Truthful and Near-Optimal Mechanism Design via Linear Programming

1. Truthful and Near-Optimal Mechanism Design via Linear Programming Ron Lavi California Institute of Technology Joint work with Chaitanya Swamy

2. Overview of the Talk • The model of Combinatorial Auctions • Definition, motivation, challenges and goals, previous results. • Our results • Plus a word on the “big picture”. • Intuition to our construction and proofs

3. Combinatorial Auctions • m indivisible non-identical items for sale • n bidders compete for subsets of these items • Each bidder i has a valuation for each set of items: vi(S) = value that i assigns to acquiring the set S • viis non-decreasing (“free disposal”) • vi () = 0 • The multi-unit case: B>1 copies of each item; no player desires more than one copy of each item • Objective: Find a partition of the items (S1…Sn) that maximizes the social welfare: ivi (Si)

4. Example t2 s1 • Each player wants a source-sink path, for some value. • Each edge is an item. We need to allocate items to players. • Each edge can be allocated to at most one player. s2 t1 V1=10 V2=4

5. Example t2 s1 • Each player wants a source-sink path, for some value. • Each edge is an item. We need to allocate items to players. • Each edge can be allocated to at most one B players. s2 t1 V1=10 V2=4 In the multi-unit case

6. Motivation • Abstracts complex resource allocation problems in systems with distributed ownership (scheduling, allocation of network resources). • Real Applications (e.g. the FCC spectrum auction).

7. Strategic issues The classic model: V1(·) S1 • Bidders aim to maximize their own utility: vi(Si) – price. • Thus a player may manipulate the alg. -- declare a false vi (·). • Wish to produce an approximately optimal outcome with respect to the true value functions. • Thus want to create an incentive to report truthfully. v2 (·) S2 ALG · · · · · · · · · · · · vn (·) Sn A game-theoretic view:

8. Mechanism Design and Truthfulness A mechanism: V1(·) S1 , P1 A truthful mechanism: No matter what the other players declare, player i will maximize his utility by reporting truthfully. ALG v2 (·) S2 , P2 · · · · · · · · · · · · Mechanism vn (·) Sn , Pn

9. Mechanism Design and Truthfulness A mechanism: V1(·) S1 , P1 A truthful mechanism: No matter what the other players declare, player i will maximize his utility by reporting truthfully. Theorem [Vickrey-Clarke-Groves, the 70’s] : If the algorithm finds the exact optimal welfare then there exist truthful prices. ALG v2 (·) S2 , P2 · · · · · · · · · · · · Mechanism vn (·) Sn , Pn

10. Mechanism Design and Truthfulness A mechanism: V1(·) S1 , P1 A truthful mechanism: No matter what the other players declare, player i will maximize his utility by reporting truthfully. Theorem [Vickrey-Clarke-Groves, the 70’s] : If the algorithm finds the exact optimal welfare then there exist truthful prices. • Unfortunately finding the exact optimum is computationally hard. ALG v2 (·) S2 , P2 · · · · · · · · · · · · Mechanism vn (·) Sn , Pn

11. Complexity Issues • Communication: input is exponential (in m). • No algorithm can approximate better than m1/(B+1) with polynomial communication [Nisan; Nisan and Segal; Dobzinski and Schapira] • Computation: • It is NP-hard to approximate better than m1/(B+1), even for short valuations [Lehmann, O'Callaghan, Shoham; Bartal, Gonen, Nisan] • There exist polynomial time O(m1/(B+1))-approximations • In particular when B=Ω(log m) there exists a (1+ε)-approximation

12. We seek truthful and computationally feasible mechanisms. In other words, are there other ways to embed truthfulness into a given algorithm?

13. Previous attempts for resolution • The “single minded” case : • √m approx. when B=1 [Lehmann, O'Callaghan, Shoham] • (1+ε)-approx. when B=Ω(log m) [Archer, Papadimitriou, Talwar, Tardos] • O(m1/B) -approx. for any B [Briest, Krysta, Vocking] • For general valuations: • O(B·m1/B-2) for B>3 [Bartal, Gonen, Nisan] • O(√m) for B=1 [Dobzinski, Nisan, Schapira] • Bundling equilibria in VCG to reduce communication (essentially a negative result). [Holzman, Kfir-Dahav, Monderer, Tennenholtz] • No result for the general case; a large gap from the best approximability results for the non-single minded case.

