1. 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

2. Combinatorial Auctions with Item Bidding A set of m indivisible items are sold by separate simultaneous single-item auctions: auction for a cell-phone auction for a tablet auction for a laptop

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 v i : 2 >0 ) auction for a cell-phone auction for a tablet auction for a laptop a bidder bid

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 = (S 1, …,S n ) is a partition of the items to the players (the sets S i are disjoint, their union is ). The demand of player i under item prices p= (p 1, …,p m ) is: D i (p) = argmax S v i (S) – p(S) ( where p(S) = x S p x ) Walrasian equilibrium (WE): allocation S=(S 1, …,S n ) and prices p= (p 1, …,p m ) such that S i D i (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 v i (S i ) ) 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. all valuations no complements marginally decreasing gross- substitutes

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 1 st -price is replaced by 2 nd -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 assigned set of items S i, under item prices p= (p 1, …,p m ) is: CD i (p, S i ) = argmax T \ Si v i (T|S i ) – 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=(S 1, …,S n ) and prices p= (p 1, …,p m ) such that (1) v i (S i ) > p(S i ), (2) CD i (p, S i ) Conceptually, CE = invisible hand with some regulation –If player i has to take at least her offered set S i, or nothing, at given prices, she will take S i 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=(S 1, …,S n ) and prices p= (p 1, …,p m ) such that (1) v i (S i ) > p(S i ), (2) CD i (p, S i ) 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=(S 1, …,S n ) and prices p= (p 1, …,p m ) such that (1) v i (S i ) > p(S i ), (2) CD i (p, S i ) 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 2 nd -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 V CE satisfies the MaxCE requirements if: All unit-demand valuations belong to V CE –(following Gul & Stacchetti 99) n > 1, any (v 1, …,v n ) (V CE ) n admits a CE (maximality) u V CE, v 1, …,v k V CE such that (v 1, …,v k ) 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 V CE that satisfies the MaxCE requirements is contained in. Lower Bound: There exists a valuation class V CE that satisfies the MaxCE requirements and contains V CE. Properties of V CE : –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, v i (S) < T S, T T v i (T) ( the cooperative (cost) game (, v i ) is totally balanced ) Lehman et al. 06: marginally decreasing fractionally subadditive subadditive

22. Supporting prices {p x } x S are supporting prices for v i (S) if (1) v i (S) = x S p x (2) T S, v i (T) > x T p x ( {p x } is in the core of the cooperative cost game (S, v i ) ) THM (Bondareva-Shapley): v i is fractionally subadditive if and only if, S, v i (S) has supporting prices. (independently formulated by Dobzinski, Nisan, Schapira 05)

23. The Flexible-Ascent auction supporting prices for v i (S): (1) v i (S) = p(S) ; (2) T S, v i (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 v i (S): (1) v i (S) = p(S) ; (2) T S, v i (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: abcbcacabcba 8466333v

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 V CE that satisfies the MaxCE requirements. Then. In particular, there exist unit-demand valuations v 1, …,v k such that (u, v 1, …,v k ) does not admit a CE.

28. Lower bound DFN (A valuation class V CE ): A valuation v V CE 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 V CE that satisfies the MaxCE requirements and contains V CE.

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 v 2, …,v n are marginally decreasing then (v 1, …,v n ) admits a CE. For two players and four items, V CE 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