Black-Box Methods for Cost-Sharing Mechanism Design

1. Black-Box Methods for Cost-Sharing Mechanism Design Chaitanya Swamy University of Waterloo Joint work with Konstantinos Georgiou University of Waterloo

2. 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

3. 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 ÞCannot 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

4. Cost-sharing mechanism-design Formally, C(S) = cost-incurred for serving set S of players (assume C(Æ)=0, C(S) ≤ C(T) if S⊆T) vi = private value/type of player i Cost-sharing mechanism M = (f, {pi}i=1…n) price charged to player i algorithm • Players report types t1,…,tn (let t=(t1,…,tn)) • Mechanism computes solution S = f(t)⊆{1,…,n}, and charges price pi(t) to each player i (usually 0 if iÎS) • utility of i = vi(S) – pi(t) (where vi(A) = vi if iÎA else 0)

5. Goals and Objectives Want mechanisms lying here: seek to understand how the 3 objectives interact/conflict Algorithms to compute solns. with near-opt. social welfare (i.e., approx. algorithms for social-cost-minimization) Truthful, cost-recovering mechanisms Truthful mechanisms

6. Goals and Objectives Recall: M = (f, {pi}), t=(t1,…,tn): reported input, S = f(t) Want to design M such that: M is truthful (strategyproof) – every i maximizes its utility by reporting its true value regardless of other players' bids S has “good” social welfare – quantify using social-cost objective [Roughgarden-Sundararajan]: SC(t, S) := C(S) + ∑iÏS ti ≤ minA SC(t, A) Cost-recovery – prices recover the cost: ∑i pi(t) ≥ C(S) Impossible to satisfy a)–c); C(.) is often NP-hard to compute, so we relax conditions suitably

7. Goals and Objectives Recall: M = (f, {pi}), t=(t1,…,tn): reported input, S = f(t) Truthful, a-approximation, b-cost-recovering mechanism Want to design M such that: M is truthful (strategyproof) – every i maximizes its utility by reporting its true value regardless of other players' bids S has “good” social welfare – quantify using social-cost objective [Roughgarden-Sundararajan (RS09)]: SC(t, S) := CM(S) + ∑iÏS ti ≤ aminA SC(t, A) Cost-recovery – prices recover the cost: ∑i pi(t) ≥ CM(S)/b Cost of solution computed by M for S 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)

8. Goals and Objectives Want mechanisms lying here: seek to understand how the 3 objectives interact/conflict Algorithms to compute solns. with near-opt. social welfare (i.e., approx. algorithms for social-cost-minimization) Moulin mechanisms Truthful, cost-recovering mechanisms VCG Truthful mechanisms

9. Example: Steiner-tree cost-sharing : Terminals ≣ Players : Root r : Node • C(S) = opt. Steiner tree cost on S∪{r} • Social-cost-minimization (SCM) problem ≣ prize-collecting Steiner tree • A truthful, a-approximation, • b-cost-recovering mechanism outputs: • a-approx. solution to SCM problem • prices that recover b-fraction of cost of output tree

10. Three types of objects Very limited understanding: most results rely on constructing cost-shares with suitable properties, which can be very challenging (or impossible!) Þ constructions are quite problem-specific and often rather intricate (A) Truthful, approximation, cost-recovering mechanisms Better understanding: nice characterization (for 1D problems) of truthful mechanisms, allows one to leverage algorithmic techniques (B) Truthful, approximation mechanisms (C) Approximation algorithms for SCM problem Good understanding: numerous techniques: LP rounding, primal-dual, ...

11. Are there reductions b/w A, B, C? (A) Truthful, approximation, cost-recovering mechanisms (C) Approximation algorithms for SCM problem (B) Truthful, approximation mechanisms

12. Are there reductions b/w A, B, C?Our work: Yes! (A) Truthful, approximation, cost-recovering mechanisms (C) Approximation algorithms for SCM problem (B) Truthful, approximation mechanisms (1) iÎf(t), iÎf(t'i, t-i)  f(t)= f(t'i, t-i) Reduction (1): B ➞ A Input: truthful, a-approximation, no-bossy mechanism Output:truthful, O(a.log n)-approx., cost-recovering mechanism  can inject cost-recovery into any no-bossy mechanism (B)

13. Are there reductions b/w A, B, C?Our work: Yes! (A) Truthful, approximation, cost-recovering mechanisms (C) Approximation algorithms for SCM problem (B) Truthful, approximation mechanisms (1) iÎf(t), iÎf(t'i, t-i)  f(t)= f(t'i, t-i) • Reduction (1): B ➞ A (works for any cost f'n.) • Input: truthful, a-approximation, no-bossy mechanism • Output:truthful, O(a.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)

