conditional equilibrium outcomes via ascending price processes

Conditional Equilibrium Outcomes via Ascending Price Processes

Joint work with Hu Fu and Robert Kleinberg (Computer Science, Cornell University)

Ron Lavi

Industrial Engineering and Management

Technion – Israel Institute of Technology

Combinatorial Auctions with Item Bidding

• A set of m indivisible items are sold by separate simultaneous single-item auctions:

auction fora cell-phone

auction fora tablet

auction fora laptop

Combinatorial Auctions with Item Bidding

• A set of m indivisible items are sold by separate simultaneous single-item auctions:

• Bidders value subsets of items (captured by a valuation function vi: 2 >0)

auction fora cell-phone

auction fora tablet

auction fora laptop

a bidder

bidbid

bid

Equilibrium of the resulting game

• Bikhchandani ’99; Hassidim, Kaplan, Nisan, Mansour ’11:model as a complete information game, and show:

THM: With first-price auctions, pure Nash eq. exists if and only if Walrasian eq. exists

Reminder: Walrasian Equilibrium (WE)

• An “allocation” S = (S1,…,Sn) is a partition of the items to the players (the sets Si are disjoint, their union is ).

• The “demand” of player i under item prices p= (p1,…,pm) is:

Di(p) = argmax S vi(S) – p(S) ( where p(S) = xS px )

• “Walrasian equilibrium” (WE): allocation S=(S1,…,Sn) and prices p= (p1,…,pm) such that Si Di(p)

• Conceptually, demonstrates the “invisible hand” principle

Three Nice Properties of WE

• The first welfare theorem: the welfare in any WE is optimal(the welfare of an allocation is i vi(Si) )

• The result of a natural ascending auction:

– start from zero prices

– raise prices of over-demanded items (given players’ demands)

– … until no item is over-demanded

THM (Gul & Stacchetti ’00, Ausubel ’06): This process terminates in a Walrasian equilibrium if valuations are “gross-substitutes”

• The second welfare theorem: the allocation with maximal welfare is supported by a WE.

A Problem: very limited existence

• Kelso & Crawford ’82: WE always exists for “gross-substitutes”

• Gul & Stacchetti ’99: gross-substitutes is the maximal such class if we want to include unit-demand valuations

• Lehman, Lehman & Nisan ’06: gross-substitutes has zero measure amongst all marginally decreasing valuations.

all valuations no complementsmarginally decreasing

gross-substitutes




nice if exists but very limited existence




nice if exists but very limited existence

THM [Christodoulou, Kovacs, Schapira ’08]: With second-price auctions, pure Nash eq. exists for all fractionally-subadditive valuations

• Which notion replaces WE when 1st-price is replaced by 2nd-price?

• What are its properties? (particularly, welfare guarantees?)

• What is a maximal existence class?

A closer look at the problematic aspect of WE

• Alternative formulation of the ascending auction [DGS’86]

– start: zero prices, empty tentative allocation

– pick a player with empty tentative allocation

– this player takes her demand; raises price of a taken item by – … until all tentative allocations equal current demands






• Gross-substitutes: demanded items whose price does not increase continue to be demanded. Implies termination in WE:

– Since all items are always allocated






• Gross-substitutes: demanded items whose price does not increase continue to be demanded. Implies termination in WE:

– Since all items are always allocated

• Without gross-substitutes, items whose price did not increase may be dropped (even with decreasing marginal valuations)

– Thus the end outcome need not be a WE, in fact a WE need not exist…

A natural modification to the auction

• Modification: a player cannot drop items currently assigned to her

• The “conditional demand” of player i, given the currently assignedset of items Si, under item prices p= (p1,…,pm) is:

CDi(p, Si) = argmax T \ Si vi(T|Si) – p(T)

• A modified auction:


– pick a player with non-empty conditional demand, (this player:)

– takes her conditional demand; raises price of a taken item by – … until all conditional demands are empty

• With gross-substitutes: the same auction as before, ends in WE.

• Without gross-substitutes ???

Conditional Equilibrium (CE)Proposition: With marginally decreasing valuations the auction

always ends in a “CE”:

• “Conditional Equilibrium” (CE): allocation S=(S1,…,Sn) and prices p= (p1,…,pm) such that (1) vi(Si) > p(Si) , (2) CDi(p, Si)

• Conceptually, CE = “invisible hand” with some regulation

– If player i has to take at least her offered set Si, or nothing, at given prices, she will take Si and will not want to expand it.

• Formally, a relaxation of WE (WE CE)




THM: With second-price auctions, pure Nash eq. with weak no-overbidding exists if and only if CE exists




THM: With second-price auctions, pure Nash eq. with weak no-overbidding exists if and only if CE exists

• Which of the “nice” properties of WE continues to hold for a CE?

Welfare Theorems for CE

• First welfare theorem (relaxed version): the welfare in any CE is at least half of the optimal welfare

Corollary: Price of Anarchy of the 2nd-price auction game is 2

– extends and simplifies a result of Bhawalkar and Roughgarden ’11 for subadditive valuations

• Second welfare theorem: the allocation with maximal welfare is supported by a CE

– holds for “fractionally subadditive” valuations

Questions

• Can a CE exist when valuations exhibit a mixture of substitutes and complements? If so, what is the largest class of valuations that always admit a CE?

• Does the existence of a CE imply that the welfare-maximizing allocation is supported by a CE? In other words, does the second welfare theorem hold whenever a CE exists?

