link from congestioned areas, keeping a large part of sectors in common 
with their competitors and to bid a lesser value. Such an airline may win the 
slot and its competitors not. 

To do a computer simulation, we need to have a lot of data about air- 
lines. For the moment, we only know their favorite slots. But we need their 
alternative slots and their willingness to pay. 

Nevertheless, this chosen mechanism leads to reach objectives of this auc- 
tion. It is a direct revealiug and efficient mechanism. The pa^inent rule, as 
for a Clarke-Groves auction, is at the origin of the incentive effects. A do- 
minant strategy for airlines is to bid then* actual ability to pay. But such a 
mechanism is complex due to multiple packages and alternatives slots. 

This auction will be repeated and in each European country a major 
ai r l i ne is at the origin of a majoritj^ of flights. Collusion problems may also 
appear. 

Moreover, with a Clarke-Groves mechanism we supposed that airlines' 
values was independent. But those values may be in fact interdependent. 
The airline's would be a function, not only of her own signal, but also of 
other airlines' signals and to "collective judgments" . A personal characteris- 
tic, useful only for itself would be the cost to operate an air link. An airline's 
abihty to pay is coimected to this cost. The competition in prices, flights 
times, frequencies and on board services would be at the origin of interde- 
pendency between airlines. For example, if two airline bid for the same air 
link, the announcement of one of them would be linked to the effect of its 
own airhne on the network of the other. 

In such a case, airlines are no more able to bid their willingness to pay, 
because it depends on the others. Moreover, the pa3Tnent rule is no more 



17 



suitable because we can not know what would be the allocation if an airline 
did not bid. 

Dasgupta and Maskin (2000) show that an efficient, but constraint equi- 
librium exists. Prom a practical point of view, the mechanism would be more 
and more complex. Now, research on ATC auction have to take into account 
this interdependency of values and to remove problems of collusion. 

A Proof of Proposition 1 

Let us show that it is a dominant strategy for airline i to bid truthfully 
for each relevant slot, whatever are bids for other airlines ft^*". We need to 
compare net surplus of airline i between the case it bids truthfully for slots 
of its air link m and the case it lies. Note that the net surplus is equal to 
the difference between the actual value of the agent and the amount it has 
to pay. 

m H is the final allocation when airline i bids truthfully for the B^ slots 
for whom she announced x;^**, whatever are bids for other airlines: 

H = Bxgmsx'£'£'£l^xh^+Y.T.^^f^ + E^^f^ (8) 

{hm }r;m;i j = l m' = l r' = l Tn'=:l r' = l r=l 

under constraints: 

E E E E (^j^^ >< l»ev- X lt=4'J ^ *»=*' V5, Vt, 

t=l m=l r=l a=l 

h^e{0,l}, Vi,Vm,Vr, 
5^4' e {0,1}, Vi,Vm. 

L{i; m) is the allocation when i gets not slot for its air link m: 



Lii;m)= argmax EEE^m'XTO;H 0) 



18 



under constxaints: 



n hfi «L' ^m' 

?4^(^m) € {oil}, Vj^i,Vm',Vr, 
C(t;m)€{0,l}, Vm'^m,Vr, 
Zj^(i;m) = 0, Vr, 






E^m'(»;"»)€{0,l}, Vt,Vm' 
If bidder i bids truthfully, when its net surplus is: 



«5n 



n M' 



ft', 



5(jy) = E<^>^^Jjr- EEE«^m^><^m"(»'"^) 

r:=l y j=l m'=l r=:l 



EEE^»^'><^^+EEC''x^iJ 






m'-l r'=l 



(10) 



• Now. let us see the case of a iiar airline i for blots of its air link m. H 
is the final allocation when airline i announces &JjJ* i^ v^ for all slots r in 

H= argmax EEE*-xftj„^ 

under constraints: 

n hP B!^ A}^ 

EEEEl'^m" X i.e^^. ^ it=t- ) ^ *^'.-*' ^^'V*' 

»=1 m=l r— 1 a=l 

/li-'eiO,!}, Vi,Vm,Vr, 

V r:=l 

Given that airlines' values are private and independent, airlines announce 
only one bid by slot and their bids are not functions of the other airlines. 



19 



Then, vector L{i]m), solution of the program (9), does not change and the 
airline i's net surplus is: 

fljn ( n Mi </ 

r=l yj=l m'=l r=l 

( n Mi <- M< ^^. \ \ (^^) 

- EEE^'^^^'^' + EEe-'x^i:; 

We compare the two net surplus (10) at (11): 

A5 = E-^^-^J^^+ EEE^^''x^^' + EEe'>^/^l;'' 

(K. ( n Mi </ M* ^m' 

Y.-t X ^jjT + E E Ee' X ^i;:' + E E^' ^ c: 
r=l \ jr=i m'=l r'=:l m'==i r'=l 

(12) 
As ^ is defined (program (8)), A5 is positive. 

Whatever are the other airlines' bids, it is a dominant strategy for an air 
link to bid truthfully for each slot of each air link relevant for her. Truthful 
bids constitute an equilibrium in dominant strategy. This mechanism is 
direct revealing. 

An efficient mechanism is such that all goods are allocated to agents with 
highest values. An efficient allocation is the one defined by H (program (8)). 
We just show that airlines bid truthfully, then the final allocation is H. This 
mechanism is also efficient. 

B Interpretation of airlines' bids in the ex- 
ample 

The first relevant slot for airline 1 is in fact for several links. Two flights 
from sectors Si and $2 go to the airline's hub in sector 54. Prom this point, 
two flights go to sectors 53 and s^. Airline I's ability to pay for those four air 
links is 70 monetary units. The airline does not try to get slots separately, 
because a part of them could miss it and then effects of its hub would be 
reduced. Alternative slots for those air links are delayed. Abilities to pay 
decrease as flight hours are put back. 

20 



Airlines 1 and 2 are in competition for a part of their first air link: sector 
5i, 56 and 54. If its first choice is not satisfied, airhne 2 prefeis to bypass the 
congestioned sector, by crossing 55, Due to the over-cost with this route, the 
ability to pay for this second choice is equal to 5 monetary units less than to 
the first one. 

Airline 3's bid is for several flights between two airports. One takes oflF 
during the period ti for the air link 54-55, and then it comes back. The same 
slot takes into account the two air links, because if the first does not begin 
at ti, the second air link is necessary delayed. The second choice for this 
round-trip is postponed. For the third choice, the slot is so delayed that the 
return is canceled. It is the reason why the ability to pay for this slot is so 
low compared to the others: 20 monetary units instead of 52. 

Airlines 1 and 3 compete for their second air link: sectors 52 at ti and Si 
at ^2. Airline 3's value is the highest. 

Sectors 53 and 5$ at ti is conamon for air links of airliaes 2 and 4. Each 
slot is made of a different complementary component for the two. For the 
whole slot airline 2's value is the highest, with 20 monetary units. 

Sector 54 at ti is asked four times. But sector capacitj^ is ^ual to two 
flights. This over-demand will lead to allocation of dots which are not "fa- 
vorite" slot. 



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