14. Nice application: taking input = VCG, get that for every cost-f'n., there is a truthful, O(log n)-approx., cost-recovering mechanism Approximation algorithms for social-cost-minimization Truthful, cost-recovering mechanisms VCG Truthful mechanisms Truthful, no-bossy mechanisms

15. Are there reductions b/w A, B, C?Our work: Yes! (A) Truthful, approximation, cost-recovering mechanisms (C) Approximation algorithms for SCM problem (B) Truthful, approximation mechanisms (2) (1) Reduction (1): B ➞ A (works for any cost-f'n.) Input: truthful, a-approximation, no-bossy mechanism Output: truthful, O(a.log n)-approx., cost-recovering mechanism Reduction (2): C ➞ B Input: LP-relative r-approx. algorithm for cost-minimization (CM) problem (find a min-cost solution for a given set of players) Output:truthful, (r+1)-approximation, no bossy mechanism Works whenever LP-relaxation of CM problem is “covering like”

16. So for a rich class of problems, can convert any LP-relative r-approximation algorithm for CM problem to truthful, O(r.log n)-approx., cost-recovering mechanism Approximation algorithms for social-cost-minimization r Truthful, cost-recovering mechanisms r+1 Truthful mechanisms Truthful, no-bossy mechanisms

17. 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

18. 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 consider Individual Competitiveness (ICT): pi(t) ≤ C({i}) "i • Makes sense to require ICT for subadditiveC(.), in which case our constructions do ensure ICT

19. 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

20. 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 subadditiveC(.)

21. Some ingredients of our results Useful characterization of truthful mechanisms (Myerson) An algorithm f is monotone if iÎf(z, t-i) and z'>z implies that iÎf(z', t-i) Suppose f is monotone. Set pi(ti, t-i) = min {z: iÎf(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). iÎf(z, t-i) i Ïf(z, t-i) z

22. Reduction 1: injecting cost-recovery Given: truthful, a-approximation, no bossy mechanism M = (g, {qi}) On input t, run Moulin mechanism initialized with output of M and with uniform cost shares. Initialize k=0, S0 = g(t) While $iÎSk s.t. ti <C(Sk)/|Sk|, set Sk+1 ={iÎSk: ti ≥C(Sk)/|Sk|}, k=k+1 Return f(t) = Sk (and prices pi(t) are set to threshold values) Why does this work? Truthfulness: Moulin construction preserves monotonicity if iÎf(t), z' > ti, then iÎg(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 iÎf(z', t-i)) Threshold of each winner i is max {qi(t), C(S0)/|S0|, ... , C(Sk)/|Sk|}

23. Reduction 1: injecting cost-recovery Given: truthful, a-approximation, no bossy mechanism M = (g, {qi}) On input t, Initialize k=0, S0 = g(t) While $iÎSk s.t. ti <C(Sk)/|Sk|, set Sk+1 ={iÎSk: ti ≥C(Sk)/|Sk|}, k=k+1 Return f(t) = Sk (and prices pi(t) are set to threshold values) Why does this work? Truthfulness: Moulin construction preserves monotonicity Threshold of each winner i is max {qi(t), C(S0)/|S0|, ... , C(Sk)/|Sk|} Cost-recovery: clear since each iÎSkpays at least C(Sk)/|Sk| Approximation: we know C(S0) + ∑iÏS0ti ≤ a(minA SC(t, A)) By our rule for rejecting players, ∑iÎS0\Skti ≤ O(log n)C(S0) so get O(a.log n)-approximation

24. Reduction 2: main idea Consider Steiner-tree cost-sharing (for simplicity) Have LP-based r=2-approximation for cost-minimization problem. Minimize ∑e cexe + ∑i vi zi (LP) subject to ∑eÎd(S) xe + zi ≥ 1for all sets S: rÏS,all iÎS x, z ≥ 0 Dual LP is of the form: Maximize ∑i,S ai,S s.t. … ∑S: rÏS, iÎSai,S ≤ vi for all i, a≥ 0 So if z*i > 0, then ∑S: rÏS, iÎSa*i,S = vi can reject all i s.t. z*i > 0 at the expense of OPTLP; serve all other i at cost ≤r.OPTLP. No bossiness: ↑vi of winner leaves z*i=0; hence LP-soln. unchanged.

25. To summarize, • Give two black-box reductions to convert (1) 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

26. Open Questions • Multidimensional cost-sharing problems • Better guarantees? • Are there similar black-box reductions? (We can show: a r-LMP approximation algorithm can be exported to a truthful-in-expect., r-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?

27. Thank You.