Maximal existence classes

A valuation class VCE satisfies the MaxCE requirements if:

• All unit-demand valuations belong to VCE

– (following Gul & Stacchetti ’99)

n > 1, any (v1,…,vn)(VCE)n admits a CE

• (maximality) uVCE, v1,…,vk VCE such that (v1,…,vk) does not admit a CE

Main Question: Describe a valuation class satisfying the MaxCE requirements. Is there a unique such class? (We know that one such class contains all fractionally subadditive valuations)

• Gul & Stacchetti ’99: gross-substitutes is the unique class that satisfies these conditions when considering WE instead of CE

Main Technical Results: Upper and Lower Bound

Upper Bound: Any valuation class VCE that satisfies the MaxCE requirements is contained in .

Lower Bound: There exists a valuation class VCE that satisfies the MaxCE requirements and contains VCE.

Properties of VCE :

– Contains all fractionally subadditive valuations.

– Contains non-subadditive valuations

Conjecture (with some supporting evidence in the paper): The unique set that satisfies the MaxCE requirements is .

We leave this as open problem.

CEV

CEV

Fractionally subadditive valuations

• (defined by Nisan’00 as XOS, the following def. is by Feige’06)

• Weights {T} T S, T are a fractional cover of S if:

xS , T s.t. xT T = 1

( these weights are “balanced” as in Bondareva-Shapley )

• Fractional subadditivity: S , fractional cover {T} of S,

vi(S) < T S, T T vi(T)

( the cooperative (cost) game (, vi) is totally balanced )

• Lehman et al. ’06:

marginally decreasing fractionally subadditive subadditive

Supporting prices

• {px}xS are supporting prices for vi(S) if

(1) vi(S) = xS px (2) T S, vi(T) > xT px

( {px} is in the core of the cooperative cost game (S, vi) )

THM (Bondareva-Shapley): vi is fractionally subadditive if

and only if, S , vi(S) has supporting prices.

(independently formulated by Dobzinski, Nisan, Schapira ’05)

The Flexible-Ascent auction• supporting prices for vi(S): (1) vi(S) = p(S) ; (2) T S, vi(T) > p(T)

• The Flexible-Ascent auction (Cristodoulou, Kovacs, Schapira ’08):

– start: zero prices, empty tentative allocations

– pick a player with non-empty conditional demand, (this player:)

– takes conditional demand; raises sum of prices of her items

– … until all conditional demands are empty

Proposition: For fractionally subadditive valuations, this auction terminates in a CE if prices are always set to be supporting prices

Proof:

• IR exists in every iteration by definition of supporting prices.

• Empty conditional demand at the end by definition of auction.

The Flexible-Ascent auction• supporting prices for vi(S): (1) vi(S) = p(S) ; (2) T S, vi(T) > p(T)

• The Flexible-Ascent auction (Cristodoulou, Kovacs, Schapira ’08):– start: zero prices, empty tentative allocations– pick a player with non-empty conditional demand, (this player:)– takes conditional demand; raises sum of prices of her items– … until all conditional demands are empty

Proposition: For fractionally subadditive valuations, this auction terminates in a CE if prices are always set to be supporting prices

Corollary: There always exists a CE for fractionally subadditive valuations.

• This is essentially the proof of [Christodoulou, Kovacs, Schapira ’08]

Can we continue to expand?

Upper bound

DFN (A valuation class ): A valuation if:

Properties:

• Contains all fractionally subadditive valuations (weights are a fractional cover)

• Does not contain all subadditive valuations, but contains non-subadditive valuations, for example:

CEV

CEVv

SxxSv

SSvSS }){\(

1||

1)(:1||,

0o/w,1||1

}\{ TSxS

abcabacbcabc

v3336648

Upper boundDFN (A valuation class ): A valuation if:

Properties:

• Contains all fractionally subadditive valuations (weights are a fractional cover)

• Does not contain all subadditive valuations, but contains non-subadditive valuations

Theorem: Fix any valuation class VCE that satisfies the MaxCE

requirements. Then .

In particular, there exist unit-demand valuations v1,…,vk such that (u, v1,…,vk) does not admit a CE.

CEV

CEVv

SxxSv

SSvSS }){\(

1||

1)(:1||,

0o/w,1||1

}\{ TSxS

CECE VV CE

Vu

Lower bound

DFN (A valuation class VCE): A valuation vVCE if and for and S (S), v(S) is fractionally subadditive.

Properties:

• Contains all fractionally subadditive valuations.

• Contains non-subadditive valuations

• Contained in

Theorem: There exists a valuation class VCE that satisfies the MaxCE requirements and contains VCE.

CEVv

CEV

What is the complete answer?

Conjecture: The unique set that satisfies theMaxCE requirements is

We leave this problem open. Additional evidence from the paper:

• When || < 3 hence the conjecture is true for this case.

• If and v2,…,vn are marginally decreasing then (v1,…,vn) admits a CE.

• For two players and four items, VCE is provably not the correct lower bound: we show one specific valuation that must be added.

CEV

CECE VV CE

Vv 1

Summary• With indivisible items, Walrasian eq. has very limited existence.

• Study a relaxed notion: “Conditional Equilibrium” (CE).

• For marginally decreasing valuations a CE exhibits:

– An approximate version of the first welfare theorem (in fact this holds for any CE regardless of the valuation class).

– A CE can be reached by a natural ascending auction.

– The second welfare theorem holds as well.

– In fact all this is true for fractionally subadditive valuations

• We study the complete characterization question:

– Show upper and lower bounds on a maximal existence class

– Implies: CE exists with a mixture of substitutes and complements

– We leave the complete characterization as an open problem

conditional equilibrium outcomes via ascending price processes

Documents