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

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Presentation on theme: "Bundling Equilibrium in Combinatorial Auctions Written by: Presented by: Ron Holzman Rica Gonen Noa Kfir-Dahav Dov Monderer Moshe Tennenholtz."— 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 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 (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 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


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