1 truthful mechanism for facility allocation: a characterization and improvement of approximation...
TRANSCRIPT
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Truthful Mechanism for Facility Allocation:
A Characterization and Improvement of Approximation Ratio
Pinyan Lu, MSR AsiaYajun Wang, MSR Asia
Yuan Zhou, Carnegie Mellon University
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Problem discussed
Design a mechanism for the following n-player gamePlayers is located on a real lineEach player report their location to the
mechanismThe mechanism decides a new location to build
the facility
x1 x2
mechanism g
y
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Problem discussed (cont’d)
Design a mechanism for the following n-player gamePlayers is located on a real lineEach player report their location to the
mechanismThe mechanism decides a new location to build
the facilityFor example, the mean func.,
mechanism
g= (x1 + x2)=2
g= (x1 + x2)=2
x1 x2y
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Problem discussed (cont’d)
Design a mechanism for the following n-player gamePlayers is located on a real lineEach player report their location to the
mechanismThe mechanism decides a new location to build
the facilityFor example, the mean func., This encourages Player 1 to report
, then becomes closer to Player 1’s real location.mechanism
g= (x1 + x2)=2
g= (x1 + x2)=2
x1 x2yx01
x01 = 2x1 ¡ x2
g(x01;x2) = x1
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Truthfulness
Design a mechanism for the following n-player gamePlayers is located on a real lineEach player report their location to the
mechanismThe mechanism decides a new location to build
the facilityTruthful mechanism does not encourage player to report untruthful locations mechanism
x1 x2
g(x1;x2) = x1 g(x1;x2) = 0
g(x1;x2) = minfx1;x2g
g= x1
6
x02
Truthfulness of
Suppose w.l.o.g. that has no incentive to lie will not change the outcome of if it misreports a value If misreports that , then the decision of will be even farther from
g(x1;x2) = minfx1;x2g
x1
g= minfx1;x2g
x1 · x2
x1
x2
x02 ¸ x1
g
x2 x02 < x1
x2
g
x2
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Truthfulness of
Suppose w.l.o.g. that has no incentive to lie will not change the outcome of if it misreports a value If misreports that , then the decision of will be even farther from
Corollary: a mechanism which outputs the leftmost (rightmost) location among players is truthful
g(x1;x2) = minfx1;x2g
x02
g0= minfx1;x02g
x1 · x2
x1
x2
x02 ¸ x1
g
x2 x02 < x1
x2
g
x2
g= minfx1;x2g
x1
n
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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. for some
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 …
g= f (xi )i 2 [n]
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g0k
x0i
Even more interesting truthful mechanisms
Suppose w.l.o.g. that has no incentive to lie can change the outcome only when it lies to be where and are on different sides of , but this makes the new outcome farther from
Corollary: outputting the median ( ) is truthful
gk(x1;x2;¢¢¢;xn) = k-th left location among inputs Mechanism:
x1 · x2 · x3 · ¢¢¢· xn
xk
xi (i 6= k)x0
i x0i
xkxi
xi
x1 xi xk xn
gk
g[n=2]
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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
Approximation ratio of mechanism
g
©(g) := maxx1 ;x2 ;¢¢¢;xn
½scx1 ;x2 ;¢¢¢;xn (g)
OPT(x1;x2;¢¢¢;xn)
¾
scx1 ;x2 ;¢¢¢;xn (g) :=nX
i=1
jxi ¡ g(x1;x2;¢¢¢;xn)j
©(median) = 1
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Approximation ratio of other mechanisms
Gap instance:
Gap instance:
©(g´ 0) = +1
0 x1;x2;¢¢¢;xn
©(outputting the leftmost player's location) = n ¡ 1
x1 x2;x3;¢¢¢;xn
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Extend to two facility game
Suppose we have more budget, and we can afford building two facilitiesEach player’s cost function: its distance to the closest facility
Good truthful approximation?
A simple tryMechanism: set facilities on the leftmost
and rightmost player’s location
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Extend to two facility game
A simple tryMechanism: set facilities on the leftmost
and rightmost player’s locationGap Instance:
x1 x2;x3;¢¢¢;xn¡ 1 xn
¡ 1 0 1
©¸ n ¡ 2
OPT = 1
sc(Mech.) = n ¡ 2©· n ¡ 2
¾) ©= n ¡ 2
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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?
