Truthful Mechanism for
Facility Allocation:
A Characterization and Improvement of
Approximation Ratio
Pinyan Lu,
MSR Asia
Yajun Wang,
MSR Asia
Yuan Zhou, Carnegie Mellon University
1
Problem discussed
Design a mechanism for the following n-player game
 Players is located on a real line
 Each player report their location to the mechanism
 The mechanism decides a new location to build the
facility
mechanism g
y
x1
x2
2
Problem discussed (cont’d)
Design a mechanism for the following n-player game
 Players is located on a real line
 Each player report their location to the mechanism
 The mechanism decides a new location to build the
facility
For example, the mean func., g = (x 1 + x 2 )=2
mechanism g = (x 1 + x 2 )=2
x1
y
x2
3
Problem discussed (cont’d)
Design a mechanism for the following n-player game
 Players is located on a real line
 Each player report their location to the mechanism
 The mechanism decides a new location to build the
facility
For example, the mean func., g = (x 1 + x 2 )=2
 This encourages Player 1 to report x 01 = 2x 1 ¡ x 2 ,
then g(x 01 ; x 2 ) = x 1 becomes closer to Player 1’s real
location.
mechanism g = (x 1 + x 2 )=2
x 01
x1
y
x2
4
Truthfulness
Design a mechanism for the following n-player game
 Players is located on a real line
 Each player report their location to the mechanism
 The mechanism decides a new location to build the
facility
Truthful mechanism does not encourage player to
report untruthful locations
g(x 1 ; x 2 ) = 0
 g(x 1 ; x 2 ) = x 1
 g(x 1 ; x 2 ) = minf x 1 ; x 2 g
mechanism g = x 1
x1
x2
5
Truthfulness of
g(x 1 ; x 2 ) = minf x 1 ; x 2 g
Suppose w.l.o.g. that x 1 · x 2
x 1 has no incentive to lie
x 2 will not change the outcome of g if it
misreports a value x 02 ¸ x 1
If x 2 misreports that x 02 < x 1 , then the decision of g
will be even farther from x 2
g = minf x 1 ; x 2 g
x1
x 02
6
Truthfulness of
g(x 1 ; x 2 ) = minf x 1 ; x 2 g
Suppose w.l.o.g. that x 1 · x 2
x 1 has no incentive to lie
x 2 will not change the outcome of g if it
misreports a value x 02 ¸ x 1
If x 2 misreports that x 02 < x 1 , then the decision of g
will be even farther from x 2
Corollary: a mechanism which outputs the leftmost
(rightmost) location among n players is truthful
gg0== minf
minf xx11;;xx202gg
x1
x 02
7
A natural question
Is there any other (non-trivial) truthful
mechanisms? Can we fully characterize the set of
truthful mechanisms?
Gibbard-Satterthwaite Theorem. If players can
give arbitrary preferences, then the only truthful
mechanisms are dictatorships, i.e. g = f (x i ) for
some i 2 [n]
In our facility game, since players are not able to
give arbitrary preferences, we have a set of
richer truthful mechanisms, such as
leftmost(rightmost), and …
8
Even more interesting truthful mechanisms
Mechanism:
gk (x 1 ; x 2 ; ¢¢¢; x n ) = k-t h left locat ion among input s
Suppose w.l.o.g. that x 1 · x 2 · x 3 · ¢¢¢· x n
x k has no incentive to lie
x i (i 6
= k) can change the outcome only when it lies to
be x 0i where x 0i and x i are on different sides of x k ,
but this makes the new outcome farther from x i
Corollary: outputting the median ( g[n =2] ) is truthful
gkk0
x1
x 0ii
xk
xn
9
Social cost and approximation ratio
Good news! Median is truthful!
 Median also optimizes the social cost, i.e. the
total distance from each player to the facility
Xn
scx 1 ;x 2 ;¢¢¢;x n (g) :=
jx i ¡ g(x 1 ; x 2 ; ¢¢¢; x n )j
i= 1
Approximation ratio of½mechanism g
©(g) :=
max
x 1 ;x 2 ;¢¢¢;x n
scx 1 ;x 2 ;¢¢¢;x n (g)
OPT(x 1 ; x 2 ; ¢¢¢; x n )
¾
©(median) = 1
10
Approximation ratio of other mechanisms
©(g ´ 0) = + 1
 Gap instance:
0
x 1 ; x 2 ; ¢¢¢; x n
©(out put t ing t he left most player's locat ion) = n ¡ 1
 Gap instance:
x1
x 2 ; x 3 ; ¢¢¢; x n
11
Extend to two facility game
Suppose we have more budget, and we can afford
building two facilities
Each player’s cost function: its distance to the
closest facility
Good truthful approximation?
