Black-Box Methods for Cost-Sharing Mechanism

Design

Chaitanya SwamyUniversity of Waterloo

Joint work with Konstantinos Georgiou

University of Waterloo

Cost-sharing mechanism design

A service provider has to decide which players to serve

•Provider incurs a publicly-known cost for serving a set of players

•Each payer has a private value for receiving the service

•Players are “selfish” only care about maximizing their utility and will “lie” if that increases their utility

Cost-sharing mechanism design

A service provider has to decide which players to serve•Provider incurs a publicly-known cost for serving a set of

players

•Each payer has a private value for receiving the service

•Players are “selfish” only care about maximizing their utility and will “lie” if that increases their utility

Cannot assume service provider knows the true private values, but we seek solution that is “good” with respect to true input

Cost-sharing mechanism = algorithm to compute a “good” solution

+ prices that induce players to act truthfully AND recover the cost incurred by the provider

Cost-sharing mechanism-design

Formally, C(S) = cost-incurred for serving set S of players

(assume C()=0, C(S) ≤ C(T) if S⊆T)

vi = private value/type of player i

Cost-sharing mechanism M = (f, {pi}i=1…n)

• Players report types t,…,tn (let t=(t,…,tn))

• Mechanism computes solution S = f(t) ⊆ {,…,n}, and charges price pi(t) to each player i (usually 0 if iS)

• utility of i = vi(S) – pi(t) (where vi(A) = vi if iA else 0)

algorithm

price charged to player i

Algorithms to compute solns. with near-opt. social welfare (i.e., approx. algorithms for social-cost-minimization)

Truthful mechanisms

Truthful, cost-recovering mechanisms

Goals and Objectives

Want mechanisms lying here: seek to understand how the 3 objectives interact/conflict

Goals and Objectives

Want to design M such that:a) M is truthful (strategyproof) – every i maximizes its utility by reporting its true value regardless of other players' bids b) S has “good” social welfare – quantify using social-cost objective [Roughgarden-Sundararajan]:

SC(t, S) := C (S) + ∑iS ti ≤ minA SC(t, A)

c) Cost-recovery – prices recover the cost: ∑i pi(t) ≥ C (S)

Recall: M = (f, {pi}), t=(t1,…,tn): reported input, S = f(t)

Impossible to satisfy a) – c); C(.) is often NP-hard to compute, so we relax conditions suitably

Goals and Objectives

Want to design M such that:a) M is truthful (strategyproof) – every i maximizes its utility by reporting its true value regardless of other players' bids b) S has “good” social welfare – quantify using social-cost objective [Roughgarden-Sundararajan (RS09)]:

SC(t, S) := CM(S) + ∑iS ti ≤ minA SC(t, A)

c) Cost-recovery – prices recover the cost: ∑i pi(t) ≥ CM(S) /

Recall: M = (f, {pi}), t=(t1,…,tn): reported input, S = f(t)

Impossible to satisfy a) – c); also C(.) often NP-hard to compute, so we relax conditions suitably(Note: VCG satifies a), b), but gives poor revenue;

Moulin mechanisms (in general) satisfy a), c) but sacrifice b)

Truthful, -approximation, -cost-recovering mechanism

Cost of solution computed by M for S

Algorithms to compute solns. with near-opt. social welfare (i.e., approx. algorithms for social-cost-minimization)

Truthful mechanisms

Truthful, cost-recovering mechanismsVCG

Moulin mechanisms

Goals and Objectives

Want mechanisms lying here: seek to understand how the 3 objectives interact/conflict

Example: Steiner-tree cost-sharing

C(S) = opt. Steiner tree cost on S∪{r}

Social-cost-minimization (SCM) problem ≣ prize-collecting Steiner tree

A truthful, -approximation, -cost-recovering mechanism outputs:• -approx. solution to SCM problem• prices that recover -fraction of cost of output tree

: Terminals ≣ Players: Root r: Node

Three types of objects

(C) Approximation algorithms

for SCM problem

(B) Truthful,approximation mechanisms

(A) Truthful, approximation, cost-

recovering mechanisms

Very limited understanding:most results rely on constructing cost-shares with suitable properties, which can be very challenging (or impossible!) constructions are quite problem-specific and often rather intricateBetter understanding: nice characterization (for D problems) of truthful mechanisms, allows one to leverage algorithmic techniquesGood understanding: numerous techniques: LP rounding, primal-dual, ...