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Multiple locations per agent
Agent controls locations Agent ‘s cost function is
Social cost:
A randomized truthful mechanismGiven , return with
probability Claim. The mechanism is truthfulTheorem. The mechanism’s approximation ratio is
i wi xi = (xi1;xi2;¢¢¢;xiwi )
iP wi
j =1 jg¡ xi j j
P ni=1
P wij =1 jg¡ xi j j
f x1;x2;¢¢¢;xng med(xi )wi =
P nj =1 wj
3¡2min1· j · n wjP n
j =1 wj
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Summary of questions.
Characterization Is there a full characterization for
deterministic truthful mechanism in one-facility game?
ApproximationUpper/lower bound for two facility game in
deterministic/randomized case?Lower bound for one facility game in
randomized case when agents control multiple locations?
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Our result and related work
Give a full characterization of one-facility deterministic truthful mechanismsSimilar result by [Moulin] and [Barbera-
Jackson]
Improve the bounds approximation ratio in several extended game settings
*: Most of previous results are due to [Procaccia-Tennenholtz]
**: In this setting, each player can control multiple locations
Setting one facility deterministic
two facilities deterministic
two facilities randomized
one facility, randomized**
Previous known*
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
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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
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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)
x1 x2
x3 c1
x1
min
max
min
max
c1
x2
x3
x1 c2
med
c3 x1 c4
med
c5 x1 c6
med
c7 x1 c8
med
x2
med
med
medstandard form
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More precise in the characterization
The image set of the mechanism can be an arbitrary closed setWe restrict the min-max function onto by finding the nearest point in
UU
U
closed set U
x1 x2
x3 c1
x1
min
max
min
max
f :
mechanism g
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mechanism g
More precise in the characterization
The image set of the mechanism can be an arbitrary closed setWe restrict the min-max function onto by finding the nearest point in
UU
U
closed set U
x1 x2
x3 c1
x1
min
max
min
max
f :
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mechanism g
More precise in the characterization
The image set of the mechanism can be an arbitrary closed setWe restrict the min-max function onto by finding the nearest point inWhat about when there are 2 nearest points ?A tie-breaking gadget takes response of that !
UU
U
x1 x2
x3 c1
x1
min
max
min
max
f :
closed set U
tie breaker
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The proof – warm-up part
Lemma. If is a truthful mechanism, then goes to the closest point in from , for allProof. For every ,
Corollary. is closed.
Now, for simplicity, assume
g g(x;x;¢¢¢;x)I (g) x x
Image set of g
y = g(x1;x2;¢¢¢;xn)jy ¡ xj = jg(x1;x2;¢¢¢;xn) ¡ xj
¸ jg(x;x2;¢¢¢;xn) ¡ xj
¸ jg(x;x;¢¢¢;xn) ¡ xj
¢¢¢
¸ jg(x;x;¢¢¢;x) ¡ xj
U = I (g)
U = I (g) = (¡ 1 ;+1 )
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Main lemma
Lemma. For each truthful mechanism , there exists a min-max function , such that is the closest point in from , for all inputsProof (sketch). Prove by induction onWhen , should output the closest
point in from : For
g
f (x) = x
g(x1;x2;¢¢¢;xn)I (g) f (x1;x2;¢¢¢;xn)
x1;x2;¢¢¢;xn 2 Rn
n = 1 g(x)xI (g)
n > 1
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Main lemma
For , define Claim 1. is truthfulClaim 2.
Claim 3. , as mechanisms for -player game, are truthful
Claim 4.