A simple try
 Mechanism: set facilities on the leftmost and
rightmost player’s location
12
Extend to two facility game
A simple try
 Mechanism: set facilities on the leftmost and
rightmost player’s location
¾
 Gap Instance: © ¸ n ¡ 2
) ©= n¡ 2
©· n¡ 2
sc(Mech.) = n ¡ 2
x1
x 2 ; x 3 ; ¢¢¢; x n ¡
¡ 1
0
OPT = 1
1
xn
1
13
Randomized mechanisms
The mechanism selects pair of locations according
to some distribution
Each player’s cost function is the expected
distance to the closest facility
Does randomness help approximation ratio?
14
Multiple locations per agent
Agent i controls wi locations x i = (x i 1 ; x i 2 ; ¢¢¢; x i w i )
P wi
Agent i ‘s cost function is j = 1 jg ¡ x i j j
Social cost:
P
P wi
n
i= 1
j = 1 jg ¡
xi j j
A randomized truthful mechanism
 Given f x 1 ; x 2 ; ¢¢¢; x n g, return med(x i ) with
Pn
probability wi = j = 1 wj
Claim. The mechanism is truthful
Theorem. The mechanism’s approximation ratio is
2 min1· j · n wj
Pn
3¡
j = 1 wj
15
Summary of questions.
Characterization
 Is there a full characterization for
deterministic truthful mechanism in one-facility
game?
Approximation
 Upper/lower bound for two facility game in
deterministic/randomized case?
 Lower bound for one facility game in
randomized case when agents control multiple
locations?
16
Our result and related work
Give a full characterization of one-facility
deterministic truthful mechanisms
 Similar result by [Moulin] and [Barbera-Jackson]
Improve the bounds approximation ratio in several
extended game settings
Setting
one facility
two facilities
deterministic deterministic
Previous known*
two facilities
randomized
one facility,
randomized**
1 vs. 1
3/2 vs. n – 1
? vs. n – 1
? vs. ?
Our result
N/A
2 vs. n – 1
1.045 vs. n – 1
1.33 vs. 3
Follow-up result
N/A
Ω(n) vs. n – 1
1.045 vs. 4
N/A
 *: Most of previous results are due to [ProcacciaTennenholtz]
 **: In this setting, each player can control
multiple locations
17
Outline
Characterization of one-facility deterministic
truthful mechanisms
Lower bound for randomized two-facility games
Lower bound for randomized one-facility games
when agents control multiple locations
Upper bound for randomized two-facility games
18
The characterization
Generally speaking, the set of one-facility
deterministic truthful mechanisms consists of
min-max functions (and its variations)
Actually we prove that all truthful mechanism can
be written in a standard min-max form with 2n
parameters (perhaps with some variation)
max
x1
standard form
min
min
x2
max
x3
med
med
x1
c1
med
c1 x1 c2
x2
x3
med
c3 x1 c4
med
med x2
c5 x1 c6
med
c7 x1 c8
19
More precise in the characterization
The image set of the mechanism can be an arbitrary
closed set U
We restrict the min-max function onto U by finding
the nearest point in U
closed set U
f :
max
min
min
x1
x2
max
x3
mechanism g
x1
c1
20
More precise in the characterization
The image set of the mechanism can be an arbitrary
closed set U
We restrict the min-max function onto U by finding
the nearest point in U
closed set U
f :
max
min
min
x1
x2
max
x3
mechanism g
x1
c1
21
More precise in the characterization
The image set of the mechanism can be an arbitrary
closed set U
We restrict the min-max function onto U by finding
the nearest point in U
 What about when there are 2 nearest points ?
