Bundling Equilibrium in Combinatorial Auctions

1. Bundling Equilibrium in Combinatorial Auctions Written by: Presented by: Ron Holzman Rica Gonen Noa Kfir-Dahav Dov Monderer Moshe Tennenholtz

2. Motivation • The Vickrey-Clarke-Groves(VCG) mechanisms are: • Central to the design of protocols with selfish participants. • In particular for combinatorial auctions. • In this paper the authors deal with a special type of VCG mechanism. • The VC mechanism. • VC mechanisms are characterized by two additional properties: • Truth telling is preferred to non participation. • The seller’s revenue is always non negative.

3. Motivation • The revelation principle in the VC mechanism follows from the truth telling property. • A mechanism with revelation principle is a direct mechanism. • Ex post equilibrium: • even if the players were told the true state, after they choose their actions, • they would not regret their actions.

4. Motivation • It was proven that every mechanism with an ex post equilibrium is economically equivalent to a direct mechanism. • The two mechanisms differ in the set of inputs that the player submits in equilibrium. • They are equivalent from the economics point of view but very different in the communication complexity.

5. About the Paper • This paper analyzes ex post equilibrium in VC mechanisms. • Let be a family of bundles of goods. • The number of bundles in represents the communication complexity of the equilibrium. • The economic efficiency of equilibrium is measured by the general social surplus. • partitions the goods in the auction.

6. About the Paper • If one partition is finer than another one, then it yields higher communication complexity as well as higher social surplus. • The paper deals with the trade of between communication complexity and economic efficiency.

7. The Problem Definition • Notations: • Seller-0 • m items • N buyers • Let be the set of all allocations of the goods. The sum of all the buyers allocations plus the goods that were left with the seller. • Valuation function of buyer i

8. Some More Definitions • The valuation function assumes: • No allocation externalities. • The buyer’s valuation does not depend on what other buyers gained. • Free disposal. • If then • Private value model. • Each buyer knows his valuation function only. • Quasi linear utilities

9. Some More Definitions • Participation constraint for every • For every possible valuation v the allocation function will allocate a bundle to buyer i, regarding the strategy of all the other buyers. This bundle worth to buyer i. If for every possible valuation buyer i has a positive utility he has the incentive to participate. • Individually rational • an ex post equilibrium that satisfies the participation constraint. • Social surplus • Is denoted by

10. Some More Definitions • Player symmetric equilibrium • where for all • a family of bundles of goods. • -valuation function • for every • And is the function that turns to • -allocation • is a -allocation if for every buyer • Bundling equilibrium • is a player-symmetric individually rational ex post equilibrium in every VC mechanism.

11. Example For that does not Induce a Bundling Equilibrium. • First we will define a valuation function: • If and • If • Else • If • for all • Consider

12. Example For that does not Induce a Bundling Equilibrium. • Buyer 2 declares • Buyer 3 declares • Consider buyer 1 with • If buyer 1 uses he declares • There exist a VC mechanism that allocates: • to buyer 2, to buyer 3 and to the seller. • The utility of buyer 1 is zero. (he gets nothing and pays nothing). • If buyer 1 did not declare • He receives and pays nothing. (according to VC payment scheme).

13. A Characterization of Bundling Equilibrium • is called a quasi field if it satisfies the following properties: • implies that ,where . • and implies that . • Theorem 1: induces a bundling equilibrium if and only if it is a quasi field.

14. A Characterization of Bundling Equilibrium • Poof: is a quasi field is an individually rational ex post equilibrium in this VC mechanism. • Assume buyer j, uses strategy . • We need to show that the best reply of buyer i with is . • Since truth revealing is a dominating strategy in every VC mechanism we will show or (1) • By the definition of (2) • By -valuation definition and free disposal assumption for every

15. A Characterization of Bundling Equilibrium • Summarizing over all valuations (3) • Define an allocation such that: • Because is a quasi field, • (4) The quality is due to definition. • Combining (2), (3), and (4) yields Therefore (1) holds.

