Game Approach for MultiChannel Allocation in MultiHop Wireless Networks

1
Game Approach for Multi-Channel Allocation in
Multi-Hop Wireless Networks

• Lin Gao and Xinbing Wnag
• Dept. Electronic Engineering
• Shanghai Jiao Tong University
• MobiHoc 2008

2
Introduction

• The authors propose the channel allocation
  mechanism for the devices using multi-radios in
  the multi-hop wireless networks
• A cooperative game approach is used to design the
  proposed mechanism
• The mechanism tries to find the optimal
  assignment of radios to available channels in the
  networks
• The mechanism focus on the performance
  improvement (e.g., bandwidth, data transmission
  rate) of the multi-hop links
• Also, the authors present the algorithm which
  enables the players in a game to find
  approximation of Nash equilibrium and is
  computationally efficient

3
Channel Allocation

• In this paper, a simple multi-radio channel
  allocation problem in multi-hop wireless networks
  is studied
• It is assume that a user has a device with 2
  radio sets, each of which contain 2k transceivers
  (k for transmitting and k for receiving)
• Users consist of data senders and relaying users
  (not target users, denoted as d above)
• The available frequency band is divided into N
  channels

(s) (r) (d)
4
Example of Channel Allocation

• A set of communication link is denoted as
• Each communication link is specified by a sender,
  the target, and relaying users between a sender
  and the target
• Each user participates in only one link L 3
• The question is how to assign radios to available
  channels to optimize the data transmission rate
  of the links
• The more users share the same channel, the
  narrower the bandwidth, so decreasing the data
  rate of the links

C c1, c2, c3, c4
5
Research Issue

• There exist research works for a game theory
  based channel allocation mechanism however, they
  focus on only single-hop communication
• In terms of a game theory, it corresponds to a
  non-cooperative game where players try to
  increase their own payoff
• i.e., each user selects available channels to
  improve the data rate of his own link (does not
  care the others)
• The non-cooperative game is not suitable when
  considering the multi-hop networks
• In a non-cooperative game, each user focuses on
  the data transmission rate of the single hop, but
  does not care the end-to-end data transmission
  rate of multi-hop links
• The authors have found that the results (i.e.,
  the solutions of non-cooperative game) does not
  always optimize the end-to-end data transmission
  rate of multi-hop links

50Mbps
10Mbps
35Mbps
25Mbps
s
r
d
s
r
d
6
Research Issue

• Finding the solutions of a cooperative game is
  computationally expensive
• A player has to check all possible strategy
  combination with the cooperating players
• It gets worse as cooperating players increase
• Cooperative game O( SxSco )
• where S is of strategies
• co is of cooperating players
• Non-cooperative game O( SxS )
• The authors propose the computationally efficient
  algorithm to find the optimal allocation strategy
  for multi-hop links
• This leads to the proposal of a cooperative game

7
Game Model

• Players are users in the network, classified into
  senders and relaying users (the target is not
  included)
• The of radios of player ui using channel c
• The total of channels used by player ui
• The total of radios using a particular channel
  c
• The strategy of player ui
• The strategy matrix (payoff table)

8
Game Model (2)

• This paper assume that CSMA/CA protocol is used
• The data rate on a channel x is decreasing as
  of used radios increases
• Total available bandwidth occupied by player ui
  on channel c
• Total available bandwidth occupied by player ui
• End-to-end data rate of communication link

9
Example of Channel Allocation
10
Example of Channel Allocation
Payoff table
11
Example of Channel Allocation
Payoff table
12
Nash Equilibria of the game
NE

• In single-hop networks, the multi-radio channel
  allocation problem is formulated as a
  non-cooperative game, which corresponds to a
  fixed channel allocation among the players
• The payoff of ui with strategy xui in strategy
  matrix
• Nash equilibrium is defined as