14. Our results Main construction: Given any alg. for general CA that also bounds the integrality gap of the LP relaxation, one can construct a randomized, truthful in expectation, mechanism that has the same approx. ratio. Immediate Applications: strategic mechanisms with approximation guarantees that match the best known non-strategic ones: • A strategic O(m1/B+1) approx. for general valuations and any B. • If B=Ω(log m) this yields a (1+ε)-approx. mechanism. • This technique applies to other “packing domains”, for example multi-parameter knapsack problems. • By moving from deterministic to randomized mechanisms, we completely close the strategic -- non-strategic gap for general CAs.

15. Truthfulness in expectation Truthfulness in expectation[Archer and Tardos]: No matter what the other players declare, player i will maximize his expected utility by reporting truthfully. • A worst case notion (the distribution is created by the mechanism, not assumed on the input). • A player need not assume anything about the rationality of others. • This implicitly implies, however, that a player is risk-neutral. • Thus weaker than deterministic truthfulness.

16. An aside – a more general view • Does deterministic truthfulness can yield such results? • For B=1, any deterministic mechanism that is also IIA cannot obtain a reasonable approximation [Lavi, Mu’alem, Nisan] • Other GT notions might yield distribution-free/worst-case results? • “Rationalizable strategies” for single-item first price auctions[Dekel and Wolinsky, Battigalli and Siniscalchi] • “Set-Nash” for online auctions [Lavi and Nisan] • “Implementation in undominated strategies” for single-value combinatorial auctions [Babaioff, Lavi, Pavlov] • What else?

17. More on VCG Truthfulness :  vi, v-i, v’i : vi(f(vi, v-i)) – pi(vi, v-i) > vi(f(v’i , v-i)) – pi (v’i, v-i) Theorem [Vickrey-Clarke-Groves] : If the algorithm finds the exact optimal welfare then there exist truthful prices. The prices: If (s1,…,sn) is the optimal allocation according to the reported types v=(v1,…,vn), set prices to pi(v) = -Σj≠ivj(sj) + hi(v-i) Proof: Suppose a player says v’i and the chosen allocation is (s’1,…,s’n). His utility isi.e. telling his true value would weakly improve his utility. vi(s’i) - pi(v’i, v-i) = vi(s’i) + Σj≠ivj(s’j) <vi(si) + Σj≠ivj(sj) = vi(si) - pi(vi, v-i)

18. The fractional case • xi,sisthe fraction of bundle S that player i gets. • The fractional case is easy to solve by an LP: • Thus we can use VCG for this case.

19. The fractional case • xi,sisthe fraction of bundle S that player i gets. • The fractional case is easy to solve by an LP: • Thus we can use VCG in this case as well. For every c>1

20. The fractional case • xi,sisthe fraction of bundle S that player i gets. • The fractional case is easy to solve by an LP: • Thus we can use VCG in this case as well. For every c>1

21. More on solving the LP • “Short” valuations (the LP is succinctly describable) • We have a (one shot) truthful in expectation mechanism. • For example k-minded players. The first strategic mechanism for this case. • General valuations: the LP is efficiently solvable with a “demand oracle” [Blumrosen-Nisan] • We have an iterative mechanism; truthfulness in expectation is ex-post Nash equilibrium. • The first strategic mechanism with polynomial communication, computation, and tight approximation bounds.

22. A randomized truthful integral mechanism Construction: • Compute a fractional solution x* (optimal w.r.t. the declared values). • Decompose x*/c = λ1·x1+…+ λL·xL where {xl}lare the integral solutions, and λ1 +…+ λL = 1. The main technical construction. Works if c is the integrality gap, and if furthermore we have an algorithm that “verifies” this.For this we extend a technique of [Carr and Vempala].

23. A randomized truthful integral mechanism Construction: • Compute a fractional solution x* (optimal w.r.t. the declared values). • Decompose x*/c = λ1·x1+…+ λL·xL where {xl}lare the integral solutions, and λ1 +…+ λL = 1. • Choose xl with probability λ1 and set the expected price to be the VCG price in the fractional setting. Claim: This is truthful in expectation

24. A randomized truthful integral mechanism Construction: • Compute a fractional solution x* (optimal w.r.t. the declared values). • Decompose x*/c = λ1·x1+…+ λL·xL where {xl}lare the integral solutions, and λ1 +…+ λL = 1. • Choose xl with probability λ1 and set the expected price to be the VCG price in the fractional setting. Claim: This is truthful in expectation Proof: Suppose that vi y* and v’i  z* . We have: vi(y*/c) – pi(vi, v-i) > vi(z*/c) – pi (v’i, v-i) As the fractional mechanism is truthful

