Exploratory Study of Demand Management
Using Auction-Based Arrival Slot Allocation
L. Le, S. Kholfi, G.L. Donohue, C-H Chen
Air Transportation Systems Engineering Laboratory
Dept. of Systems Engineering & Operations Research
George Mason University
Fairfax, VA
Outlines
qSlot auction model
qAuctioneer optimization model
qAirline optimization model
qIllustration : a case study
qSummary and future work
Problem Identification
Asynchronous non-uniform scheduling
ATL – Atlanta Airport
(FAA Airport Capacity Benchmark Report 2001)
TOTAL SCHEDULED OPERATIONS AND CURRENT
OPTIMUM RATE BOUNDARIES
Queuing delay
Safety violation
60
50
40
Safety violation?
30
20
10
0
7
8
Underutilization
9
10
11
12
13
14
Schedule
15
16
17
Facility Est.
18
19
20
21
Problem Identification
Small aircraft makes inefficient use of slots
aircraft in
small
category
Aircraft Size (seats)
Cumulative Seat Share
LAX arrivals, 1998 (Mark Hansen, Berkeley)
25% flight share
for 5% seat share
Cumulative Flight Share
Auction Model Design issues
q Feasibility
− package slot allocation for departure and arrival slots
− incremental (airports and slots to auction off)
q Optimality
− efficiency : throughput (enplanement opportunity) and delay
− regulatory standards: safety, flight priorities
− equity:
§ stability in schedule
§ airlines’ need to leverage investments
§ airlines’ competitiveness : new-entrants vs. incumbents
q Flexibility
− primary market at strategic level
− secondary market at tactical level
Design approach
q Objective:
− provide an optimum fleet mix at optimum safe arrival capacity
− ensure fair market access opportunity
− reduce queuing delay
q Assumptions
− airlines could make use of slots they bid
q Auction process:
− an interactive and iterative process to enable flexibility and optimization
− a mixture of simultaneous auction model and package model
q Auction rules: Bidders are ranked using a linear combination of:
− flight OD pair
− #seats
− airline’s prior investment
− historical slot occupancy rate
− bid
Background on auction models
q Simultaneous multiple-round auction
− have discrete, successive rounds, with length of each round
announced in advance. After each round closes, round results
are processed and made public
à Account for departure/arrival slots interdependence but
subject to aggregation risks
q Package bidding
− bidders submit bids for multiple combinations of lots rather
than just individual lots. Package biddings are either accepted
or rejected in their entirety
à Eliminate aggregation risks
Auction Model Process
Determine factor weights
Determine time-windows to auction off
Incremental
auction prioritizing
most congested
periods
another
time window?
No
End auction process
Yes
Call for bids for slots in
the time window
Submit information
and bid
Yes
No
Winners
determined?
Yes
Auctioneer’s action
Airline’s action
Auctioneer Optimization Model
Airlines are ranked by a linear combination of :
−#seats
−flight OD pair
−airline’s prior investment
BID
−historical slot occupancy rate
−monetary offer
Ranking function:
τ ( Ba ,s ) = W ⋅ Ba ,s
T
ST
A
WT
Ba,s
slot vector, |ST|=AAR
airline vector
vector of factor weights
bid of airline a for slot s
Auctioneer Optimization Model
Airlines are ranked by a linear combination of :
−#seats
−flight OD pair
−airline’s prior investment
BID
−historical slot occupancy rate
−monetary offer
Ranking function:
τ ( Ba ,s ) = W ⋅ Ba ,s
slot vector, |ST|=AAR
airline vector
vector of factor weights
bid of airline a for slot s
ST
A
WT
Ba,s
T
1
( X)a,s =
0
Bidding matrix X=A.ST
Slots
Minimum Bid
AC1
AC2
AC3
AC4
AC5
AC6
AC7
if airline a is ranked highest for
slot s after a round
otherwise
8:00:00
8:02:30
8:05:00
8:07:30
8:10:00
8:12:30
8:15:00
8:17:30
5000
7500
8600
10000
12000
8300
6600
5200
Auctioneer Optimization Model
à Objective function: Allocate slots to the
highest ranked airlines
∑∑ τ ( B
Max
a,s
a
s
Subject to:
∑ (X )
a,s
) ⋅ ( X )a , s
=1
∀s
One
slot – one
Minimum
Airlinebid
aircraft
constraint
constraint
Variables:
a
( M T ) s ⋅ ( X ) a, s <= ( Ba,s )5
∑(X )
s
a ,s
<= 1
∀a, s
ST
A
WT
Ba,s
slot vector, |ST|=AAR
airline vector
vector of factor weights
bid of airline a for slot s
X= A*ST
bidding matrix
1
( X)a,s =
0
if airline a is ranked highest for
slot s after a round
otherwise
Airline Optimization Model
Objective function: Maximize revenue and ultimately maximize profit
Maximize
∑ (P − B )
s
s
s
Subject to:
Bs ≤ M ⋅ y s
( B0T ) s ≤ Bs + M ⋅ (1 − y s )