Strategy of ui a set of strategies of users
except ui i.e., the others strategies do not
change
No players can increase their payoff without
changing others strategies
13
Specific example of finding NE
Lets assume that U u1,u2,u3 and S
s1,s2 For user 1, there are 2 possible
strategies xu1 ?s1, s2 There are 4 possible
others strategy sets X-u1 xu2, xu3
? s1,s1, s1,s2, s2,s1, s2,s2 Finding
xui which satisfies the condition abovefor each
X-u1 is the same as finding a user 1s strategy
which gives the largest payoff among all possible
strategies
14
Specific example of finding NE (2)
For each of others strategy sets, we need 1
comparison to determine the user 1s strategy
which gives the largest payoff For u1 When u2.s1
and u3.s1 Ru1( s1 ) v.s. Ru1( s2 )
and mark the larger one (See e.g.) When u2.s1
and u3.s2 Ru1( s1 ) v.s. Ru1( s2 ) When u2.s2
and u3.s1 Ru1( s1 ) v.s. Ru1( s2 ) When u2.s2
and u3.s2 Ru1( s1 ) v.s. Ru1( s2 )
and mark the larger one (See e.g.)
15
Specific example of finding NE (2)
User 1 evaluates 1 comparisons for 4 others
strategy sets In general, it requires
comparisons for others strategy
sets The same process is performed for other
users Finally, identify a strategy set, which is
represented by the cell where the all payoff
values are marked. The identified strategy set is
NE (see)
SU-1
SC2
16
Imbalance of payoff

• Non-cooperative game is not suitable for
  multi-hops networks in terms of the data rate of
  the multi-hops links
• For example
• Assume that the game found Nash equilibrium as
  follows
• U1 and U2 are in the same link
• Pu1 1.0, Pu2 1.5
• The data rate of the link is 1.0
• If u1 uses c4 and u2 uses c1 instead
• Then Pu1 1.17, Pu2 1.33
• The data rate of the link is 1.17

17
Coalition-Proof Nash Equilibrium

• For multi-hop networks, coalition is defined as
  a set of users in the same communication link,
  denoted by coi
• To overcome the payoff imbalance, the definition
  of Nash equilibrium is modified and called CPNE
• In this paper, Min-Max CPNE, which is CPNE
  specific to the considering problem (i.e.,
  channel allocation) is proposed

18
Min-Max Coalition-Proof Nash Equilibrium (MMCPNE)
is NE when considering the concept of cooperation
with other users In the process of finding
MMCPNE, it is allowed that a user tries to
increase its coalitions payoff rather than its
own payoff A user is a player U u1, u2, ,
uk A user strategy is a strategy selected by the
user from a set of strategies S s1, s2, ,
ss e.g., Xu3 s2 indicates a user 3 selects
a strategy 2 A coalition is a set of users who
are cooperating each other and it is denoted as Q
co1, co2, , com A coalition strategy is a
strategy combination of the users in the same
coalition Xcoi Xui ui ?coi e.g., Xco2
Xu2, Xu4 s3, s2 indicates a user 2 and 4
are a coalition 2 and select strategy 3 and 2
19
MMCP
Min-Max CPNE A set of coalition strategy Xco1,
Xco2, , Xcom is said to be MMCPNE
Xcoi a coi s coalition strategy in
MMCPNE X-coi a set of coalition strategies of
the coalitions except coi Xcoi a possible
coalition strategy of coalition
coi Ru(Xcoi,X-coi) a payoff that a user u can
gain when a cois coalition strategy is Xcoi (a
user u is a member of coi) and the other
coalitions coalition strategies are X-coi
20
MMCP
Min-Max CPNE A set of coalition strategy Xco1,
Xco2, , Xcom is said to be MMCPNE
is a coalition payoffwhich is the payoff
that the users in a coalition coi can gain when a
cois coalition strategy is Xcoi and the other
coalitions coalition strategies are X-coi Note
The design of coalition payoff depends on the
problem we are considering. In this paper, the
end-to-end data transmission rate of multi-hops
link is considered. Simply, the minimum Ru among
users in a coalition is set to a coalition payoff.
21

