Pinyan Lu, MSR Asia Yajun Wang, MSR Asia

Truthful Mechanism for Facility Allocation:

A Characterization and Improvement of Approximation Ratio

Pinyan Lu, MSR AsiaYajun Wang, MSR Asia

Yuan Zhou, Carnegie Mellon University

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

mechanism g

Problem discussed (cont’d)

the facilityFor example, the mean func.,

mechanism

g= (x1 + x2)=2

x1 x2y

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

x1 x2yx01

x01 = 2x1 ¡ x2

g(x01;x2) = x1

Truthfulness

the facilityTruthful mechanism does not encourage player to report untruthful locations mechanism

g(x1;x2) = x1 g(x1;x2) = 0g(x1;x2) = minfx1;x2g

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

g= minfx1;x2g

x1 · x2x1x2

x02 ¸ x1

x2 x02 < x1

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

g0= minfx1;x02g

x1 · x2x1x2

x02 ¸ x1

x2 x02 < x1

g= minfx1;x2g

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]

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 · ¢¢¢· xnxkxi (i 6= k)

x0i x0

i xkxixi

x1 xi xk xn

g[n=2]

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) := max

x1 ;x2 ;¢¢¢;xn

½ scx1 ;x2 ;¢¢¢;xn (g)OPT(x1;x2;¢¢¢;xn)

scx1 ;x2 ;¢¢¢;xn (g) :=nX

i=1jxi ¡ g(x1;x2;¢¢¢;xn)j

©(median) = 1

Approximation ratio of other mechanisms

Gap instance:

©(g´ 0) = +1

0 x1;x2;¢¢¢;xn

©(outputting the leftmost player's location) = n ¡ 1

x1 x2;x3;¢¢¢;xn

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

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

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?

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 )i P 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 nj =1 wj

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?

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

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

The characterizationGenerally 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)

c3 x1 c4

c5 x1 c6

c7 x1 c8

medstandard form

More precise in the characterizationThe image set of the mechanism can be an arbitrary closed setWe restrict the min-max function onto by finding the nearest point in

closed set U

f :mechanism g

mechanism g

More precise in the characterizationThe image set of the mechanism can be an arbitrary closed setWe restrict the min-max function onto by finding the nearest point in

closed set U

mechanism g

More precise in the characterizationThe 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 !

closed set U

tie breaker

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 )

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

f (x) = x

g(x1;x2;¢¢¢;xn)I (g) f (x1;x2;¢¢¢;xn)

x1;x2;¢¢¢;xn 2 Rn

n = 1 g(x)xI (g)

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)

9L = L(x1;¢¢¢;xn¡ 1);R = R(x1;¢¢¢;xn¡ 1);s:t: I (g0

x1 ;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

R l Rr

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 )

I (g0x1 ;x2 ;¢¢¢;xn ¡ 1 )

L(x1;x2;¢¢¢;xn¡ 1) R(x1;x2;¢¢¢;xn¡ 1)I (L )

L l L r

R l Rr

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

xnmed med

Main lemma

L l L r R l Rr

xnmed med

1 player:

g(1) : x1

2 players:g(2) :

c1 c2 c3 c4

x2med med

g(1)1 g(1)

L l L r R l Rr

xnmed med

g(2) :

c1 c2 c3 c4

x2med med

Main lemma

L l L r R l Rr

xnmed med

1 player:

g(1) : x1x1x1

2 players:g(2) :

c1 c2 c3 c4

x2med med

3 players:

c1 c2 c3 c4

x2med med

x1 x1 c5 c6 c7 c8

x2med med

Main lemma

med medx3

c9 c10 c11 c12

g(3) :

g(2)1 g(2)

1 player:

g(1) : x1x1x1

2 players:g(2) :

c1 c2 c3 c4

x2med med

3 players:

c1 c2 c3 c4

x2med med

x1 x1 c5 c6 c7 c8

x2med med

Main lemma

med medx3

c9 c10 c11 c12

g(3) :

1 player:

g(1) : x1x1x1

2 players:g(2) :

c1 c2 c3 c4

x2med med

3 players:

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

g0x1;¢¢¢;xi ¡ 1 ;xi + 1 ;¢¢¢;xn (xi )

i xj (j 6= i)

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.

