1 Truthful Mechanism for Facility Allocation: A Characterization and Improvement of Approximation Ratio Pinyan Lu, MSR Asia Yajun Wang, MSR Asia Yuan Zhou,

1

Truthful Mechanism for Facility Allocation:

A Characterization and Improvement of Approximation Ratio

Pinyan Lu, MSR AsiaYajun Wang, MSR Asia

Yuan Zhou, Carnegie Mellon University

2

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

3

Problem discussed (cont’d)



the facilityFor example, the mean func.,

mechanism

g= (x1 + x2)=2

g= (x1 + x2)=2

x1 x2y

4

Problem discussed (cont’d)



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

5

Truthfulness



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


x1

g= minfx1;x2g

x1 · x2

x1

x2

x02 ¸ x1

g

x2 x02 < x1

x2

g

x2

7

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


x02

g0= minfx1;x02g

x1 · x2

x1

x2

x02 ¸ x1

g

x2 x02 < x1

x2

g

x2

g= minfx1;x2g

x1

n

8

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]

9

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]

10

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

12

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

14

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?

15

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


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 :

22

mechanism g


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

35



0 1

0 1

0 1

Instance 1

Instance 2

Instance 3

Player 1Player 2

g = D1

g = D2

g = D3


For Player 1 jx ¡ 0jx2D 1 · jx ¡ 0jx2D 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

36



0 1

0 1

0 1

Instance 1

Instance 2

Instance 3

Player 1Player 2

g = D1

g = D2

g = D3




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

37


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

38


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)

39


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



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

42

x0n

1+ ®



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

43

x0n

1+ ®


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

]

44

x0n

1+ ®


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

45

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

46


Proof of truthfulness


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

47


Proof of truthfulness (cont’d)


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

48

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

49

Open problems

CharacterizationDeterministic 2-facility game?Randomized 1-facility game?

ApproximationStill some gaps…Randomized 3-facility game?

50

Thank you!

Documents

1 Truthful Mechanism for Facility Allocation: A Characterization and Improvement of Approximation Ratio Pinyan Lu, MSR Asia Yajun Wang, MSR Asia Yuan Zhou,