• Example of a coalition payoff
• U u1, u2, u3, u4, u5 and S s1, s2
• Q co1u1, u2, co2u3, co3u4, u5
• Lets focus on coalition 1, and assume that
• Xco1 Xu1, Xu2 s1, s2
• X-co1 Xco2, Xco3 Xu3, Xu4, Xu5 s1,
  s2, s2
• min Ru1(Xco1, X-co1), Ru2(Xco1,
  X-co1)
• min Ru1(s1, s2, s1, s2, s2), Ru2(s1,
  s2, s1, s2, s2

22

• More specific example of finding MMCPNE
• U u1, u2, u3 and S s1, s2
• Q co1u1, u2, co2u3
• A payoff table for users (see)
• Lets focus on coalition 1, and assume that
• Xco1 Xu1, Xu2 s1, s2
• X-co1 Xco2 Xu3 s1
• min Ru1(Xco1, X-co1), Ru2(Xco1, X-co1)
• min Ru1(s1, s2, s1), Ru2(s1, s2,
  s1 (see)
• min d, e

23
More specific example of finding MMCPNE (2)
(1)
Now, coalition co1s size is 2, the size of
strategies S is 2 There are 22 possible
coalition strategies Xco1 ? s1, s1, s1, s2,
s2, s1, s2, s2To evaluate the condition (1)
for all coalition strategies is the same as
finding Xcoi, which gives the largest coalition
payoff among the all possible coalition
strategies when the other coalitions coalition
strategies are X-coi In this example, X-coi
Xu3, i.e, s1 or s2 So, find the largest coalition
payoff when Xu3 is s1 or s2 (see)
24
More specific example of finding MMCPNE (3)
(1)
How many comparisons will be required? There
are 4 different coalition strategies for co1
It requires 4C2 comparisons (the worst case) to
find Xco1, which gives the largest payoff, for
each X-co1 In this example, X-co1 Xu3
1 the of different coalition strategies
for X-co1 is SXu3 2 So, the total of
comparisons is 21 x 4C2 12
NC2 where NScoi
SU-k x NC2
where kcoi
and NSk
25
More specific example of finding MMCPNE (4)
(1)
In summary Let Y1 Xco1, X-co1 Xco1,
Xco2 Xu1, Xu2, Xu3 s1, s1, s1
Y2 s1, s2, s1, Y3 s2, s1, s1, Y4 s2,
s2, s1 For a coalition 1 and Xu3 s1 , we need
to evaluate min Ru1(Y1), Ru2(Y1) v.s.
min Ru1(Y2), Ru2(Y2) min Ru1(Y1),
Ru2(Y1) v.s. min Ru1(Y3), Ru2(Y3)
min Ru1(Y3), Ru2(Y3) v.s. min Ru1(Y4),
Ru2(Y4)
Mark a cell for the largest
26
More specific example of finding MMCPNE (5)
(1)
Let Y5 s1, s1, s2, Y6 s1, s2, s2,
Y7 s2, s1, s2, Y8 s2, s2, s2 For a
coalition 1 and Xu3 s2 , we need to evaluate
min Ru1(Y5), Ru2(Y5) v.s. min Ru1(Y6),
Ru2(Y6) min Ru1(Y5), Ru2(Y5) v.s. min
Ru1(Y7), Ru2(Y7) min Ru1(Y7), Ru2(Y7)
v.s. min Ru1(Y8), Ru2(Y8)
Mark a cell for the largest
27
More specific example of finding MMCPNE (5)
(1)
For a coalition 2, find Xco2 Xu3 ? s1, s2
with the largest coalition payoff, and there are
4 possible X-co2 Xco1 ? s1, s1, s1, s2,
s2, s1, s2, s2So, we need to evaluate
min Ru3(Y1) v.s. min Ru3(Y5) min
Ru3(Y2) v.s. min Ru3(Y6) min Ru3(Y3)
v.s. min Ru3(Y7) min Ru3(Y4) v.s. min
Ru3(Y8)
Y1 s1, s1, s1Y2 s1, s2, s1Y3 s2, s1,
s1Y4 s2, s2, s1Y5 s1, s1, s2Y6 s1,
s2, s2Y7 s2, s1, s2Y8 s2, s2, s2 (See)
Mark a cell
Mark a cell
Mark a cell
Mark a cell
28
More specific example of finding MMCPNE (5)
(1)
Finally, find the cell where all the payoff
values are marked A set of strategies, which is
represented by the cell is saidto be MMCPNE (See)
29
Approximate solutions for MMCPNE