 A tie-breaking gadget takes response of that !
closed set U
f :
max
min
min
x1
mechanismtiegbreaker
x2
max
x3
x1
c1
22
The proof – warm-up part
Image set of g
Lemma. If g is a truthful mechanism, then g(x; x; ¢¢¢; x)
goes to the closest point in I (g) from x , for all x
Proof. For every y = g(x 1 ; x 2 ; ¢¢¢; x n ),
jy ¡ xj = jg(x 1 ; x 2 ; ¢¢¢; x n ) ¡ xj
¸
jg(x; x 2 ; ¢¢¢; x n ) ¡ xj
¸
jg(x; x; ¢¢¢; x n ) ¡ xj
¢¢¢
¸
jg(x; x; ¢¢¢; x) ¡ xj
Corollary. U = I (g) is closed.
Now, for simplicity, assume U = I (g) = (¡ 1 ; + 1 )
23
Main lemma
Lemma. For each truthful mechanism g, there exists
a min-max function , such that g(x 1 ; x 2 ; ¢¢¢; x n ) is the
closest point in I (g) from f (x 1 ; x 2 ; ¢¢¢; x n ) , for all
inputs x 1 ; x 2 ; ¢¢¢; x n 2 R
Proof (sketch). Prove by induction on n
 When n = 1, g(x) should output the closest point
in I (g) from x : f (x) = x
 For n > 1
24
Main lemma
0
g
n
>
1
For
, define x 1 ;x 2 ;¢¢¢;x n ¡ 1 (x n ) = g(x 1 ; x 2 ; ¢¢¢; x n )
 Claim 1. g0 is truthful
 Claim 2. 9L = L(x 1 ; ¢¢¢; x n ¡ 1 ); R = R(x 1 ; ¢¢¢; x n ¡ 1 );
s:t: I (gx01 ;x 2 ;¢¢¢;x n ¡ 1 ) = I (g) \ [L ; R]
 Claim 3. L; R , as mechanisms for (n ¡ 1) -player
game, are truthful
 Claim 4. 9L l ; L r : I (L ) = I (g) \ [L l ; L r ];
9Rl ; Rr : I (R) = I (g) \ [Rl ; Rr ]
I (g)
I (gx01 ;x 2 ;¢¢¢;x n ¡ 1 )
L (x 1 ; x 2 ; ¢¢¢; x n ¡ 1 )
I (g)
I (L )
Ll
R(x 1 ; x 2 ; ¢¢¢; x n ¡ 1 )
I (R)
Lr
Rl
Rr
25
Main lemma
Thus,
g(x 1 ; x 2 ; ¢¢¢; x n ) = gx01 ;x 2 ;¢¢¢;x n ¡
1
= med(L ; x n ; R)
L = med(L l ; g1 (x 1 ; x 2 ; ¢¢¢; x n ¡ 1 ); L r )
R = med(Rl ; g2 (x 1 ; x 2 ; ¢¢¢; x n ¡ 1 ); Rr )
g:
I (g)
med
xn
L
I (gx01 ;x 2 ;¢¢¢;x n ¡ 1 )
L (x 1 ; x 2 ; ¢¢¢; x n ¡ 1 )
I (g)
I (L )
Ll
R
R(x 1 ; x 2 ; ¢¢¢; x n ¡ 1 )
I (R)
Lr
Rl
Rr
26
Main lemma
Thus,
g(x 1 ; x 2 ; ¢¢¢; x n ) = gx01 ;x 2 ;¢¢¢;x n ¡
1
= med(L ; x n ; R)
L = med(L l ; g1 (x 1 ; x 2 ; ¢¢¢; x n ¡ 1 ); L r )
R = med(Rl ; g2 (x 1 ; x 2 ; ¢¢¢; x n ¡ 1 ); Rr )
g:
xn
med
Ll
g1
med
Lr
med
R l g2
Rr
27
Main lemma
g:
med
1 player:
xn
med
Ll
g1
med
R l g2
Lr
g( 1) :
x1
Rr
2 players:
g( 2) :
med
x2
med
c1 g( 1)
1
c2
med
c3 g( 1)
2
c4
28
Main lemma
g:
med
1 player:
xn
med
3 players:
Ll
g1
med
R l g2
Lr
g( 1) :
x1
Rr
2 players:
g( 2) :
med
x2
med
c1 x 1
c2
med
c3 x 1
c4
29
Main lemma
2 players:
1 player:
( 1)
g
g( 2) :
x1
:
3 players:
( 3)
g
c1 x 1
c9 med
( 2) c10
g1
med x 2 med
c1 x 1
c2 c3 x 1
x3
x2
med
: med
med
med
c2
med
c3 x 1
c4
med
c11 med
c12
( 2)
g2
med x 2 med
c4 c5 x 1
c6 c7 x 1
c8
30
Main lemma
2 players:
1 player:
( 1)
g
g( 2) :
x1
:
3 players:
( 3)
g
c1 x 1
c9 med c10
med x 2 med
c1 x 1
c2 c3 x 1
x3
x2
med
: med
med
med
c2
med
c3 x 1
c4
med
c11 med c12
med x 2 med
c4 c5 x 1
c6 c7 x 1
c8
31
The reverse direction
Lemma. Every min-max function is truthful
 Observation. To prove a n -player mechanism
is truthful, only need to prove the 1-player
mechanisms gx01 ;¢¢¢;x i ¡ 1 ;x i + 1 ;¢¢¢;x n (x i ) are truthful for
every i and x j (j 6= i )
Theorem. The characterization is full
32
Multiple locations per agent
Theorem. Any randomized truthful mechanism of
the one facility game has an approximation ration
at least 1.33 in the setting that each agent
controls multiple locations.
