Combinatorial Auctions with Item Bidding: Equilibria and Dynamics - PowerPoint PPT Presentation

Combinatorial Auctions with Item Bidding: Equilibria and Dynamics Thomas Kesselheim Max Planck Institute for Informatics Based on joint work with Paul D¨ utting WINE 2016

Thomas Kesselheim 2/38

Combinatorial Auctions with Item Bidding v 4 ( { 1 } ) = 10 v 4 ( { 2 } ) = 20 v 4 ( { 1 , 2 } ) = 20 . . . n bidders m items 1 2 3 4 5 6 Each item is sold in a separate second-price auction. Bidders usually cannot express their preferences. Might have to pay for multiple items although they only want one. Thomas Kesselheim 3/38

Setting [Christodoulou/Kov´ acs/Schapira, JACM 2016] Set of n bidders N , set of m items M Each bidder i has valuation function v i : 2 M → R ≥ 0 Each bidder i reports a bid b i , j ≥ 0 for every item j Each item j is sold to bidder i that maximizes b i , j Has to pay 2nd highest bid: max i ′ � = i b i ′ , j Each bidder i tries to maximize his/her utility � u i ( b ) = v i ( S i ) − max i ′ � = i b i ′ , j , j ∈ S i where S i is the set of items bidder i wins under b Thomas Kesselheim 4/38

Example Two bidders, two items v 1 ( { 1 } ) = 2, v 1 ( { 2 } ) = 1, v 1 ( { 1 , 2 } ) = 2 0 1 1 1 1 0 v 2 ( { 1 } ) = 1, v 2 ( { 2 } ) = 2, v 2 ( { 1 , 2 } ) = 2 b 1 , 1 = 0, b 1 , 2 = 1 b 2 , 1 = 1, b 2 , 2 = 0 1 2 Bidder 1 wins item 2; bidder 2 wins item 1. No bidder wants to unilaterally deviate ⇒ pure Nash equilibrium Thomas Kesselheim 5/38

Equilibrium Concepts Definition A bid profile b is a pure Nash equilibrium if for all bidders i and all b ′ i u i ( b ) ≥ u i ( b ′ i , b − i ) Other equilibrium concepts: mixed Nash correlated Bayes-Nash Thomas Kesselheim 6/38

Questions How good are (pure Nash, mixed Nash, correlated, Bayes-Nash, . . . ) equilibria? Do they always exist? If so, can they be computed in polynomial time? If so, can they be reached by simple dynamics? Thomas Kesselheim 7/38

Outline 1 Price of Anarchy 2 Complexity of Equilibria Best-Response Dynamics 3 Open Problems 4 Thomas Kesselheim 8/38

Outline 1 Price of Anarchy 2 Complexity of Equilibria Best-Response Dynamics 3 Open Problems 4 Thomas Kesselheim 9/38

Price of Anarchy Given b call SW ( b ) = � i ∈ N v i ( S i ) social welfare of b i ∈ N v i ( S ∗ � Compare to OPT ( v ) = max ( S ∗ i ) 1 , . . . , S ∗ n ) is partition Price of Anarchy OPT ( v ) PoA = max v 1 ,..., v n max SW ( b ) b ∈ PNE Two bidders, one item: v 1 = 0, v 2 = 1 b 1 = 1, b 2 = 0 is pure Nash equilibrium, SW ( b ) = 0, OPT ( v ) = 1 Therefore restrict attention to equilibria with weak no-overbidding : � j ∈ S b i , j ≤ v i ( S ) if bidder i wins set S Thomas Kesselheim 10/38

Classes of valuation functions A function v i : 2 M → R ≥ 0 is . . . additive if v i ( S ) = � j ∈ S v i , j for some v i , j ≥ 0 unit demand if v i ( S ) = max j ∈ S v i , j for some v i , j ≥ 0 fractionally subadditive or XOS if j ∈ S v ℓ i , j for some v ℓ v i ( S ) = max ℓ � i , j ≥ 0 subadditive if v i ( S ∪ T ) ≤ v i ( S ) + v i ( T ) Additive Unit demand XOS Subadditive Thomas Kesselheim 11/38