 '
 Bs +


Bs ≤ α ⋅ Ps
max (τ (B )) −τ (B
a,s
a
(W )5
To
bid or
not
Upper
bound
Lower
bound
to bid
for
forbids
bids
Variables:
A,s )


T
⋅ (B0 ) s  ⋅ y s ≤ α ⋅ Ps ⋅ y s



(τ ( Ba,s )) −τ ( BA,s )

 ' max

T
a
B
+
⋅
(
B
)
s
0
s

 ≤ Bs + M ⋅ (1 − y s )
(
W
)


5


Airlines’ package bidding constraints
{Bs}
{Ps}
M
ys
1
ys = 
0
BoT
α
Bs’
set of monetary bids
airline expected profit by using a slot
big positive value
binary value
if airline bids for slot s
otherwise
airport threshold vector
airline threshold fraction
old bid for slot s in previous round
Illustration : an example
Winner-determining factors
Factors
Weight
Number of seats
Previous Airline infrastructure investment
0,32
0,25
Historic slot occupancy frequency
0,19
OD-Pair
Bid
0,13
0,11
Ranking function :
τ ( Ba ,s ) = W T ⋅ Ba ,s
Bidding matrix, initial round
Slots
Min Monetery Bid
NW (0.25)
8:00:00
8:02:30
8:05:00
8:07:30
8:10:00
8:12:30
8:15:00
8:17:30
5000
7500
8600
10000
12000
8300
6600
5200
S (30) 0 (0.4)
AA (0.19)
H (205) 1 (0.25)
L (147) 0 (0.5)
UA (0.15)
CA (0.13)
H (283) 1 (0.4)
S (18) 0 (0.25)
S (30) 0 (0.35)
SW (0)
S (30) 0 (0.35)
DAL (0.11)
S (18) 0 (0.2)
%investment
L (291) 1 (0.35)
H (392) 1 (0.3)
S (30) 0 (0.15)
S (18) 0 (0.15) S (30) 0 (0.06)
US (0.21)
L (128) 0 (0.35)
S (18) 0 (0.2)
L (150) 0 (0.35)
S (18) 0 (0.1)
L (120) 0 (0.5) L (85) 0 (0.2)
S (30) 0 (0.4)
S (30) 0 (0.45)
H (202) 1 (0.3)
S (30) 0 (0.4)
Aircraft type (#seats) OD pair (Slot occupancy rate)
Illustration: an example
Bid score matrix, initial round
Slots
Min Monetery Bid
NW (0.25)
8:00:00
8:02:30
8:05:00
8:07:30
8:10:00
8:12:30
8:15:00
8:17:30
5000
7500
8600
10000
12000
8300
6600
5200
1.48
AA (0.19)
3.23
3.44
UA (0.15)
CA (0.13)
3.94
1.04
1.18
SW (0)
1.34
DAL (0.11)
0.87
3.98
7.75
0.96
0.62
US (0.21)
3.18
0.83
2.96
2.2
0.62
1.68
Round
1
NW (0.25)
4.3
8:00:00
8:02:30
8:05:00
8:07:30
8:10:00
5000
7500
8600
10000
12000
5000
AA (0.19)
64109
7500
UA (0.15)
CA (0.13)
161590
SW (0)
199773
8600
149482
US (0.21)
11364
DAL (0.11)
32727
0.62
1.11
Air carrier monetary bids
Slots
Min Monetery Bid
0.96
9200
0
164945
8100
1.1
Withdraw
8:15:00
from
auction
8300
6600
for these6600
slots 139800
8:12:30
8:17:30
5200
5200
8300
0
185291
164036
90545
297745
131400
Illustration: an example
Bid score matrix, initial round
Slots
Min Monetery Bid
NW (0.25)
8:00:00
8:02:30
8:05:00
8:07:30
8:10:00
8:12:30
8:15:00
8:17:30
5000
7500
8600
10000
12000
8300
6600
5200
1.48
AA (0.19)
3.23
3.44
UA (0.15)
CA (0.13)
3.94
1.04
1.18
SW (0)
1.34
DAL (0.11)
0.87
3.98
7.75
0.96
0.62
US (0.21)
3.18
0.83
2.96
2.2
0.62
1.68
Round
1
NW (0.25)
4.3
8:00:00
8:02:30
8:05:00
5000
7500
8600
S (30) 0 (0.4)
AA (0.19)
8:07:30
8:10:00
10000
12000
H (205) 1 (0.25)
L (147) 0 (0.5)
H (283) 1 (0.4)
UA (0.15)
CA (0.13)
S (18) 0 (0.25)
S (30) 0 (0.35)
SW (0)
S (18) 0 (0.15) S (30) 0 (0.06)
US (0.21)
S (30) 0 (0.35)
DAL (0.11)
S (18) 0 (0.2)
0.62
1.11
Air carrier monetary bids
Slots
Min Monetery Bid
0.96
H (392) 1 (0.3)
S (30) 0 (0.15)
1.1
Withdraw
8:15:00
from
auction
8300
6600
L (128) 0 (0.35)
for these
slots
S (18) 0 (0.2)
8:12:30
8:17:30
5200
L (291) 1 (0.35)
L (150) 0 (0.35)
S (18) 0 (0.1)
L (120) 0 (0.5) L (85) 0 (0.2)