n > 1 g0x1;x2 ;¢¢¢;xn ¡ 1
(xn) = g(x1;x2;¢¢¢;xn)
g0
9L = L(x1;¢¢¢;xn¡ 1);R = R(x1;¢¢¢;xn¡ 1);
s:t: I (g0x1 ;x2 ;¢¢¢;xn ¡ 1
) = I (g) \ [L ;R]
L;R (n ¡ 1)
9L l ;L r : I (L) = I (g) \ [L l ;L r ];
9R l ;Rr : I (R) = I (g) \ [R l ;Rr ]
I (g0x1 ;x2 ;¢¢¢;xn ¡ 1
)
L(x1;x2;¢¢¢;xn¡ 1) R(x1;x2;¢¢¢;xn¡ 1)
I (L )
L l L r
I (R)
R l Rr
I (g)
I (g)
I (g)
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med
L Rxn
Main lemma
Thus,g(x1;x2;¢¢¢;xn) = g0
x1 ;x2 ;¢¢¢;xn ¡ 1= med(L;xn ;R)
L = med(L l ;g1(x1;x2;¢¢¢;xn¡ 1);L r )R = med(R l ;g2(x1;x2;¢¢¢;xn¡ 1);Rr )
g :
I (g0x1 ;x2 ;¢¢¢;xn ¡ 1
)
L(x1;x2;¢¢¢;xn¡ 1) R(x1;x2;¢¢¢;xn¡ 1)
I (L )
L l L r
I (R)
R l Rr
I (g)
I (g)
I (g)
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Main lemma
Thus,g(x1;x2;¢¢¢;xn) = g0
x1 ;x2 ;¢¢¢;xn ¡ 1= med(L;xn ;R)
L = med(L l ;g1(x1;x2;¢¢¢;xn¡ 1);L r )R = med(R l ;g2(x1;x2;¢¢¢;xn¡ 1);Rr )
L l L r R l Rr
med
xnmed med
g :
g1 g2
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Main lemma
L l L r R l Rr
med
xnmed med
g :
g1 g2
1 player:
g(1) : x1
2 players:
g(2) :
c1 c2 c3 c4
med
x2med med
g(1)1 g(1)
2
L l L r R l Rr
med
xnmed med
g1 g2
x1x1
g(2) :
c1 c2 c3 c4
med
x2med med
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Main lemma
L l L r R l Rr
med
xnmed med
g :
g1 g2
1 player:
g(1) : x1x1x1
2 players:
g(2) :
c1 c2 c3 c4
med
x2med med
x1 x1
3 players:
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c1 c2 c3 c4
med
x2med med
x1 x1 c5 c6 c7 c8
med
x2med med
x1 x1
Main lemma
med
med medx3
c9 c10 c11 c12
g(3) :
g(2)1 g(2)
2
1 player:
g(1) : x1x1x1
2 players:
g(2) :
c1 c2 c3 c4
med
x2med med
x1 x1
3 players:
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c1 c2 c3 c4
med
x2med med
x1 x1 c5 c6 c7 c8
med
x2med med
x1 x1
Main lemma
med
med medx3
c9 c10 c11 c12
g(3) :
1 player:
g(1) : x1x1x1
2 players:
g(2) :
c1 c2 c3 c4
med
x2med med
x1 x1
3 players:
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The reverse direction
Lemma. Every min-max function is truthfulObservation. To prove a -player
mechanism is truthful, only need to prove the -player mechanisms are truthful for every and
Theorem. The characterization is full
n1
g0x1 ;¢¢¢;xi ¡ 1 ;xi + 1 ;¢¢¢;xn
(xi )i xj (j 6= i)
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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.
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Multiple locations per agent (cont’d)
Proof. (weaker version)
0 1
0 1
0 1
Instance 1
Instance 2
Instance 3
Player 1Player 2
g = D1
g = D2
g = D3
L2 = Ex2D 2 [jx ¡ 0j ¢1x· 0] R2 = Ex2D 2 [jx ¡ 1j ¢1x¸ 1]
For Player 1 at Instance 1 (compared to Instance 2)
2¢jx ¡ 0jx2D 1 + jx ¡ 1jx2D 1 · 2¢jx ¡ 0jx2D 2 + jx ¡ 1jx2D 2
jx ¡ 0jx2D 1 · jx ¡ 0jx2D 2 + 2(L2 + R2))For Player 2 at Instance 3 (compared to Instance 2)
2¢jx ¡ 1jx2D 3 + jx ¡ 0jx2D 3 · 2¢jx ¡ 1jx2D 2 + jx ¡ 0jx2D 2
jx ¡ 1jx2D 3 · jx ¡ 1jx2D 2 + 2(L2 + R2))
For Player 1
For Player 2
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Multiple locations per agent (cont’d)
Proof. (weaker version)
0 1
0 1
0 1
Instance 1
Instance 2
Instance 3
Player 1Player 2
g = D1
g = D2
g = D3
L2 = Ex2D 2 [jx ¡ 0j ¢1x· 0] R2 = Ex2D 2 [jx ¡ 1j ¢1x¸ 1]
For Player 1 jx ¡ 0jx2D 1 · jx ¡ 0jx2D 2 + 2(L2 + R2)
For Player 2 jx ¡ 1jx2D 3 · jx ¡ 1jx2D 2 + 2(L2 + R2)
Assume <1.2 approx.