Examples v i ( { 1 } ) = 2, v i ( { 2 } ) = 1, v i ( { 1 , 2 } ) = 2 is unit demand Every submodular function is XOS, e.g. v i ( S ) = min { c i , � j ∈ S v i , j }  0 if | S | = 0   v i ( S ) = if | S | = 1 or | S | = 2 1  if | S | = 3  2 is subadditive but not XOS Thomas Kesselheim 12/38

Price of Anarchy: Bound for XOS Valuations [Christodoulou/Kov´ acs/Schapira, JACM 2016] Theorem Consider XOS valuations v. Let b be a pure Nash equilibrium. Then SW ( b ) ≥ 1 2 OPT ( v ) . Proof for unit-demand valuations: Let j i be the item that bidder i gets in OPT ( v ) . Bidder i could deviate to b ′ i , j such that b ′ i , j = v i , j if j = j i and 0 otherwise. u i ( b ) ≥ u i ( b ′ i , b − i ) ≥ v i , j i − max i ′ b i ′ , j i . ⇒ � i ∈ N u i ( b ) + � j ∈ M max i ′ b i ′ , j ≥ � i ∈ N v i , j i = OPT ( v ) � i ∈ N u i ( b ) ≤ SW ( b ) by definition, � j ∈ M max i ′ b i ′ , j ≤ SW ( b ) by no-overbidding Thomas Kesselheim 13/38

Price of Anarchy: Bound for XOS Valuations [Christodoulou/Kov´ acs/Schapira, JACM 2016] Theorem Consider XOS valuations v. Let b be a pure Nash equilibrium. Then SW ( b ) ≥ 1 2 OPT ( v ) . Proof for unit-demand valuations: Let j i be the item that bidder i gets in OPT ( v ) . Bidder i could deviate to b ′ i , j such that b ′ i , j = v i , j if j = j i and 0 otherwise. u i ( b ) ≥ u i ( b ′ i , b − i ) ≥ v i , j i − max i ′ b i ′ , j i . “Smoothness” proof : Deviation does not depend on b ⇒ � i ∈ N u i ( b ) + � j ∈ M max i ′ b i ′ , j ≥ � i ∈ N v i , j i = OPT ( v ) ⇒ extends to mixed Nash, correlated, Bayes-Nash equilibria � i ∈ N u i ( b ) ≤ SW ( b ) by definition, � j ∈ M max i ′ b i ′ , j ≤ SW ( b ) by no-overbidding Thomas Kesselheim 13/38

Bound is tight Two bidders, two items v 1 ( { 1 } ) = 2, v 1 ( { 2 } ) = 1, v 1 ( { 1 , 2 } ) = 2 0 1 1 0 v 2 ( { 1 } ) = 1, v 2 ( { 2 } ) = 2, v 2 ( { 1 , 2 } ) = 2 b 1 , 1 = 0, b 1 , 2 = 1 b 2 , 1 = 1, b 2 , 2 = 0 1 2 SW ( b ) = 2, OPT ( v ) = 4 Thomas Kesselheim 14/38

Further Results Roughgarden, STOC 2009, Syrgkanis/Tardos, STOC 2013, . . . : General smoothness framework for Price of Anarchy Bhawalkar/Roughgarden, SODA 2011: Subadditive valuations: PoA = 2 for pure Nash, PoA = O ( log m ) via smoothness Feldman/Fu/Gravin/Lucier, STOC 2013: Subadditive valuations: constant PoA for Bayes-Nash equilibria, not a smoothness proof More results on simultaneous first-price auctions, generalized second price, greedy auctions, . . . Thomas Kesselheim 15/38

Outline 1 Price of Anarchy 2 Complexity of Equilibria Best-Response Dynamics 3 Open Problems 4 Thomas Kesselheim 16/38