Are there reductions b/w A, B, C?

(C) Approximation

algorithms for SCM problem

(B) Truthful,

approximation

mechanisms

(A) Truthful, approximati

on, cost-recovering

mechanisms

Are there reductions b/w A, B, C?

Our work: YES! (C) Approximation

algorithms for SCM problem

(B) Truthful,

approximation

mechanisms

(A) Truthful, approximati

on, cost-recovering

mechanisms

Reduction (): B ➞ AInput: truthful, -approximation, no-bossy mechanismOutput: truthful, O(.log n)-approx., cost-recovering mechanism can inject cost-recovery into any no-bossy mechanism (B)

()

i f(t), i f(t'i, t-i) f(t)= f(t'i, t-i)

Are there reductions b/w A, B, C?

Our work: YES! (C) Approximation

algorithms for SCM problem

(B) Truthful,

approximation

mechanisms

(A) Truthful, approximati

on, cost-recovering

mechanisms

Reduction (): B ➞ A (works for any cost f'n.)Input: truthful, -approximation, no-bossy mechanismOutput: truthful, O(.log n)-approx., cost-recovering mechanism can inject cost-recovery into any no-bossy mechanism (B)– First reduction for general costs (subadditive C(.): Bleischwitz et al.)– log n factor matches the lower bound of Dobzinski et al. (D+08)

()

i f(t), i f(t'i, t-i) f(t)= f(t'i, t-i)

Approximation algorithms for social-cost-minimization

Truthful mechanisms

Truthful, cost-recovering mechanismsVCG

Truthful, no-bossy mechanisms

Nice application: taking input = VCG, get that for every cost-f'n., there is a truthful, O(log n)-approx., cost-recovering mechanism

Are there reductions b/w A, B, C?

Our work: YES! (C) Approximation

algorithms for SCM problem

(B) Truthful,

approximation

mechanisms

(A) Truthful, approximati

on, cost-recovering

mechanisms

Reduction (): B ➞ A (works for any cost-f'n.)Input: truthful, -approximation, no-bossy mechanismOutput: truthful, O(.log n)-approx., cost-recovering mechanism

Reduction (2): C ➞ BInput: LP-relative -approx. algorithm for cost-minimization (CM) problem (find a min-cost solution for a given set of players)Output: truthful, ()-approximation, no bossy mechanism Works whenever LP-relaxation of CM problem is “covering like”

() (2)

Approximation algorithms for social-cost-minimization

Truthful mechanisms

Truthful, cost-recovering mechanisms

Truthful, no-bossy mechanisms

+

So for a rich class of problems, can convert any LP-relative -approximation algorithm for CM problem to truthful, O(.log n)-approx., cost-recovering mechanism

Reductions find numerous applications. • First guarantees for:

– {edge, vertex, element}-disjoint survivable network design: C(S) = cost of connecting set S of (si, ti) pairs (allow edges with multiplicity)

– makespan minimization on unrelated machines: C(S) = makespan for scheduling set S of jobs

– soft-capacitated facility location (FL): C(S) = cost of serving set S of clients

• Improved guarantees (approx. improves to O(log n)) for:– Steiner {tree, forest}– multicommodity connected FL

• For many problems, D+08 gives matching log n lower bound

Previous work gives stronger notions of truthfulness: group-strategyproofness (GSP) and its variants

Two departures from earlier work

• Focus on truthfulness, so we are not considering the effect of coalitions

• Do not impose any upper bound on revenue (like ∑i pi(t) ≤ C(S)): – usual rationale for upper bound: otherwise

players in S may collude and secede from the mechanism

– We do not consider coalitions, so do not impose this; instead we project this condition to individual players and considerIndividual Competitiveness (ICT): pi(t) ≤ C({i}) i

• Makes sense to require ICT for subadditive C(.), in which case our constructions do ensure ICT

Related Work• Moulin and Moulin-Shenker introduced Moulin

mechanisms – show that cost-shares having certain properties yield GSP, cost-recovering mechanisms

• Roughgarden-Sundararajan (RS09) introduced social-cost objective, identified another property of cost-shares which yields good approximation for Moulin mechanisms

• Lots of work on devising suitable cost shares for various problems – methods are problem-specific and often intricate

• Immorlica et al. exposed an inherent limitation of this approach – designing suitable cost shares may be impossible

• Mehta et al. modified Moulin mechanisms – require weaker properties of cost-shares and yield weakly-GSP mechanisms

Related Work (contd.)