S (30) 0 (0.4)
S (30) 0 (0.45)
H (202) 1 (0.3)
S (30) 0 (0.4)
Illustration: an example
Bid score matrix, initial round
Slots
Min Monetery Bid
NW (0.25)
8:00:00
8:02:30
8:05:00
8:07:30
8:10:00
8:12:30
8:15:00
8:17:30
5000
7500
8600
10000
12000
8300
6600
5200
1.48
AA (0.19)
3.23
3.44
UA (0.15)
CA (0.13)
3.18
3.94
3.98
1.04
1.18
SW (0)
0.96
0.62
US (0.21)
1.34
DAL (0.11)
0.87
7.75
2.2
0.83
2.96
0.62
1.68
Round
1
NW (0.25)
4.3
8:00:00
8:02:30
8:05:00
8:07:30
8:10:00
5000
7500
8600
10000
12000
5000
AA (0.19)
64109
7500
UA (0.15)
CA (0.13)
161590
SW (0)
199773
8600
149482
US (0.21)
11364
DAL (0.11)
32727
0.62
1.11
Air carrier monetary bids
Slots
Min Monetery Bid
0.96
9200
0
Withdraw
8:15:00
from
auction
8300
6600
for these6600
slots 139800
8:12:30
8:17:30
5200
5200
8300
164945
8100
1.1
0
185291
164036
90545
297745
131400
Bid score matrix
Slots
Min Monetery
Break
tiesBidby
NW (0.25)
adding 10%
AA (0.19)
Round UA of
(0.15)airline
CA (0.13)
1
profit
to bids
SW (0)
8:00:00
8:02:30
8:05:00
8:07:30
8:10:00
8:12:30
8:15:00
8:17:30
5000
7500
8600
10000
12000
8300
6600
5200
1.48
3.94
3.44
3.18
3.94
2.96
3.98
7.75
3.44
3.44
US (0.21)
1.48
DAL (0.11)
1.48
3.18
2.2
2.96
2.96
3.98
3.94
2.2
7.75
3.18
Illustration: an example
Air carrier monetary bids
Slots
Min Monetery Bid
NW (0.25)
8:00:00
8:02:30
8:05:00
8:07:30
8:10:00
8:12:30
8:15:00
8:17:30
5000
7500
8600
10000
12000
8300
6600
5200
187500
AA (0.19)
Round
12
(final)
48000
58620
609800
15440
UA (0.15)
CA (0.13)
201040
103340
SW (0)
US (0.21)
48900
DAL (0.11)
Bid score matrix
Slots
Min Monetery Bid
NW (0.25)
Round
12
(final)
8:00:00
8:02:30
8:05:00
8:07:30
8:10:00
8:12:30
8:15:00
8:17:30
5000
7500
8600
10000
12000
8300
6600
5200
5.5
AA (0.19)
3.87
4.19
11.63
UA (0.15)
CA (0.13)
4.19
10.05
3.45
SW (0)
US (0.21)
3.51
DAL (0.11)
Auction winners
Slots
Minimum Bid
NW (0.25)
8:00:00
8:02:30
8:05:00
8:07:30
8:10:00
8:12:30
8:15:00
8:17:30
5000
7500
8600
10000
12000
8300
6600
5200
S (30) 0 (0.4)
AA (0.19)
H (205) 1 (0.25)
L (147) 0 (0.5)
UA (0.15)
CA (0.13)
H (283) 1 (0.4)
S (18) 0 (0.25)
S (30) 0 (0.35)
SW (0)
S (30) 0 (0.35)
DAL (0.11)
S (18) 0 (0.2)
L (291) 1 (0.35)
H (392) 1 (0.3)
S (30) 0 (0.15)
S (18) 0 (0.15) S (30) 0 (0.06)
US (0.21)
L (128) 0 (0.35)
S (18) 0 (0.2)
L (150) 0 (0.35)
S (18) 0 (0.1)
L (120) 0 (0.5) L (85) 0 (0.2)
S (30) 0 (0.4)
S (30) 0 (0.45)
H (202) 1 (0.3)
S (30) 0 (0.4)
Summary and future work
q Summary
− Generic market-based approach to airport demand
management allows to evaluate many alternatives
by varying the weighting factors,
− Legal and procedural challenges would require
rigorous cost-benefit evaluation.
q Future work
− Modeling airlines’ behavior,
− Cross-airport package bidding,
− Conduct sensitivity analysis for different auction
formats (with different values of factor weights)
back-up slides
Problem Identification
Lack of competition
§Hirschman-Herfindahl Index
(HHI) is standard measure of
market concentration
§Department of Justice uses to
measure the competition within
a market place
§HHI=S(100*si)2 w/ si is market
share of airline i
§Ranging between 100 (perfect
competitiveness) and 10000
(perfect monopoly)
§In a market place with an
