Understanding VCG Mechanisms and Combinatorial Auctions: A Comprehensive Overview

1. Issues on the border of economics and computationנושאים בגבול כלכלה וחישוב Speaker: Dr. Michael Schapira Topic: VCG and Combinatorial Auctions II

2. QuickRecap

3. Mechanism Design Scheme types reports t1 r1 t2 r2 outcome payments t3 r3 Social planner p1,p2,p3,p4 t4 r4

4. VCG Basic Idea • You can maximize efficiency by: • Choosing the efficient outcome (given the bids) • Each player pays his “social cost” (how much his existence hurts the others). pi = Optimal welfare (for the other players) if player i was not participating. Welfare of the other playersfrom the chosen outcome

5. VCG: Formal Definition • Bidders are asked to report their private values ti • Terminology: (given the reportedti’s) • w*outcome that maximizes the efficiency. • Let w*-ibe the efficient outcome when i is not playing. • The VCG mechanism: • Outcome w* is chosen. • Each bidder pays: The total value for the others when player i is not participating The total value for the others when i participates

6. Truthfulness Theorem (Vickrey-Clarke-Groves): In the VCG mechanism, truth-telling is a dominant strategy for all players. • Conclusion:welfare maximization can always be achieved in dominant strategies. • No Bayesian distributional assumptions. • No real multiple-equilibria problem as in Nash. • Very simple strategy for the bidders.

7. Combinatorial Auctions • Set M of m indivisible items • Set N of n bidders • Preferences are on subsets S – bundles – of items • Valuation function vi: 2M R • vi(S) – bidder i’s value for bundle S • monotone: vi(S) not decreasing in S • normalized: vi() = 0 Allocation: mutually-disjoint subsets S1, S2, … Sn Social welfare of allocation: ivi(Si)

8. Single Minded Auctions • A valuation v is single minded if there is a bundle of items S* and value a such that • v(S) = a if S contains S* • v(S) = 0 for all other S • Very simple to represent: (S*, a) • Allocation problem for single minded bidders: • Given bids {(Si*, ai)}i for bidders i=1..n • Find a feasible subset W of winning bids with maximum social welfarej in Waj*

9. What Do We Want? • “Good” (w.r.t. efficiency) outcomes (preferably optimal) • Incentive compatibility (preferably in dominant strategies) • Low running time (in the “natural parameters”: n and m)

10. Cannot Simply Use VCG! • Finding optimal allocation is computationally (=NP) hard! • Cannot compute “approximate” VCG payments. • The “clash” between Econ and CS. What can we do?

11. Approximating the Best Allocation • Allocation S1,..,Sn is a g-approximation if: • Even approximating optimal allocation of items in single-minded auctions within factor of is NP-hard!

12. Incentive-CompatibleMechanism forSingle-Minded Auctions

13. Mechanism for Single-Minded Auctions • Approximation factor of (m is #items) • Incentive compatible in dominant strategies • Efficiently computable (obvious)

14. Proof of Incentive Compatibility • Lemma: A mechanism for single minded bidders in which losers pay 0 is incentive compatible iffit satisfies: • Monotonicity: if a bid (S,a) is a winning bid, the bid (S*,a*), where S* is contained in S, or a*>a, is also winning. • Critical payment: A bidder who wins with bid (S,a) pays the minimum needed for winning: the infimum of all values b such that (S,b) wins • The two conditions are met by the greedy algorithm. Why?

15. Proof of Incentive Compatibility • Monotonicity • Critical payment

16. Proof of Incentive Compatibility • We prove that the two conditions imply incentive compatibility (in dominant strategies). • Exercise: Prove the reverse direction. • Let B=(S,a) be the true input of a bidder, and let B*=(S*, a*) be a possible bid • If B* loses or S* does not contain S, it makes no sense to bid B* • Let p be the bidder’s critical payment for bid B, and p* be the critical payment for bid B* • Critical payment: for every x < p, the bid (S,x) loses • Monotonicity: so, for every x < p, the bid (S*,x) also loses • Hence: p≤p* • Bidding (S, a*) instead of B*=(S*, a*) is no worse • But, B=(S, a) is no worse than (S, a*) • If B wins payment is always p • If B loses, a < p and therefore itis not worth to win

17. Proof of Approximation Ratio Theorem: Let OPT be allocation maximizingiOPTvi* and let W be the output of the greedy algorithm. Then iOPTvi* < √m(jWvj*) Proof: • For eachi in W letOPTi={jOPT,i≤j| Si*Sj*≠} • the set of elements in OPT that did not enter W “because” ofi (also including i) • Observe that OPT iWOPTi • Will show: jOPTivj*≤ (√m)vi* for all i in W

18. Proof of Approximation Ratio • For alljOPTiwe know thatvj*≤vi*√(|Sj*|/|Si*|) • Hence, jOPTivj*≤ (vi*/√|Si*|)(jOPTi√|Sj*|) • Using the Cauchy-Schwartz inequality we get that:jOPTi √|Sj*| ≤ (√|OPTi|)(√jOPTi|Sj*|) • For jOPTi, Si*Sj*≠ • Since OPT is an allocation: • these intersections are disjoint and so |OPTi| ≤ |Si*| • jOPTi |Sj*| ≤m • jOPTi √|Sj*| ≤ √|Si*|√m • Plugging into first inequality: jOPTivj* ≤ (√m)vi*

19. Other Interesting Combinatorial Auctions

20. Natural Restrictions on Bidders • Defn: A valuation v is subadditive (complement-free) if for all S,TM,v(ST) ≤ v(S) + v(T). • Defn: A valuation v is submodular if for all S,TM,v(ST) ≤ v(S) + v(T). • Equivalent definition of submodularity: for all STM, and j not in T,v(T{j})-v(T) ≤ v(S{j})-v(S)(decreasing marginal utilites) • Fact: Submodularity implies subadditivity.