25. A randomized truthful integral mechanism Construction: • Compute a fractional solution x* (optimal w.r.t. the declared values). • Decompose x*/c = λ1·x1+…+ λL·xL where {xl}lare the integral solutions, and λ1 +…+ λL = 1. • Choose xl with probability λ1 and set the expected price to be the VCG price in the fractional setting. Claim: This is truthful in expectation Proof: Suppose that vi y* and v’i  z* . We have: vi(y*/c) – pi(vi, v-i) > vi(z*/c) – pi (v’i, v-i) [λy1·vi(x1)+…+ λyL·vi(xL)]– pi(vi, v-i) >[λz1·vi(x1)+…+ λzL·vi(xL)] – pi (v’i, v-i) By the decomposition

26. A randomized truthful integral mechanism Construction: • Compute a fractional solution x* (optimal w.r.t. the declared values). • Decompose x*/c = λ1·x1+…+ λL·xL where {xl}lare the integral solutions, and λ1 +…+ λL = 1. • Choose xl with probability λ1 and set the expected price to be the VCG price in the fractional setting. Claim: This is truthful in expectation Proof: Suppose that vi y* and v’i  z* . We have: vi(y*/c) – pi(vi, v-i) > vi(z*/c) – pi (v’i, v-i) [λy1·vi(x1)+…+ λyL·vi(xL)]– pi(vi, v-i) >[λz1·vi(x1)+…+ λzL·vi(xL)] – pi (v’i, v-i) E[ vi(f(vi, v-i)) – pi(vi, v-i) ] > E[ vi(f(v’i , v-i)) – pi (v’i, v-i) ] By construction

27. The decomposition (1) Claim: Given a c-approx. algorithm to the optimal fractional solution, one can decompose any fractional point x*/c to a convex combination of integral points, i.e. x*/c = λ1·x1+…+ λL·xL(where xl is integral), in polynomial time. Remark: The alg. should “work” for any weights {wi,s} Method (based on [Carr and Vempala]):

28. The decomposition (1) Claim: Given a c-approx. algorithm to the optimal fractional solution, one can decompose any fractional point x*/c to a convex combination of integral points, i.e. x*/c = λ1·x1+…+ λL·xL(where xl is integral), in polynomial time. Remark: The alg. should “work” for any weights {wi,s} Method (based on [Carr and Vempala]): xwi,s xz

29. The decomposition (2) Observation: If (wi,s , z) is feasible then

30. The decomposition (2) Observation: If (wi,s , z) is feasible then Proof: Suppose o/w. (1/c) Σi,s x*i,s · wi,s > 1 - z

31. The decomposition (2) Observation: If (wi,s , z) is feasible then Proof: Suppose o/w. Using A, find xl s.t. contradicting feasibility. Σi,s wi,s·xli,s > (1/c) Σi,s x*i,s · wi,s > 1 - z

32. The decomposition (2) Observation: If (wi,s , z) is feasible then Proof: Suppose o/w. Using A, find xl s.t. contradicting feasibility. Implications: • The optimal solution is 1, as we need. Σi,s wi,s·xli,s > (1/c) Σi,s x*i,s · wi,s > 1 - z

33. The decomposition (2) Observation: If (wi,s , z) is feasible then Proof: Suppose o/w. Using A, find xl s.t. contradicting feasibility. Implications: • The optimal solution is 1, as we need. • We can use the ellipsoid method to find it in polynomial time: • A separation oracle is implemented as above. • This yields a dual program of polynomial size. Its dual will give us the convex decomposition. Σi,s wi,s·xli,s > (1/c) Σi,s x*i,s · wi,s > 1 - z

34. Summary • Studied the clash between computational and game-theoretic considerations. • For a variety of domains, we give a technique to embed truthfulness in existing algorithmic methods, via randomization and Linear Programming. • Our technique closes the existing large approximation gaps in the literature, providing several new and tight results. • CAs, multi-parameter knapsack problems, Routing and flow problems. • Still open: • Deterministic truthfulness in CAs. • Truthfulness for special cases of CAs (e.g. sub-modularity of value functions). • Other methods for truthful constructions?