Conditional Equilibrium Outcomes via Ascending Price Processes

1. Conditional Equilibrium Outcomes via Ascending Price Processes Ron Lavi Industrial Engineering and Management Technion – Israel Institute of Technology Joint work with Hu Fu and Robert Kleinberg(Computer Science, Cornell University)

2. 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 laptop auction fora tablet

3. 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 laptop auction fora tablet bid bid bid a bidder

4. 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

5. 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) = xS px ) • “Walrasian equilibrium” (WE): allocation S=(S1,…,Sn) and prices p= (p1,…,pm) such that Si  Di(p) • Conceptually, demonstrates the “invisible hand” principle

6. 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.

7. 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. gross-substitutes no complements all valuations marginally decreasing

8. 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  nice if exists but very limited existence

9. 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  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?

10. 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

11. 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

12. 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 • 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…

13. 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   \ Sivi(T|Si) – p(T) • A modified auction: • start: zero prices, empty tentative allocation • 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 ???

14. 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)

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

16. 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) 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?

17. 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

18. 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?

19. 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)  uVCE,  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

20. 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.

21. 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: xS ,  T s.t. xT T = 1 (  these weights are “balanced” as in Bondareva-Shapley ) • Fractional subadditivity: S  ,  fractional cover {T} of S,vi(S) < T S, TTvi(T) (  the cooperative (cost) game (, vi) is totally balanced ) • Lehman et al. ’06:marginally decreasing  fractionally subadditive  subadditive

22. Supporting prices • {px}xS are supporting prices for vi(S) if(1) vi(S) = xS px(2) T  S, vi(T) > xT px (  {px} is in the core of the cooperative cost game (S, vi) ) THM (Bondareva-Shapley): vi is fractionally subadditive ifand only if, S  , vi(S)hassupporting prices. (independently formulated by Dobzinski, Nisan, Schapira ’05)

23. 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.

24. 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]

25. Can we continue to expand?

26. 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:

27. 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 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.

28. Lower bound DFN (A valuation class VCE): A valuation vVCE 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.

29. What is the complete answer? Conjecture: The unique set that satisfies the MaxCE 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.

30. 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