Theorem (weaker). Any randomized truthful
mechanism of the one facility game has an
approximation ration at least 1.2 in the setting
that each agent controls multiple locations.
33
Multiple locations per agent (cont’d)
Player 1
Player 2
Proof. (weaker version)
Instance 1
g = D1
Instance 2
g = D2
0
1
L 2 = Ex 2 D 2 [jx ¡ 0j ¢1x · 0 ]
Instance 3
For
For
g = D3
R2 = Ex 2 D 2 [jx ¡ 1j ¢1x ¸ 1 ]
0
1
0
1
Player 1 at Instance 1 (compared to Instance 2)
2 ¢jx ¡ 0j x 2 D 1 + jx ¡ 1j x 2 D 1 · 2 ¢jx ¡ 0j x 2 D 2 + jx ¡ 1j x 2 D 2
)
jx ¡ 0j x 2 D 1 · jx ¡ 0j x 2 D 2 + 2(L 2 + R2 )
Player 2 at Instance 3 (compared to Instance 2)
2 ¢jx ¡ 1j x 2 D 3 + jx ¡ 0j x 2 D 3 · 2 ¢jx ¡ 1j x 2 D 2 + jx ¡ 0j x 2 D 2
)
jx ¡ 1j x 2 D 3 · jx ¡ 1j x 2 D 2 + 2(L 2 + R2 )
34
Multiple locations per agent (cont’d)
Player 1
Player 2
Proof. (weaker version)
Instance 1
g = D1
Instance 2
g = D2
0
1
L 2 = Ex 2 D 2 [jx ¡ 0j ¢1x · 0 ]
Instance 3
g = D3
R2 = Ex 2 D 2 [jx ¡ 1j ¢1x ¸ 1 ]
0
1
0
1
For
Player 1
jx ¡ 0j x 2 D 1 · jx ¡ 0j x 2 D 2 + 2(L 2 + R2 )
For
Player 2
jx ¡ 1j x 2 D 3 · jx ¡ 1j x 2 D 2 + 2(L 2 + R2 )
Assume <1.2 approx.
For Inst. 1
2 ¢jx ¡ 0j x 2 D 1 + 4 ¢jx ¡ 1j x 2 D 1 < 1:2 ¢2
For Inst. 2
3 ¢jx ¡ 0j x 2 D 2 + 3 ¢jx ¡ 1j x 2 D 2 < 1:2 ¢3
For Inst. 3
4 ¢jx ¡ 0j x 2 D 3 + 2 ¢jx ¡ 1j x 2 D 3 < 1:2 ¢2
35
Multiple locations per agent (cont’d)
Player 1
Player 2
Proof. (weaker version)
Instance 1
g = D1
Instance 2
g = D2
0
1
L 2 = Ex 2 D 2 [jx ¡ 0j ¢1x · 0 ]
Instance 3
g = D3
For
Player 1
For
Player 2
R2 = Ex 2 D 2 [jx ¡ 1j ¢1x ¸ 1 ]
0
1
0
1
jx ¡ 0j x 2 D 1 · jx ¡ 0j x 2 D 2 + 2(L 2 + R2 )
1.6 < jx ¡ 1j
x 2 D 3 · jx ¡ 1j x 2 D 2 + 2(L 2 + R2 )
Assume <1.2 approx.