• However, the computation to find MMCPNE is
  expensive
• The worst case, a single user needs
• O(Scoi) evaluations
• Exponential to the size of coalition coi
• The authors introduces approximate solutions to
  reduce the computation of finding MMCPNE
• MCPNE
• ACPNE
• ICPNE

30
MCP
MCPNE A set of coalition strategy Xu1, Xu2, ,
Xun is said to be MCPNE
xu a users strategy in MCPNE X-u a set of
strategies of the users except a user u xu a
possible strategy of a user u Ru(xu,X-u) a
payoff that a user u can gain when a users
strategy is xu and the other users strategies
are X-u
31
MCP
MCPNE A set of coalition strategy Xu1, Xu2, ,
Xun is said to be MCPNE
Similar to the process of finding MMCPNE,a
coalition payoff is considered How to find
MCPNE is also similar to MMCPNE However, the
number of comparisons may be reduced
MMCPNE MCPNE
32
Approximate solutions (AP) for MMCPNE

• Who play the game is different
• In MM, a player is a coalition, a set of users
  cooperating each other while in AP, a player is a
  user
• So, a players strategy is different
• MM uses a coalition strategy, a set of users
  strategies in the coalition while AP uses a user
  strategy
• For each player, finding NE is the same as
  finding the players strategy which makes the
  coalition payoff the largest among all other
  possible players strategies
• The of possible players strategies is
  different
• of possible coalition strategies in MM is
  larger or equal to of possible user strategies
  in AP
• So, the of comparisons to find the players
  strategy which makes the coalition payoff the
  largest is different

