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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

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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

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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

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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

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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

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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.

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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

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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

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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?

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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

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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

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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 …

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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 ???

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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)

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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

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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?

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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

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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?

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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

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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.

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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

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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)

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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.

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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]

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Can we continue to expand?

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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 v

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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.

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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.

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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.

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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

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