21. Computational Hardness • Thm: Finding an optimal allocation in combinatorial auctions with submodular bidders is NP-hard. • We now prove the theorem.

22. Proof • We show a reduction of the PARTITION problem: We are given k real numbers {a1,…,ak} and the goal is to determined whether they can be partitioned into two disjoint subsets, W1 and W2, so that iW1 ai = jW2 ai • Given an instance of PARTITION, we construct an auction with two identical bidders with valuation function:v(S) = min{jSaj, ½iai} • Observe that this valuation is submodular. • Observe that a social welfare of iai is achievable iff it is possible to partition {a1,…,ak} as desired.

23. Approximating the Optimum? • Thm: A 2-approximation to the optimal allocation in combinatorial auctions with submodular bidders can be computed in a computationally-efficient manner. • How?

24. Greedy Algorithm for Submodular Auctions • Set S1=S2=…=Sn= • Go over the items in some order, WLOG, j=1,…,m • Let k be the bidder for which the marginal value for item j, i.e., vi(Si{j})-vi(Si), is maximized. • Allocated item j to bidder k, i.e., set Sk=Sk{j}

25. Approximability for Submodular Bidders • Thm: The greedy algorithm outputs a2-approximation tothe optimal allocation in combinatorial auctions with submodular bidders. • Remark: There exists a (different!)2-approximation algorithm for the more general case of subadditive bidders. • We now prove the theorem.

26. Proof • We prove by induction on the number of items. Suppose that the statement is true for m-1 items. • Let ALG(I) be the allocation the algorithm outputs for a given instance I of a combinatorial auction with submodular bidders. Let OPT(I) be the optimal allocation for instance I. • We will abuse notation and use ALG(I) and OPT(I) to denote both allocations and social-welfare of allocations. • Let k be the bidder to which item 1 is allocated in ALG(I). Let I* denote the instance derived from instance I by removing item 1 and setting v’k(S)=vk(S{1})-vk({1}) for all S • Observe that the bidders remain submodular! • Let ALG(I*) and OPT(I*) denote the algorithm’s output and optimal allocation for instance I*, respectively

27. Proof • Clearly ALG(I)=ALG(I*)+vk({1}) • We will now show that OPT(I) ≤ OPT(I*)+2vk({1}) • We will then use the fact that OPT(I*) ≤ 2ALG(I*) • the induction hypothesis • To conclude that:OPT(I) ≤ OPT(I*)+2vk({1}) ≤ 2ALG(I*)+2vk({1}) =2ALG(I) • So, let’s prove that OPT(I) ≤ OPT(I*)+2vk({1})

28. Proof • We wish to show that OPT(I) ≤ OPT(I*)+2vk({1}) • Let OPT(I)={O1,…,On}. Suppose that item 1 is in Or. Let T1,…,Tn be the allocation of items {2,…,m} as in OPT(I). • T1,…,Tn is a possible solution to I*. We now compare its value to OPT(I). • All bidders but r get the exact same bundle in T1,…,Tn and in OPT(I). All bidders but k have the exact same valuation function in I and in I*. • How much does bidder r lose? vr(Or)-vr(Tr) = vr(Tr{1})-vr(Tr) ≤ vr({1}) ≤ vk({1}) • How much does bidder k lose?vk(Ok)-(vk(Tk{1})-vk({1}) = vk(Ok)-vk(Ok{1}+vk({1}) ≤ vk({1} • So, OPT(I) ≤ OPT(I*)+2vk({1})

29. So… • We have a 2-approximation algorithm for combinatorial auctions with submodular bidders. • The analysis for this algorithm is tight • better approximation ratios are achievable. • Is this algorithm incentive compatible?

30. Simple Example • 2 items, 2 bidders: • v1(1)=1+e, v1(2)=2-e, v1({1,2})=2-e • v2(1)=1, v2(2)=1, v1({1,2})=1 • What will the algorithm do? • Is this incentive compatible? • Thm: The greedy algorithm cannot be rendered incentive compatible (via any payment rule).

31. Proof • Lemma: If an algorithm A is incentive compatible in dominant strategies then: pi(v, v-i) = pi (a, v-i), where A(v) = a. • Proposition:(incentive compatibility  weak monotonicity): • Suppose A(vi,v-i) = a and A(ui,v-i) = b. Then pi(a,v-i) - vi(a) >pi(b,v-i) - vi(b),(otherwise bidder i would declare ui instead of vi).And, pi(b,v-i) - ui(b) >pi(a,v-i) - ui(a),(otherwise bidder i would declare ui instead of vi).vi (a) + ui(b) ≤ui(a) + vi (b).

32. Proof • Now, let us revisiting the 2-item 2-bidder example: • v1(1)=1+e, v1(2)=2-e, v1({1,2})=2-e • v2(1)=1, v2(2)=1, v1({1,2})=1 • Now, consider v1 above and the following u1: u1(1)=0, u1(2)=2-e, u1({1,2})=2-e • Observe that weak monotonicity does not hold!