Bleischwitz et al. (B+07), Brenner-Schafer propose some black-box reductions converting algorithms (C) to cost-recovering mechanisms (A)– both results require various conditions on the approximation algorithm and cost f'n., which seem

much more restrictive (and slightly unnatural) compared to our condition of LP-relative approx.– B+07 also give a O(log n)-approx., cost-recovering weakly GSP mechanism for any subadditive C(.)

Some ingredients of our results

Useful characterization of truthful mechanisms (Myerson)An algorithm f is monotone if

i f(z, t-i) and z' > z implies that i f(z', t-i)

Suppose f is monotone. Set pi(ti, t-i) = min {z: i f(z, t-i)} if i wins, and 0 otherwise, for every i.

Then, (f, p) is a truthful mechanism and players' utilities are nonnegative (when they bid truthfully).So we concentrate on designing monotone algorithms with desirable properties (prices always set as above).

i f(z, t-i)

zi f(z, t-i)

Reduction : injecting cost-recovery

Given: truthful, -approximation, no bossy mechanism M = (g, {qi})

On input t, run Moulin mechanism initialized with output of M and with uniform cost shares.1. Initialize k=0, S0 = g(t)

2. While i Sk s.t. ti < C(Sk)/ |Sk|, set Sk+1 = {i Sk: ti ≥ C(Sk)/ |Sk|},

k=k+3. Return f(t) = Sk (and prices pi(t) are set to threshold values)

Why does this work? Truthfulness: Moulin construction preserves monotonicityif if(t), z' > ti, then i g(z', t-i) g(t) = g(z', t-i) (M truthful, no-bossy)so runs on t and (z', t-i) are identical f(t) = f(z', t-i) (and i f(z', t-i))

Threshold of each winner i is max {qi(t), C(S0)/|S0|, ... , C(Sk)/|Sk|}

Reduction : injecting cost-recovery

Given: truthful, -approximation, no bossy mechanism M = (g, {qi})

On input t,1. Initialize k=0, S0 = g(t)

2. While i Sk s.t. ti < C(Sk)/ |Sk|, set Sk+1 = {i Sk: ti ≥ C(Sk)/ |Sk|},

k=k+3. Return f(t) = Sk (and prices pi(t) are set to threshold values)

Why does this work? Truthfulness: Moulin construction preserves monotonicityThreshold of each winner i is max {qi(t), C(S0)/|S0|, ... , C(Sk)/|Sk|}

Cost-recovery: clear since each i Sk pays at least C(Sk)/|Sk|

Approximation: we know C(S0) + ∑iS0 ti ≤ (minA SC(t,

A))By our rule for rejecting players, ∑iS0\Sk ti ≤ O(log

n)C(S0) so get O(.log n)-approximation

Reduction 2: main ideaConsider Steiner-tree cost-sharing (for simplicity)Have LP-based =2-approximation for cost-minimization problem.

Minimize ∑e cexe + ∑i vi zi (LP)subject to

∑e(S) xe + zi ≥ 1 for all sets S: rS, all iS

x, z ≥ 0

Dual LP is of the form:Maximize∑i,S i,S s.t. … ∑S: rS, iS i,S ≤ vi for all i, ≥ 0

So if z*i > 0, then ∑S: rS, iS *

i,S = vi can reject all i s.t. z*

i > 0 at the expense of OPTLP ; serve all other i at cost ≤ .OPTLP .

No bossiness: ↑ vi of winner leaves z*i=0; hence LP-

soln. unchanged.

To summarize,

• Give two black-box reductions to convert() Truthful, approximation, no-bossy mechanisms

➡ cost-sharing mechanisms(2) LP-relative approximation algorithms

➡ truthful, approximation, no-bossy mechanisms

Reduce cost-sharing mechanism design to algorithm design

• Various applications: first / improved / matching guarantees for SNDP, FL, makespan-minimization, ...

• Also, have some extensions to multidimensional settings (players own multiple elements, or require multiple levels of service) but guarantees degrade with dimensionality

Open Questions

• Multidimensional cost-sharing problems – Better guarantees?– Are there similar black-box reductions? (We

can show: a -LMP approximation algorithm can be exported to a truthful-in-expect., -approx. mechanism, but do not know how to inject cost recovery.)

– Can one avoid no-bossiness in first reduction?

• Black-box reductions with other notions of incentive-compatibility?

Thank You.