Multiple locations per agent (cont’d)Proof. (weaker version)

Instance 1

Instance 2

Instance 3

Player 1Player 2g = 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

Instance 1

Instance 2

Instance 3

g = D2

g = D3

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¢23¢jx ¡ 0jx2D 2 + 3¢jx ¡ 1jx2D 2 < 1:2¢3

For Inst. 1For Inst. 2For Inst. 3

Instance 1

Instance 2

Instance 3

g = D2

g = D3

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 < 0:1

For Inst. 1For Inst. 2For Inst. 3

) jx ¡ 0jx2D 1 > 0:8

) jx ¡ 1jx2D 3 > 0:8jx ¡ 0jx2D 2 + jx ¡ 1jx2D 2 < 1:2

< 1.61.6 <Contradiction

Multiple locations per agent (cont’d)Proof. (stronger version)

Instance 1

Instance 2

Instance 3

Player 1Player 2

Instance 4

Instance 5

Multiple locations per agent (cont’d)Proof. (stronger version)

Instance

Player 1Player 2

Instance

i(1 · i · K )

2K + 2¡ i(K ¸ i ¸ 1)

£(2K + 1) £(2K + 1)

£(K + i) £(2K + 1)£ (K + 1¡ i)

£ (2K + 1)£ (K + 1¡ i)

£(K + i)

Multiple locations per agent (cont’d)Linear Programming

Take K = 500: ®> 1:33

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¡ 1n ¡ 3 n ¸ 30

I 1 ¡ 1 n ¡ 20 1 1

¡ 1 0 1x1 x2;x3;¢¢¢;xn¡ 1 xn

e1 e2 e3

e1 + e2 + e3 ¸ 1e2 · 2=(n ¡ 2)

e3 ¸ 1=2¡ 1=(n ¡ 2)

411+ ®x0

players at , player at Another instance : player at ,

players at , player at

1:045¡ 1n ¡ 3 n ¸ 30

I 1 ¡ 1 n ¡ 20 1 1

¡ 1 0 1x1 x2;x3;¢¢¢;xn¡ 1 xn

e1 e2 e3

e3 ¸ 1=2¡ 1=(n ¡ 2)

I 0 1 ¡ 1 n ¡ 20 1 1+ ®

players at , player at Another instance : player at ,

players at , player at By truthfulness:

1:045¡ 1n ¡ 3 n ¸ 30

I 1 ¡ 1 n ¡ 20 1 1

¡ 1 0 1x1 x2;x3;¢¢¢;xn¡ 1 xn

e1 e2 e3

e3 ¸ 1=2¡ 1=(n ¡ 2)

I 0 1 ¡ 1 n ¡ 20 1 1+ ®

e03 ¸ 1=2¡ 1=(n ¡ 2) ¡ ®

Theorem. For any 2-facility randomized truthful mechanism, the approximation ratio is at least a , where is the number of playersProof

1:045¡ 1n ¡ 3 n ¸ 30

¡ 1 0 1x1 x2;x3;¢¢¢;xn¡ 1 xn

e1 e2 e3 e03

e03 ¸ 1=2¡ 1=(n ¡ 2) ¡ ®

sc(I 0) = e1 + (n ¡ 2)e2 + e03 ¸ Pr[yr · ¡ 1

n ¡ 2]¢1+ Pr[yr ¸ 1n ¡ 2]¢1+ e0

¸ 1¡ Pr[¡ 1n ¡ 2 < yr < 1

n ¡ 2]+ 12 ¡ 1

n ¡ 2 ¡ ®

sc(I 0) ¸ Pr[yr · ¡ 1n ¡ 2]¢1+ Pr[yr ¸ 1

n ¡ 2]¢1+ Pr[¡ 1n ¡ 2 < yr < 1

n ¡ 2]¢(1+ ®)

= 1+®¢Pr[¡ 1n ¡ 2 < yr < 1

n ¡ 2]

Theorem. For any 2-facility randomized truthful mechanism, the approximation ratio is at least a , where is the number of playersProof

1:045¡ 1n ¡ 3 n ¸ 30

¡ 1 0 1x1 x2;x3;¢¢¢;xn¡ 1 xn

e1 e2 e3 e03

sc(I 0) ¸ 1+p 2¡ 1

12¡ 2p 2 ¡ 1n ¡ 2 > 1:045¡ 1

n ¡ 2opt(I 0) = 1

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

Proof of truthfulness

with probability

Proof. For player , when misreporting to ,

cost =P

j 6=i minf jxj ¡ xi j; jc¡ xi jgjxj ¡ cjPj 6=i jxj ¡ cj + jxi ¡ cj

cost0=P

j 6=i minf jxj ¡ xi j; jc¡ xi jgjxj ¡ cj + minf jxi ¡ cj; jxi ¡ x0i jgjx0

i ¡ cjPj 6=i jxj ¡ cj + jx0

i ¡ cj

= SA + b

= S + minfb;jxi ¡ x0i jgb0

A + b0 ¸ S + minfb; jb¡ b0jgb0

A + b0

bb b’

S · Abjxi ¡ x0

i j ¸ jb¡ b0j

Proof of truthfulness (cont’d)

with probability

Proof.

S · Abjxi ¡ x0

i j ¸ jb¡ b0j

cost0¡ cost ¸ S + minfb; jb¡ b0jgb0

A + b0 ¡ SA + 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)

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

Open problems

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

ApproximationStill some gaps…Randomized 3-facility game?

Thank you!

Pinyan Lu, MSR Asia Yajun Wang, MSR Asia

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