33
Specific example of finding MCPNE
Again, lets consider the game where U u1,
u2, u3 S s1, s2 Q co1u1, u2,
co2u3 A payoff table is defined as (see)
34
Specific example of finding MCPNE (2)
For user 1, there are 2 possible strategies xu1
? s1, s2 and 4 possible sets of the other
users strategies X-u1 xu2, xu3 ? s1, s1,
s1, s2, s2, s1, s2, s2 A user 1 is a
member of co1 u1, u2, so we need to find xu,
which gives the largest payoff among all
possible xu1 for each of X-u1 min
Ru1(xu1, X-u1), Ru2(xu2, X-u2)
35
Specific example of finding MCPNE (3)
For user 1 and each X-u1? s1, s1, s1, s2,
s2, s1, s2, s2 We need to evaluate when
X-u1 xu2, xu3 s1, s1 min Ru1(Y1),
Ru2(Y1) v.s. min Ru1(Y2), Ru2(Y2) where Y1
xu1, xu2, xu3 s1, s1, s1 Y2 s2, s1,
s1
Mark a cell
(See)
36
Specific example of finding MCPNE (4)
Do the same process for other strategies set
min Ru1(Y1), Ru2(Y1) v.s. min Ru1(Y2),
Ru2(Y2) min Ru1(Y3), Ru2(Y3) v.s. min
Ru1(Y4), Ru2(Y4) min Ru1(Y5), Ru2(Y5) v.s.
min Ru1(Y6), Ru2(Y6) min Ru1(Y7), Ru2(Y7)
v.s. min Ru1(Y8), Ru2(Y8)
Mark a cell
Mark a cell
Mark a cell
Mark a cell
where Y1 s1, s1, s1 Y2 s1, s2, s1Y3
s2, s1, s1 Y4 s2, s2, s1Y5 s1, s1,
s2 Y6 s1, s2, s2Y7 s2, s1, s2 Y8
s2, s2, s2
(See)
37
Specific example of finding MCPNE (5)
4 comparisons are required for each user So, in
total, 12 comparisons are required
U x SU-1 x SC2
In general,
Q x SU-k x NC2
where NScoi, kcoi
For MMCPNE
38
ACP
ACPNE A set of coalition strategy Xu1, Xu2, ,
Xun is said to be ACPNE
ACPNE extends MCPNE by adding tie braking
rule. The number of candidate NEs is reduced
(i.e., during a process to find NE, less number
of cells are marked)
39
(No Transcript)
40
ACP
ACPNE A set of coalition strategy Xu1, Xu2, ,
Xun is said to be ACPNE
xu a users strategy in MCPNE X-u a set of
strategies of the users except a user u xu a
possible strategy of a user u Ru(xu,X-u) a
payoff that a user u can gain when a users
strategy is xu and the other users strategies
are X-u
41
ACP
The process of finding ACPNE is very similar to
find the MCPNE (almost the same) Only the
difference is that if two or more different
strategies, which give the same coalition payoff,
are found, then the strategy, which gives the
highest average (or total) payoff of the users in
the same coalition, is considered
42
Specific example of finding ACPNE
Again, lets consider the game where U u1,
u2, u3 S s1, s2 Q co1u1, u2,
co2u3
43
Specific example of finding ACPNE (2)
For user 1, perform the same process as MCPNE
case However, mark (i.e., pick) the largest
average coalition payoff when two or more
coalition payoff are the same (See) Perform the
same process for user 2 and user 3 ACPNE is a set
of strategies, which is represented by the cell
where all of payoff values are marked of
comparisons is the same as MCPNE
U x SU-1 x SC2
44
ICP
ICPNE A set of coalition strategy Xu1, Xu2, ,
Xun is said to be ICPNE
xu a users strategy in ICPNE X-u a set of
strategies of the users except a user u xu a
possible strategy of a user u Ru(xu,X-u) a
payoff that a user u can gain when a users
strategy is xu and the other users strategies
are X-u
45
ICP
The process of finding ICPNE is also very similar
to find the MCPNE (almost the same) Only the
difference is that if two or more different
strategies, which give the same coalition payoff
value, are found, then the strategy, which makes
the users own payoff the highest, is considered
46
Specific example of finding ICPNE
Again, lets consider the game where U u1,
u2, u3 S s1, s2 Q co1u1, u2,
co2u3
47
Specific example of finding ICPNE (2)
- For user 1, perform the same process as MCPNE
case However, mark (i.e., pick) the largest
users payoff when two or more coalition payoff
are the same (See) - Perform the same process for
user 2 and user 3 - ICPNE is a set of strategies,
which is represented by the cell where all of
payoff values are marked - of comparisons is
the same as MCPNE
U x SU-1 x SC2
48
Simulation Results

• The authors evaluate the performance of links
  that channels are allocated based on six
  different approaches
• NE, CPNE, MMCPNE, DCP-M, DCP-A, DCP-I
• Simulation configuration
• Assume 8 channels, 4 transceivers, 5 users

49
Simulation Results

• Performance measurement
• Coalition Utility The ability of coalition to
  gain utility for the channel bandwidth
• Coalition Usage Factor The utilization of
  bandwidth occupied by coalition

Total bandwidth occupied by coi
Average bandwidth per user
The actual bandwidth usagefor the coalition
50
Average coalition utility
CPNE and MMCP (coalition-proof) show almost the
same result, and best DCP-A shows higher
coalition utility than the other DCP-x algorithm
- In ACPNE, all players in the coalition are
willing to improve the total bandwidth like
CPNE and MMCP MMCP is little higher than DCP-A
due to the cooperation
51
Average coalition factor
The optimal value of CF is 0.5 when the of
users in co is 2
MMCP and DCP-M show better usage factor than the
others, and converge to almost optimal point,
0.5 - The difference between MMCP and DCP-M
is 1 CPNE and DCP-A show low usage factor -
Because they occupy high total bandwidth (slide
22)
52
Conclusion