16. A Characterization of Bundling Equilibrium • Proof: induces a bundling equilibrium is a quasi field. • First we will show • Let , assume to contradiction • Let • And let , definition and • There exist a VC mechanism that allocates B to buyer 2 and to the seller. • If buyer 1 declares his true valuation VC allocates to him and he pays nothing. • is not a bundling equilibrium. Contradiction.

17. A Characterization of Bundling Equilibrium • Next we will show • By the first part of the proof it is suffices to show • Assume • Consider • There exist VC mechanism that will allocate B to buyer 2, C to buyer 3 and to the seller. • If buyer 1 will say he will receive and will pay 0. • is not a bundling equilibrium. Contradiction.

18. Partition-Based Equilibrium • A partition of A into k non empty parts. • for every • is a field generated by • If • If • If • A corollary of Theorem 1: • Corollary 1: is a bundling equilibrium.

19. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • Communication Complexity of the equilibrium is • Buyer has to submit numbers to the seller. • Note: Communication Complexity of the equilibrium is • Because of the field characteristic. • Define: and • Let H be a group of A’s indexes. • such that: • for every • for every (The number of groups that includes the index l of is less or equal to the number of goods in )

20. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • Theorem 2: For every partition , where is the number of sets in . • Proof: • Let be the union of all the set such that (this is the minimum set since are disjoint). • Let be a partition of the buyers to r subsets. • The allocations of the buyers in each subset are disjoint. • Assume r is minimal. • is a group of indexes of sets A that intersect with the allocations of buyers in I.

21. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • The partition we defined is a . • otherwise we can join I and J in contradiction to the minimality of r. • There are no more than buyers to goods. No more than buyers Therefore no more than different sets will “include”

22. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • Since there was extension of allocation for every buyer i. • Since every defines a allocation for every • We showed that • For the next direction it is suffusion to show

23. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • Poof: • Associate a set of goods with : • One good from every in is in . • We can build such because of the second condition of H. • If and because

24. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • The maximum number of is n. • Each is associated with a set of buyers I. • The maximum number of buyers groups is n (we have only n buyers). • Let us take n=s buyers • Let • If we allocated to every buyer i • Let buyer i use then: • since • are not disjoint in pairs therefore only one buyer will be allocated and

25. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • Proposition 1: Let and let be a partition of A into k non empty sets. • If k=1 then • If k=2 then • If k=3 then • Poof: k=1 • Build Hi that includes only A (partition into one group). • We can build m such identical Hi. • A can be included in |A| set of Hi due to the second condition of H.

26. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • Proof k=2: • Without the lose of generality assume is larger than . • Add to every Hi you build. • sets of Hi can be build due to the second condition of H. • Proof k=3: • First side: • If there is a set Hi that includes only then there are at least sets of Hi that include too (due to the second condition of H). • Therefore

27. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • Otherwise all the option for Hi are: • = and together. • = and together. • = and together. • = and and together. • We have the following inequalities:

28. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • By adding these inequalities we obtain: • Which implies: • Second side: • If one of the is maximal then we can build all the Hi to include . • From the second condition of H we will have sets of Hi. • We need to prove

29. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • We will build maximum by building minimum Hi sets. • Each Hi will include exactly two s. • Hi that includes only one will not help. According to the second condition of H the rest of Hi will include two .

30. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • is maximal since : • Since

31. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • Theorem 3 bounds . • Theorem 3: Let be a partition of A into k non empty sets. Then Where And • is increasing and then decreasing. The maximum will occur exactly before decreasing.

32. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • Since does not have to be an integer number • In the special case where all sets in have equal size then:

33. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • Consider the following case: (q is non negative integer) • The number of sets in the partition • for • In this case Since and

34. Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium. • Theorem 4 claims that for infinitely many of these cases this upper bound is tight. • Theorem 4: Let be a partition that satisfies and for some q which is either 0 or 1 or of the form where p is a prime number and l is a positive integer. Then:

35. Summary • We talked about that induces a bundling equilibrium if and only if it is a quasi field. • We defined as a partition of A into k non empty parts. • We then conclude that is a bundling equilibrium. • We tried to bound the Communication Efficiency vs. Economic Efficiency in Partition-Based Equilibrium, by bounding • For k=1 we bound by • For k=2 we bound by for k=3 we bound by • And for the general case we bound by • For a special case it can be bound by