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

MotivationThe 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.

MotivationThe 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.

MotivationIt 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.

About the PaperThis 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.

About the PaperIf 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.

The Problem DefinitionNotations: 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

1,..., mA a a 1m 1,2,...,N n 1n

0ii N

: 2Aiv R

0iv

Some More DefinitionsThe 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

, , 2AB C B C i iv B v C

i i i iu v c

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 rationalan ex post equilibrium that satisfies the

participation constraint. Social surplus

Is denoted by

0i i iv d b v c b v

v V

iv

id

, i ii N

S v v

max max ,S v S v

Some More DefinitionsPlayer 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.

i i Nb b

i jb b ,i j N

2A

,maxi iC C B

v B v C

2AB

i if v v iv iv

i N i

f

Example For that does not Induce a Bundling Equilibrium.

First we will define a valuation function: If and

If

Else If

for all

Consider

f

2AB B B C

1B C 0B C

B 0B C 2AC

, , ,A a b c d , , , , ,a d bcd abc A

Example For that does not Induce a Bundling Equilibrium.

Buyer 2 declaresBuyer 3 declaresConsider 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).

f

2 1a a a av a abc A 3 1d d d dv d bcd A

1 1bc bc bc bc bcv bc bcd abc A

' ' '1 1 1 1v bcd v abc v A

f

a d bc

f

bc

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.

2A

B cB \cB A B,B C B C B C


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

f

f j i

iviv

i i i ii i

v c v c max , , ,i i i iS v v S v v

,i id v v

maxS

max , , ,i i i iS v v S v v

i iv B v B 2AB


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.

max, , , , ,i i i i i iS v v S v v S v v

j

,j j j jd v v

j j j jv v

ci j i j

i

max, , , , ,i i i i i iS v v S v v S v v

iv

max max max, , , , ,i i i i i i i iS v v S v v S v v S v v


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.

cB B cB A B

B A cB 2 1v B 2 2 1A

B B Bv B B 1 2 1c c c

c c A

B B Bv B B

1 0c

c

Bv D B

1v

1 0cv B

cB

cBcB

f


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.

,B C B C B C

cB C cB C

1 1cB C

v

1 0cB C

v

2 2 1Bv v 3 3 1Cv v

cB C

1v cB C

f

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.

1,..., kA A

iA

i jA A i j

,i j i jA A A A ,i j i jA A A A

ci iA A

f

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 )

max ,S v S v

maxsupv V

S vr

S v

f

f

2

1,..., sH H

i jH H 1 ,i j s : i li l H A 1 l k

iH lAlA


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.

maxr s

s s

maxmax max

S vr s s

S v

i lA l iA

lA

IH


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”

I JH H

lA lA

lA i lA

lA IH

i lA

S r


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

max i ii N

S v v

,IH I

i ii I

v S v

I

max i i i ii N I i I I

S v v v S v rS v

min maxr S r S

maxr S


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

maxmax max

S vr s s

S v

iB

IHlAIH

iB

i jB B

iB

,i jC C ,i i j jB C B C i jC C

I JH H


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

IH

IH

iB maxS v s f

' 1ii B i iv v C

1ii B i iv v B

i iB C

iCiC 1S v


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.

1,..., kA A A m

r m

1 2max ,r A A

1 2 3max , , ,2

mr A A A


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

1A

2A

2A

2A

1 2 3max , , ,2

ms A A A

ls A

lAlAlA


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:

1A 2A

3A12s13s23s123s

1A

2A 3A

1A 2A 3A

12 13 123 1s s s A 12 23 123 2s s s A

13 23 123 3s s s A


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

12 13 23 1232 3s s s s m

12 13 23 123 2

ms s s s s

1 2 3max , , ,2

ms A A A

2

ms

lAlA

lA


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 .

s

lA

lAlA

1 2 312 2

A A As

1 3 213 2

A A As

2 3 123 2

A A As


in Partition-Based Equilibrium. is maximal since :

Since

s 12 13 1s s A

12 23 2s s A

13 23 3s s A

1 2 3A A A m 12 13 23 2

ms s s s


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.

r

1,..., kA A r k

1max ,..., kA A

1,...,

max min ,j k

kk j

j

min ,kjj


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:

max min ,k

k kk

k

k k

m

k m mr k k

k k


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

2 1k q q 1lA q 1,...,l k

2 1

1

q qk

q

2 1min ,

q qq q

q

2 21 1min 1,

1 1

q q q qq

q q

2 1

11

q qr k q k

q


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:

1,..., kA A 2 1k q q 1lA q

lp

r k

SummaryWe 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

1,..., kA A

f

maxsupv V

S vr

S v

r m

1 2max ,r A A

1 2 3max , , ,2

mr A A A

r k

r k

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Bundling Equilibrium in Combinatorial Auctions

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