index over 1800, the market
begins to demonstrate a lack of
competition
HHI Index
6000
5000
4000
3000
HHI Index
2000
1000
0
ATL EWR LAS
LAX LGA MSP ORD PHX SEA STL
Airport
Potential solutions
q Congestion pricing
− monitoring and updating constraints
q Promote use of larger aircraft
− airline economic constraints
q Improve local infrastructure
− environmental constraints
q Rerouting flights
− market constraints
Design scope
q Scope:
− Strategic auction for arrival slots at individual airports
q Objective:
− Provide an optimum fleet mix at optimum safe capacity
− Ensure fair market access opportunity
− Reduce queuing delay
q Definitions:
− Slot: The concession or the entitlement to use runway capacity of a
certain airport by an air carrier on a specific date and at a specific
time [CITE: EEC COM(2001)335]
− Grandfather right: Air carriers having historical use rate of a slot of
at least 80% will have precedence.
Auctioneer Optimization Model
Ranking function:
τ (Ba , s ) = W T * Ba , s
Objective function: Max
∑∑τ ( B
Variables:
ST
A
WT
Ba,s
X= A*ST
1
( X)a,s =
0
MT
slot vector, |ST|=AAR
airline vector
vector of factor weights
bid of airline a for slot s
bidding matrix
a ,s
a
Subject to:
∑ (X )
a ,s
=1
) * ( X )a, s
s
∀s
a
( M T ) s * ( X ) a ,s <= ( Ba , s ) 5
if airline a is ranked highest for slot s
after a round
∀a , s
Safe separation b/w leading and trailing
aircraft:
otherwise
initial monetary offer vector
Mean of the calculated inter-arrival times by aircraft weight categories
Trailing
Leading
Small
Large
Heavy
B757
Small
1.33
1.13
1.07
1.1
Large
2.74
1.21
1.07
1.1
Heavy
3.91
2.467
1.73
2.26
B757
3.35
1.92
1.69
1.7
(Mark Hansen)
Auctioneer Optimization Model
Ranking function:
τ (Ba , s ) = W T * Ba , s
Objective function: Max
∑∑τ ( B
Variables:
ST
A
WT
Ba,s
X= A*ST
1
( X)a,s =
0
MT
slot vector, |ST|=AAR
airline vector
vector of factor weights
bid of airline a for slot s
bidding matrix
a ,s
a
Subject to:
∑ (X )
a ,s
=1
) * ( X )a, s
s
∀s
a
( M T ) s * ( X ) a ,s <= ( Ba , s ) 5
if airline a is ranked highest for slot s
after a round
∀a , s
Safe separation b/w leading and trailing
aircraft:
otherwise
∑ (X )
initial monetary offer vector
a ,s
∗ f ( AT ( a, s), AT (a' , s + 1)) <= 60
s
Mean of the calculated inter-arrival times by aircraft weight categories
Trailing
Leading
Small
Large
Heavy
B757
Small
1.33
1.13
1.07
1.1
Large
2.74
1.21
1.07
1.1
Heavy
3.91
2.467
1.73
2.26
B757
3.35
1.92
1.69
1.7
(Mark Hansen)
Auctioneer Optimization Model
Ranking function:
Objective function:
Max
∑∑τ ( Ba ,s ) * ( X )a ,s
a
τ ( Ba,s ) = W T * Ba ,s
s
Subject to:
Capacity constraint: One slot allocated
to one flight
∑ (X )
a,s
a
=1
∀s
Variables:
ST
A
WT
Ba,s
slot vector, |ST|=AAR
airline vector
vector of factor weights
bid of airline a for slot s
X= A*ST
bidding matrix
1
( X)a,s =
0
if airline a is ranked highest for
slot s after a round
otherwise
Auctioneer Optimization Model
Ranking function:
Objective function:
Max
∑∑τ ( Ba ,s ) * ( X )a ,s
a
Subject to:
s
∑ (X )
a ,s
=1
τ ( Ba,s ) = W T * Ba ,s
∀s
a
Minimum bid: At any round, monetary offers
should be equal or greater than
initial thresholds