2¢jx ¡ 0jx2D 1 + 4¢jx ¡ 1jx2D 1 < 1:2¢2
4¢jx ¡ 0jx2D 3 + 2¢jx ¡ 1jx2D 3 < 1:2¢2
3¢jx ¡ 0jx2D 2 + 3¢jx ¡ 1jx2D 2 < 1:2¢3For Inst. 1
For Inst. 2
For Inst. 3
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Multiple locations per agent (cont’d)
Proof. (weaker version)
0 1
0 1
0 1
Instance 1
Instance 2
Instance 3
Player 1Player 2
g = D1
g = D2
g = D3
L2 = Ex2D 2 [jx ¡ 0j ¢1x· 0] R2 = Ex2D 2 [jx ¡ 1j ¢1x¸ 1]
For Player 1 jx ¡ 0jx2D 1 · jx ¡ 0jx2D 2 + 2(L2 + R2)
For Player 2 jx ¡ 1jx2D 3 · jx ¡ 1jx2D 2 + 2(L2 + R2)
Assume <1.2 approx.
jx ¡ 1jx2D 1 < 0:2
jx ¡ 0jx2D 3 < 0:2
) L2 + R2 < 0:1For Inst. 1
For Inst. 2
For Inst. 3
) jx ¡ 0jx2D 1 > 0:8
) jx ¡ 1jx2D 3 > 0:8
jx ¡ 0jx2D 2 + jx ¡ 1jx2D 2 < 1:2
< 1.61.6 <
Contradiction
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Multiple locations per agent (cont’d)
Proof. (stronger version)
0 1
0 1
0 1
Instance 1
Instance 2
Instance 3
Player 1Player 2
0 1
0 1
Instance 4
Instance 5
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Multiple locations per agent (cont’d)
Proof. (stronger version)
0 1
0 1
Instance
Instance
Player 1Player 2
0 1
Instance
i(1 · i · K )
K + 1
2K + 2¡ i(K ¸ i ¸ 1)
£(2K + 1) £(2K + 1)
£(K + i) £(2K + 1)£(K + 1¡ i)
£ (2K + 1)£(K + 1¡ i)
£(K + i)
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Multiple locations per agent (cont’d)
Linear Programming
Take K = 500: ®> 1:33
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Lower bound for 2-facility randomized case
Theorem. For any 2-facility randomized truthful mechanism, the approximation ratio is at least a , where is the number of playersProofConsider instance : player at ,
players at , player at For mechanisms within 2-approx. :Assume w.l.o.g.:
1:045¡1
n ¡ 3n ¸ 30
I 1 ¡ 1 n ¡ 20 1 1
¡ 1 0 1
x1 x2;x3;¢¢¢;xn¡ 1 xn
e1 e2 e3
e1 + e2 + e3 ¸ 1
e2 · 2=(n ¡ 2)e3 ¸ 1=2¡ 1=(n ¡ 2)
yl yr
411+ ®
x0n
Lower bound for 2-facility randomized case
Theorem. For any 2-facility randomized truthful mechanism, the approximation ratio is at least a , where is the number of playersProofConsider instance : player at ,
players at , player at Another instance : player at ,
players at , player at
1:045¡1
n ¡ 3n ¸ 30
I 1 ¡ 1 n ¡ 20 1 1
¡ 1 0 1
x1 x2;x3;¢¢¢;xn¡ 1 xn
e1 e2 e3
e3 ¸ 1=2¡ 1=(n ¡ 2)
I 0 1 ¡ 1 n ¡ 20 1 1+ ®
yl yr
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x0n
1+ ®
Lower bound for 2-facility randomized case
Theorem. For any 2-facility randomized truthful mechanism, the approximation ratio is at least a , where is the number of playersProofConsider instance : player at ,
players at , player at Another instance : player at ,
players at , player at By truthfulness:
1:045¡1
n ¡ 3n ¸ 30
I 1 ¡ 1 n ¡ 20 1 1
¡ 1 0 1
x1 x2;x3;¢¢¢;xn¡ 1 xn
e1 e2 e3
e3 ¸ 1=2¡ 1=(n ¡ 2)
I 0 1 ¡ 1 n ¡ 20 1 1+ ®
e03
e03 ¸ 1=2¡ 1=(n ¡ 2) ¡ ®
yl yr
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x0n
1+ ®
Lower bound for 2-facility randomized case
Theorem. For any 2-facility randomized truthful mechanism, the approximation ratio is at least a , where is the number of playersProof
1:045¡1
n ¡ 3n ¸ 30
¡ 1 0 1
x1 x2;x3;¢¢¢;xn¡ 1 xn
e1 e2 e3 e03
e03 ¸ 1=2¡ 1=(n ¡ 2) ¡ ®
yl yr
sc(I 0) = e1 + (n ¡ 2)e2 + e03 ¸ Pr[yr · ¡
1n ¡ 2
]¢1+ Pr[yr ¸1
n ¡ 2]¢1+ e0
3
¸ 1¡ Pr[¡1
n ¡ 2< yr <
1n ¡ 2
]+12
¡1
n ¡ 2¡ ®
sc(I 0) ¸ Pr[yr · ¡1
n ¡ 2]¢1+ Pr[yr ¸
1n ¡ 2
]¢1+ Pr[¡1
n ¡ 2< yr <
1n ¡ 2
]¢(1+ ®)
= 1+ ®¢Pr[¡1
n ¡ 2< yr <
1n ¡ 2
]
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x0n
1+ ®
Lower bound for 2-facility randomized case
Theorem. For any 2-facility randomized truthful mechanism, the approximation ratio is at least a , where is the number of playersProof
Done.
