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: v i (S) = value that i assigns to acquiring the set S –v i is non-decreasing (“free disposal”) –v i (  ) = 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 (S 1 …S n ) that maximizes the social welfare:  i v i (S i )

4. Example 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. V 1 =10 V 2 =4 t1t1 s1s1 s2s2 t2t2

5. Example 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. V 1 =10 V 2 =4 t1t1 s1s1 s2s2 t2t2 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 and Truthfulness Study rational bidders that aim to maximize v i (S i ) – price A mechanism is M = (A, p 1, p 2, , p n ), where A is an algorithm, and p i : V  R is the payment function of player i. We would like a truthful mechanism, i.e.  v i, v -i, v’ i : v i (A(v i, v -i )) – p i (v i, v -i ) > v i (A(v’ i, v -i )) – p i (v’ i, v -i ) Theorem [Vickrey-Clarke-Groves, the 70’s] : If the algorithm finds the exact optimal welfare then there exist truthful prices. –This is true for any problem domain. Unfortunately, since finding the exact optimum is computationally hard, we cannot use this.

8. Strategic issues Bidders aim to maximize their own utility: v i (S i ) – price. –Thus a player may manipulate the alg. -- declare a false v i (·). Wish to produce an approximately optimal outcome with respect to the true value functions. –Thus want to create an incentive to report truthfully. ALG V 1 (·) v n (·) v 2 (·) S1S1 SnSn S2S2 ······ ······ ······ ······ The classic model: A game-theoretic view:

9. Mechanism Design and Truthfulness A truthful mechanism: No matter what the other players declare, player i will maximize his utility by reporting truthfully. V 1 (·) v n (·) v 2 (·) S 1, P 1 S 2, P 2 ······ ······ ······ ······ S n, P n A mechanism: ALG Mechanism

10. Mechanism Design and Truthfulness 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. Mechanism V 1 (·) v n (·) v 2 (·) S 1, P 1 S 2, P 2 ······ ······ ······ ······ S n, P n A mechanism: ALG

11. Mechanism Design and Truthfulness 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. V 1 (·) v n (·) v 2 (·) S 1, P 1 S 2, P 2 ······ ······ ······ ······ S n, P n A mechanism: ALG Mechanism

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

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

14. 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(m 1/B ) -approx. for any B [Briest, Krysta, Vocking] For general valuations: – O(B·m 1/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.

15. 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(m 1/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.

16. 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.

17. 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?

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

19. The fractional case x i,s is the 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.

20. The fractional case x i,s is the 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. The fractional case x i,s is the 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

22. 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.

23. A randomized truthful integral mechanism Construction: 1.Compute a fractional solution x * (optimal w.r.t. the declared values). 2.Decompose x * /c = λ 1 ·x 1 +…+ λ L ·x L where {x l } l are the integral solutions, and λ 1 +…+ λ L = 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].

24. A randomized truthful integral mechanism Construction: 1.Compute a fractional solution x * (optimal w.r.t. the declared values). 2.Decompose x * /c = λ 1 ·x 1 +…+ λ L ·x L where {x l } l are the integral solutions, and λ 1 +…+ λ L = 1. 3.Choose x l with probability λ 1 and set the expected price to be the VCG price in the fractional setting. Claim: This is truthful in expectation

25. A randomized truthful integral mechanism Construction: 1.Compute a fractional solution x * (optimal w.r.t. the declared values). 2.Decompose x * /c = λ 1 ·x 1 +…+ λ L ·x L where {x l } l are the integral solutions, and λ 1 +…+ λ L = 1. 3.Choose x l with probability λ 1 and set the expected price to be the VCG price in the fractional setting. Claim: This is truthful in expectation Proof: Suppose that v i  y* and v’ i  z*. We have: v i (y*/c) – p i (v i, v -i ) > v i (z*/c) – p i (v’ i, v -i ) As the fractional mechanism is truthful

26. A randomized truthful integral mechanism Construction: 1.Compute a fractional solution x * (optimal w.r.t. the declared values). 2.Decompose x * /c = λ 1 ·x 1 +…+ λ L ·x L where {x l } l are the integral solutions, and λ 1 +…+ λ L = 1. 3.Choose x l with probability λ 1 and set the expected price to be the VCG price in the fractional setting. Claim: This is truthful in expectation Proof: Suppose that v i  y* and v’ i  z*. We have: v i (y*/c) – p i (v i, v -i ) > v i (z*/c) – p i (v’ i, v -i ) [λ y 1 ·v i (x 1 )+…+ λ y L ·v i (x L )] – p i (v i, v -i ) > [λ z 1 ·v i (x 1 )+…+ λ z L ·v i (x L )] – p i (v’ i, v -i ) By the decomposition

27. A randomized truthful integral mechanism Construction: 1.Compute a fractional solution x * (optimal w.r.t. the declared values). 2.Decompose x * /c = λ 1 ·x 1 +…+ λ L ·x L where {x l } l are the integral solutions, and λ 1 +…+ λ L = 1. 3.Choose x l with probability λ 1 and set the expected price to be the VCG price in the fractional setting. Claim: This is truthful in expectation Proof: Suppose that v i  y* and v’ i  z*. We have: v i (y*/c) – p i (v i, v -i ) > v i (z*/c) – p i (v’ i, v -i ) [λ y 1 ·v i (x 1 )+…+ λ y L ·v i (x L )] – p i (v i, v -i ) > [λ z 1 ·v i (x 1 )+…+ λ z L ·v i (x L )] – p i (v’ i, v -i ) E[ v i (f(v i, v -i )) – p i (v i, v -i ) ] > E[ v i (f(v’ i, v -i )) – p i (v’ i, v -i ) ] By construction

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 = λ 1 ·x 1 +…+ λ L ·x L (where x l is integral), in polynomial time. Remark: The alg. should “work” for any weights {w i,s } Method (based on [Carr and Vempala]):

29. 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 = λ 1 ·x 1 +…+ λ L ·x L (where x l is integral), in polynomial time. Remark: The alg. should “work” for any weights {w i,s } Method (based on [Carr and Vempala]): x w i,s x zx z

30. The decomposition (2) Observation: If (w i,s, z) is feasible then

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

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

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

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

35. 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?