< 1.6
Contradiction
For Inst. 1
jx ¡ 1j x 2 D 1 < 0:2
For Inst. 2
jx ¡ 0j x 2 D 2 + jx ¡ 1j x 2 D 2 < 1:2
For Inst. 3
jx ¡ 0j x 2 D 3 < 0:2
) jx ¡ 0j x 2 D 1 > 0:8
) L 2 + R2 < 0:1
) jx ¡ 1j x 2 D 3 > 0:8
36
Multiple locations per agent (cont’d)
Player 1
Player 2
Proof. (stronger version)
Instance 1
Instance 2
0
1
Instance 3
0
1
Instance 4
0
1
Instance 5
0
1
0
1
37
Multiple locations per agent (cont’d)
Player 1
Player 2
Proof. (stronger version)
Instance i (1 · i · K )
£ (K + i )
Instance K + 1
0
£ (2K + 1)
0
Instance 2K + 2 ¡ i (K ¸ i ¸ 1)
£ (K + 1 ¡ i )
£ (2K + 1)
0
£ (K + 1 ¡ i )
£ (2K + 1)
1
£ (2K + 1)
1
£ (K + i )
1
38
Multiple locations per agent (cont’d)
Linear Programming
Take K = 500 : ® > 1:33
39
Lower bound for 2-facility randomized case
Theorem. For any 2-facility randomized truthful
mechanism, the approximation ratio is at least
1
a ¡
, where n ¸ 30 is the number of players
1:045
n¡ 3
Proof
 Consider instanceI :1 player at ¡ 1 , n ¡ 2 players
at 0 , 1 player at 1
 e1 + e2 + e3 ¸ 1
 For mechanisms within 2-approx. : e2 · 2=(n ¡ 2)
 Assume w.l.o.g.: e3 ¸ 1=2 ¡ 1=(n ¡ 2)
e1
x1
yl ¡ 1
e2
x 2 ; x 3 ; ¢¢¢; x n ¡
0 yr
e3
1
xn
1
40
Lower bound for 2-facility randomized case
Theorem. For any 2-facility randomized truthful
mechanism, the approximation ratio is at least
1
a ¡
, where n ¸ 30 is the number of players
1:045
n¡ 3
e3 ¸ 1=2 ¡ 1=(n ¡ 2)
Proof
 Consider instanceI :1 player at ¡ 1 , n ¡ 2 players
at 0 , 1 player at 1
 Another instance I 0: 1 player at ¡ 1, n ¡ 2 players
at 0 , 1 player at 1 + ®
e1
x1
yl ¡ 1
e2
x 2 ; x 3 ; ¢¢¢; x n ¡
0 yr
e3
1
xx 0nn
1 +1 ®
41
Lower bound for 2-facility randomized case
Theorem. For any 2-facility randomized truthful
mechanism, the approximation ratio is at least
1
a ¡
, where n ¸ 30 is the number of players
1:045
n¡ 3
e3 ¸ 1=2 ¡ 1=(n ¡ 2)
Proof
 Consider instanceI :1 player at ¡ 1 , n ¡ 2 players
at 0 , 1 player at 1
 Another instance I 0: 1 player at ¡ 1, n ¡ 2 players
at 0 , 1 player at 1 + ®
 By truthfulness: e03 ¸ 1=2 ¡ 1=(n ¡ 2) ¡ ®
e1
x1
yl ¡ 1
e2
x 2 ; x 3 ; ¢¢¢; x n ¡
0 yr
e3
1
e03
xn
x 0n
1
1+ ®
42
Lower bound for 2-facility randomized case
Theorem. For any 2-facility randomized truthful
mechanism, the approximation ratio is at least
1
a ¡
, where n ¸ 30 is the number of players
1:045
n¡ 3
e03 ¸ 1=2 ¡ 1=(n ¡ 2) ¡ ®
Proof
sc(I 0)
=
¸
0
sc(I )
¸
=
1
1
] ¢1 + Pr[yr ¸
] ¢1 + e03
n¡ 2
n¡ 2
1
1
1
1
1 ¡ Pr[¡
< yr <
]+ ¡
¡ ®
n¡ 2
n¡ 2
2 n¡ 2
e1 + (n ¡ 2)e2 + e03 ¸ Pr[yr · ¡
1
1
1
1
Pr[yr · ¡
] ¢1 + Pr[yr ¸
] ¢1 + Pr[¡
< yr <
] ¢(1 + ®)
n¡ 2
n¡ 2
n¡ 2
n¡ 2
1
1
1 + ® ¢Pr[¡
< yr <
]
n¡ 2
n¡ 2
e1
x1
yl ¡ 1
e2
x 2 ; x 3 ; ¢¢¢; x n ¡
0 yr
e3
1
e03
xn
x 0n
1
1+ ®
43
Lower bound for 2-facility randomized case
Theorem. For any 2-facility randomized truthful
mechanism, the approximation ratio is at least
1
a ¡
, where n ¸ 30 is the number of players
1:045
n¡ 3
Proof
p
2¡ 1
1
1
0
p ¡
sc(I ) ¸ 1 +
> 1:045 ¡
n¡ 2
12 ¡ 2 2 n ¡ 2
opt(I 0) = 1
Done.