( M T ) s * ( X )a ,s <= ( Ba ,s ) 5
∀a, s
Variables:
ST
A
WT
Ba,s
slot vector, |ST|=AAR
airline vector
vector of factor weights
bid of airline a for slot s
X= A*ST
bidding matrix
1
( X)a,s =
0
MT
if airline a is ranked highest for
slot s after a round
otherwise
initial monetary offer vector
Auctioneer Optimization Model
Ranking function:
Objective function:
Max
∑∑τ ( Ba ,s ) * ( X )a ,s
a
s
Subject to:
∑ (X )
a ,s
=1
a
τ ( Ba,s ) = W T * Ba ,s
∀s
( M )s * ( X ) a,s <= ( Ba, s )5
T
∀a, s
Airline constraints: Only one slot is needed
in a set of adjacent slots
∑(X )
s
a, s
<= 1
Variables:
ST
A
WT
Ba,s
slot vector, |ST|=AAR
airline vector
vector of factor weights
bid of airline a for slot s
X= A*ST
bidding matrix
1
( X)a,s =
0
MT
if airline a is ranked highest for
slot s after a round
otherwise
initial monetary offer vector
Auctioneer Optimization Model
Ranking function:
Objective function:
Max
∑∑τ ( Ba ,s ) * ( X )a ,s
a
τ ( Ba,s ) = W T * Ba ,s
s
Subject to:
∑ (X )
a,s
=1
∀s
Variables:
a
(M ) s * ( X )a , s <= ( Ba, s )5
T
∀a, s
Airlines’ package bidding constraints
ST
A
WT
Ba,s
slot vector, |ST|=AAR
airline vector
vector of factor weights
bid of airline a for slot s
X= A*ST
bidding matrix
1
( X)a,s =
0
MT
if airline a is ranked highest for
slot s after a round
otherwise
initial monetary offer vector
Airline Optimization Model
Objective function: Maximize profit
Maximize
∑ (P − B )
s
s
s
Subject to:
Airline either bids for slot s (monetary
offer greater than minimum threshold)
or not.
Bs ≤ M * y s
( B0T ) s ≤ Bs + M * (1 − y s )
Variables:
{Bs}
{Ps}
M
ys
Bo
1
ys = 
0
T
set of monetary bids
airline expected profit by using a slot
big positive value
binary value
if airline bids for slot s
otherwise
airport threshold vector
Airline Optimization Model
Objective function: Maximize profit
Maximize
∑ (P − B )
s
s
s
Subject to:
Bs ≤ M * y s
( B0T ) s ≤ Bs + M * (1 − ys )
When offering bid for slot s, monetary
amount should not pass a threshold
Bs ≤ α * Ps
Variables:
{Bs}
{Ps}
M
ys
1
ys = 
0
T
Bo
a
set of monetary bids
airline expected profit by using a slot
big positive value
binary value
if airline bids for slot s
otherwise
airport threshold vector
airline threshold fraction
Airline Optimization Model
Objective function: Maximize profit
Maximize
∑ (P − B )
s
s
s
Subject to:
Bs ≤ M * y s
( B0T ) s ≤ Bs + M * (1 − ys )
Bs ≤ α * Ps
Given Bs’ the bid airline offered in the previous
round, in order to be ranked highest in this
round, airline has to increase at least :
max (τ ( B
a,s
( B0T ) s
(W ) 5
T
0 s
max
)) − τ ( B A,s )
a
(W ) 5
*(B )
a
(τ ( B a,s )) − τ ( B A,s )
Variables:
{Bs}
{Ps}
M
ys
1
ys = 
0
Bo
a
Bs’
T
set of monetary bids
airline expected profit by using a slot
big positive value
binary value
if airline bids for slot s
otherwise
airport threshold vector
airline threshold fraction
old bid for slot s in previous round
Airline Optimization Model
Objective function: Maximize profit
Maximize
∑ (P − B )
s
s
s
Subject to:
Bs ≤ M * y s
( B0T ) s ≤ Bs + M * (1 − ys )
Bs ≤ α * Ps
Given Bs’ the bid airline offered in the previous
round, in order to be ranked highest in this
round, airline has to increase at least :

