1 / 35

Bundling Equilibrium in Combinatorial Auctions

Bundling Equilibrium in Combinatorial Auctions. Written by: Presented by: Ron Holzman Rica Gonen Noa Kfir-Dahav Dov Monderer Moshe Tennenholtz. Motivation. The Vickrey-Clarke-Groves(VCG) mechanisms are: Central to the design of protocols with selfish participants.

orde
Download Presentation

Bundling Equilibrium in Combinatorial Auctions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related