• This paper proposes a game approach for
  multi-radio channel allocation in multi-hop
  networks
• The authors introduce the concept of coalition
  and apply a cooperative game for finding Nash
  equilibrium as coalition (called coalition-proof
  Nash equilibrium, CPNE)
• The authors proposes three approximate solutions
  to efficiently find CPNE (they are DCP-M, -A, -I)
• The simulation results show that DCP-M algorithm
  achieves similar result to Min-Max CPNE
• 1 difference in usage factor

53
APPENDIX
54
Distributed CP algorithm

• The authors presents the Distributed algorithm
  for the three approximate solutions of Min-Max
  CPNE
• It is called, DCP-M, DCP-A, DCP-I, respectively.
• It enables users to converge to an approximate
  solutions of MMCPNE
• The simulation section show the result of DCP
  algorithm is very close to the actual result of
  MMCPNE
• The computation is reduce from exponential to
  linear increasing with coi

55
DCP-M
For each user
Counter is assigned to a user to avoid all
usersperforming the algorithm at the same
time If the counter is not 0 yet, then decrease
it by 1
For each radio
For each unused channel
A set of channels, unused by ui
Compute the coalition payoffgain by changing
channel b to c Record the payoff for all
possible c
Select a channel (say a) with the largest
payoffamong the recorded payoffs Change radio
js channel from b to a
56
Average coalition efficiency
CPNE and DCP-A converge with low coalition
efficiency (i.e. low data rate) - They may
result in imbalanced allocation although the
average is high But, MMCP still shows the highest
coalition efficiency DCP-M shows the best among
other DCP-x - The difference from MMCP is 5
Keep the minimum utilityof coalition high
57
S1
u2
u3
u1
S2
u3
u2
u1
58
S1
u2
u3
u1
S2
u3
u2
u1
59

• co1u1, u2
• co2u3
• 3 users and 2 coalitions
• A user selects one of 2 strategies c1, c2,
  which indicates a channel the user uses to send a
  packet