e1
x1
yl ¡ 1
e2
x 2 ; x 3 ; ¢¢¢; x n ¡
0 yr
e3
1
e03
xn
x 0n
1
1+ ®
44
A 4-approx. randomized mechanism for 2-facility game
Mechanism. Choose i 2 f 1; 2; ¢¢¢; ng by random, then
choose j 2 f 1; 2; ¢¢¢; ngwith probability
jx i ¡ x j j
Pn
j 0= 1 jx i ¡ x j 0 j
set two facilities at x i ; x j
Truthfulness: only need to prove the following 2facility mechanism is truthful
 Set one facility at c, and the other facility at x i
with probability
P
jc ¡ x i j
n
j = 1 jc ¡ x j j
45
Proof of truthfulness
Truthfulness: only need to prove the following 2facility mechanism is truthful
 Set one facility at c, and the other facility at x i
with probability jc ¡ x i j
P
n
j = 1 jc ¡
Proof. For player i ,
P
j6
=i
cost =
0
cost =
j6
=i
jx i ¡ x 0ij ¸ jb¡ b0j
S
minf jx j ¡ x i j; jc ¡ x i jgjx j ¡ cj
S
P
=
A+ b
j6
= i jx j ¡ cj + jx i ¡ cj
A
when misreporting to x 0i,
P
S · Ab
xj j
b
S
b
b’
minf jx j ¡ x i j; jc ¡ x i jgjx j ¡ cj + minf jx i ¡ cj; jx i ¡ x 0ijgjx 0i ¡ cj
P
0
j6
= i jx j ¡ cj + jx i ¡ cj
S + minf b; jx i ¡
=
A + b0
A
x 0ijgb0
0
b’
S + minf b; jb¡ b jgb0
¸
A + b0
46
Proof of truthfulness (cont’d)
Truthfulness: only need to prove the following 2facility mechanism is truthful
 Set one facility at c, and the other facility at x i
with probability jc ¡ x i j
P
Proof.
n
j = 1 jc ¡
xj j
S · Ab
jx i ¡ x 0ij ¸ jb¡ b0j
S + minf b; jb¡ b0jgb0
S
cost ¡ cost ¸
¡
A + b0 ³
A+ b
´
1
0
0
0
=
minf b; jb¡ b jgb (A + b) ¡ S(b ¡ b)
(A + b0)(A + b)
³
´
(assume b0 ¸ b)
1
0
0
0
¸
minf b; jb¡ b jgb (A + b) ¡ Ab(b ¡ b)
(A + b0)(A + b)
³
´
1
0
0
bb (A + b) ¡ Ab(b ¡ b) ¸ 0
½ (when b < jb¡ b0j) =
(A + b0)(A + b)
³
´
1
0
0
0
0
(when b · jb¡ b j) =
(b ¡ b)b (A + b) ¡ Ab(b ¡ b) ¸ 0
(A + b0)(A + b)
0
47
Approximation ratio
Claim. The mechanism approximates the optimal
social cost within a factor of 4.
Intuition
 When locations are “sparse”, opt is also bad
x1
x2
¢¢¢
¢¢¢
¢¢¢
xn
 When locations fall into two groups, opt is small,
but Mechanism behaves very similar to opt
x 1 ; x 2 ; ¢¢¢; x n =2
x n =2+ 1 ; x n =2+ 2 ; ¢¢¢; x n
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Open problems
Characterization
 Deterministic 2-facility game?
 Randomized 1-facility game?
Approximation
 Still some gaps…
 Randomized 3-facility game?
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Thank you!
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