1:045¡1
n ¡ 3n ¸ 30
¡ 1 0 1
x1 x2;x3;¢¢¢;xn¡ 1 xn
e1 e2 e3 e03
yl yr
sc(I 0) ¸ 1+
p2¡ 1
12¡ 2p
2¡
1n ¡ 2
> 1:045¡1
n ¡ 2
opt(I 0) = 1
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jc¡ xi jP nj =1 jc¡ xj j
A 4-approx. randomized mechanism for 2-facility game
Mechanism. Choose by random, then choose with probability
set two facilities at
Truthfulness: only need to prove the following 2-facility mechanism is truthfulSet one facility at , and the other facility at
with probability
i 2 f1;2;¢¢¢;ng
xi ;xj
jxi ¡ xj jP nj 0=1 jxi ¡ xj 0j
j 2 f1;2;¢¢¢;ng
c xi
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jc¡ xi jP nj =1 jc¡ xj j
Proof of truthfulness
Truthfulness: only need to prove the following 2-facility mechanism is truthfulSet one facility at , and the other facility at
with probability
Proof. For player , when misreporting to ,
c xi
i
cost =
Pj 6=i minf jxj ¡ xi j; jc¡ xi jgjxj ¡ cj
Pj 6=i jxj ¡ cj + jxi ¡ cj
x0i
cost0=
Pj 6=i minf jxj ¡ xi j; jc¡ xi jgjxj ¡ cj + minf jxi ¡ cj; jxi ¡ x0
i jgjx0i ¡ cj
Pj 6=i jxj ¡ cj + jx0
i ¡ cj
=S
A + b
=S + minfb;jxi ¡ x0
i jgb0
A + b0 ¸S + minfb; jb¡ b0jgb0
A + b0
S
S
A
A
b
b b’
b’
S · Ab
jxi ¡ x0i j ¸ jb¡ b0j
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jc¡ xi jP nj =1 jc¡ xj j
Proof of truthfulness (cont’d)
Truthfulness: only need to prove the following 2-facility mechanism is truthfulSet one facility at , and the other facility at
with probability
Proof.
c xi
S · Ab
jxi ¡ x0i j ¸ jb¡ b0j
cost0¡ cost ¸S + minfb; jb¡ b0jgb0
A + b0 ¡S
A + b
=1
(A + b0)(A + b)
³minfb; jb¡ b0jgb0(A + b) ¡ S(b0¡ b)
´
(assume b0¸ b)¸
1(A + b0)(A + b)
³minfb; jb¡ b0jgb0(A + b) ¡ Ab(b0¡ b)
´
(when b< jb¡ b0j) =1
(A + b0)(A + b)
³bb0(A + b) ¡ Ab(b0¡ b)
´¸ 0
=1
(A + b0)(A + b)
³(b0¡ b)b0(A + b) ¡ Ab(b0¡ b)
´¸ 0(when b· jb¡ b0j)
½
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Approximation ratio
Claim. The mechanism approximates the optimal social cost within a factor of 4.
IntuitionWhen locations are “sparse”, opt is also bad
When locations fall into two groups, opt is small, but Mechanism behaves very similar to opt
x1 xnx2 ¢¢¢ ¢¢¢ ¢¢¢
x1;x2;¢¢¢;xn=2 xn=2+1;xn=2+2;¢¢¢;xn
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Open problems
CharacterizationDeterministic 2-facility game?Randomized 1-facility game?
ApproximationStill some gaps…Randomized 3-facility game?
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Thank you!