u1
u2
d1
u3
d2
u1
u2
u1
u1
u2
u3
u2
u3
u3
u3
u2
u1
C1 C2
C1 C2
C1 C2
C1 C2
u1
u2
u1
u1
u2
u3
u3
u2
u1
u2
u3
u3
C1 C2
C1 C2
C1 C2
C1 C2
60
S1
u2
u3
u1
S2
u3
u2
u1
61
back
S1
u2
u3
u1
Ru1( s1 ) v.s. Ru1( s2 )
S2
u3
u2
u1
62
For xu2, xu3 s1,s2
S1
u2
u3
u1
Ru1( s1 ) v.s. Ru1( s2 )
S2
u3
u2
u1
63
For xu2, xu3 s2,s1
S1
u2
u3
u1
Ru1( s1 ) v.s. Ru1( s2 )
S2
u3
u2
u1
64
back
For xu2, xu3 s2,s1
S1
u2
u3
u1
Ru1( s1 ) v.s. Ru1( s2 )
S2
u3
u2
u1
65
back
S1
u2
u3
u1
xu1, xu2, xu3 s1, s2, s2 is NE
S2
u3
u1
66
S1
u2
u3
u1
S2
u3
u2
u1
67
back
S1
u2
u3
u1
S2
u3
u2
u1
68
back
S1
u2
u3
u1
min Ru1(s1, s2, s1), Ru2(s1, s2, s1 min
d, e
69
When Xu3 s1
S1
u2
u3
u1
S2
u3
u2
u1
70
When Xu3 s1
u2
S1
u3
u1
S2
u3
u2
u1
71
When Xu3 s2
S1
u2
u3
u1
S2
u3
u2
u1
72
back
When Xu3 s2
S1
u2
u3
u1
u2
S2
u3
u1
73
When X-co2 Xco1 s1, s1
S1
u2
u3
u1
u2
S2
u3
u1
74
back
When X-co2 Xco1 s1, s2
S1
u2
u3
u1
u2
S2
u3
u1
75
back
Similarly for X-co2 s2, s1 and s2, s2
S1
u2
u3
u1
u2
S2
u3
u1
76
back
S1
u2
u3
u1
X s1, s2, s2 is MMCPNE
S2
u3
u1
77
S1
u2
u3
u1
We need to evaluate for X-u1 xu2, xu3
s1, s1 min Ru1(Y1), Ru2(Y1) v.s. min
Ru1(Y2), Ru2(Y2) where Y1 xu1, xu2, xu3
s1, s1, s1 Y2 s2, s1, s1
78
S1
u2
u3
u1
We need to evaluate for X-u1 xu2, xu3
s1, s1 min Ru1(Y1), Ru2(Y1) v.s. min
Ru1(Y2), Ru2(Y2) where Y1 xu1, xu2, xu3
s1, s1, s1 Y2 s2, s1, s1
79
S1
u2
u3
u1
We need to evaluate for X-u1 xu2, xu3
s1, s1 min Ru1(Y1), Ru2(Y1) v.s. min
Ru1(Y2), Ru2(Y2) where Y1 xu1, xu2, xu3
s1, s1, s1 Y2 s2, s1, s1
80
back
S1
u2
u3
u1
We need to evaluate for X-u1 xu2, xu3
s1, s1 min Ru1(Y1), Ru2(Y1) v.s. min
Ru1(Y2), Ru2(Y2) Mark the largeset value
81
For user 1
S1
u2
u3
u1
S2
u3
u2
u1
82
For user 2
S1
u2
u3
u1
S2
u3
u2
u1
83
For user 3
S1
u2
u3
u1
S2
u3
u2
u1
84
back
Identify MCPNE
S1
u2
u3
u1
S2
u3
u2
u1
85
back
S1
u2
u3
u1
S2
u3
u2
u1
86
For user 1
S1
u2
u3
u1
1 v.s. 2
S2
u3
u2
u1
87
For user 1
S1
u2
u3
u1
2 v.s. 2 ? 3.5 v.s. 2.5
S2
u3
u2
u1
88
For user 1
S1
u2
u3
u1
S2
u3
u2
u1
89
For user 2
S1
u2
u3
u1
2 v.s. 2 ? 2.5 v.s. 3.5
S2
u3
u1
90
For user 2
S1
u2
u3
u1
S2
u3
u2
u1
91
For user 3
S1
u2
u3
u1
S2
u3
u2
u1
92
For user 3
S1
u2
u3
u1
S2
u3
u2
u1
93
ACPNE
back
S1
u2
u3
u1
S2
u3
u2
u1
94
back
S1
u2
u3
u1
S2
u3
u2
u1
95
For user 1
S1
u2
u3
u1
1) 2 v.s. 3
S2
u3
u2
u1
96
For user 1
S1
u2
u3
u1
1) 1 v.s. 1 2) 1 v.s. 9
S2
u3
u2
u1
97
For user 1
S1
u2
u3
u1
S2
u3
u2
u1
98
For user 2
S1
u2
u3
u1
S2
u3
u2
u1
99
For user 2
S1
u2
u3
u1
1) 3 v.s. 3 2) 3 v.s. 4
S2
u3
u2
u1
100
For user 2
S1
u2
u3
u1
S2
u3
u2
u1
101
For user 3
S1
u2
u3
u1
S2
u3
u2
u1
102
For user 3
S1
u2
u3
u1
S2
u3
u2
u1
103
ICPNE
back
S1
u2
u3
u1
S2
u3
u2
u1