(τ (Ba,i )) − τ ( B A,i )

 ' max

a
* ( B0T )i  * yi ≤ α * Pi * yi
 Bi +
(W )5




Old bid
Minimum increase
Variables:
{Bs}
{Ps}
M
ys
1
ys = 
0
Bo
a
Bs’
T
set of monetary bids
airline expected profit by using a slot
big positive value
binary value
if airline bids for slot s
otherwise
airport threshold vector
airline threshold fraction
old bid for slot s in previous round
Airline Optimization Model
Objective function: Maximize profit
Maximize
∑ (P − B )
s
s
s
Subject to:
Bs ≤ M * y s
( B0T ) s ≤ Bs + M * (1 − ys )
Bs ≤ α * Ps

(τ ( Ba,i )) − τ ( BA,i )

 ' max

T
a
B
+
*
(
B
)
0 i  * yi ≤ α * Pi * yi
 i
(
W
)


5


Old bid
Minimum increase

(τ ( Ba,i )) − τ ( B A,i )

 ' max

T
a
B
+
*
(
B
)
0 i  ≤ Bi + M * (1 − y i )
 i
(
W
)

5



Variables:
{Bs}
{Ps}
M
ys
1
ys = 
0
Bo
a
Bs’
T
Minimum new bid
set of monetary bids
airline expected profit by using a slot
big positive value
binary value
if airline bids for slot s
otherwise
airport threshold vector
airline threshold fraction
old bid for slot s in previous round
Airline Optimization Model
Objective function: Maximize profit
Maximize
∑ (P − B )
s
s
s
Subject to:
Bs ≤ M * y s
( B0T ) s ≤ Bs + M * (1 − ys )
Bs ≤ α * Ps

(τ ( Ba,i )) − τ ( BA,i )

 ' max

T
a
B
+
*
(
B
)
0 i  * yi ≤ α * Pi * yi
 i
(W ) 5





(τ ( Ba,i )) − τ ( B A,i )

 ' max

T
a
B
+
*
(
B
)
0 i  ≤ Bi + M * (1 − y i )
 i
(
W
)

5



Variables:
{Bs}
{Ps}
M
ys
1
ys = 
0
Bo
a
Bs’
T
Airlines’ package bidding constraints
set of monetary bids
airline expected profit by using a slot
big positive value
binary value
if airline bids for slot s
otherwise
airport threshold vector
airline threshold fraction
old bid for slot s in previous round