Complexity of Equilibria (1/3) [Dobzinski/Fu/Kleinberg, SODA 2015] Submodular valuations: Computing an equilibrium with good welfare is essentially as easy as computing an allocation with good welfare. Subadditive valuations: Computing an equilibrium requires exponential communication. XOS valuations: “If there exists an efficient algorithm that finds an equilibrium, it must use techniques that are very different from our current ones.” Thomas Kesselheim 17/38

Complexity of Equilibria (2/3) [Cai/Papadimitriou, EC 2014] One unit-demand bidder, others additive: Computing Bayes-Nash equilibrium in such auctions is PP-hard Finding an approximate Bayes-Nash equilibrium is NP-hard Recognizing a Bayes-Nash equilibrium is intractable Thomas Kesselheim 18/38

Complexity of Equilibria (3/3) [Daskalakis/Syrgkanis, FOCS 2016] Unit-demand valuations: There are no polynomial-time no-regret learning algorithms, unless RP ⊇ NP Reason: Huge strategy spaces Alternative concept: No-envy learning. Only decide which items to buy but not the bids Thomas Kesselheim 19/38

Outline 1 Price of Anarchy 2 Complexity of Equilibria Best-Response Dynamics 3 Open Problems 4 Thomas Kesselheim 20/38

Best-Response Dynamics b i is best response to b − i if u i ( b i , b − i ) ≥ u i ( b ′ for all b ′ i , b − i ) i Best-Response Dynamics with Round-Robin Activation Activate bidders in order 1 , 2 , . . . , n , 1 , 2 , . . . , n , 1 , 2 , . . . Every bidder switches to a best response Best responses usually not unique: Two bidders, one item. If b 1 = 1 and v 2 = 2, then every b 2 > 1 is a best response to b 1 Thomas Kesselheim 21/38

Potential Procedure [Christodoulou/Kov´ acs/Schapira, JACM 2016] All valuation functions are XOS, that is, j ∈ S v ℓ i , j for some v ℓ v i ( S ) = max ℓ � i , j ≥ 0 When bidder i gets activated: Determine S that maximizes v i ( S ) − � j ∈ S max k � = i b k , j j ∈ S v ℓ Let ℓ be such that v i ( S ) = � i , j . b i , j = v ℓ i , j if j ∈ S and 0 otherwise Note: Updates fulfill strong no-overbidding: j ∈ S b t For every S ⊆ M and every i and t : � i , j ≤ v i ( S ) . Thomas Kesselheim 22/38

Potential Procedure: Convergence [Christodoulou/Kov´ acs/Schapira, JACM 2016] Theorem The Potential Procedure reaches a fixed point (pure Nash equilibrium) after finitely many steps. Thomas Kesselheim 23/38

Core Lemma Define declared welfare : DW ( b ) = � j ∈ M max i ∈ N b i , j . Lemma If i makes an improvement step from b t to b t + 1 , then DW ( b t + 1 ) − DW ( b t ) ≥ u i ( b t + 1 ) − u i ( b t ) . Proof. Suppose i previously won set S , now wins S ′ . j ∈ S ′ b t + 1 j ∈ S b t = v i ( S ′ ) By choice of updates: � i , j ≤ v i ( S ) � i , j DW ( b t + 1 ) − DW ( b t ) j ∈ S ′ ( b t + 1 − max i ′ � = i b t + 1 i , j − max i ′ � = i b t + 1 = � i ′ , j ) − � j ∈ S ( b t i ′ , j ) i , j � � j ∈ S ′ max i ′ � = i b t + 1 j ∈ S max i ′ � = i b t + 1 ≥ v i ( S ′ ) − � i ′ , j − v i ( S ) − � i ′ , j = u i ( b t + 1 ) − u i ( b t ) Thomas Kesselheim 24/38

Potential Procedure: Convergence [Christodoulou/Kov´ acs/Schapira, JACM 2016] Theorem The Potential Procedure reaches a fixed point (pure Nash equilibrium) after finitely many steps. Proof. Define declared welfare : DW ( b ) = � j ∈ M max i ∈ N b i , j . If i makes an improvement step from b t to b t + 1 , then DW ( b t + 1 ) − DW ( b t ) ≥ u i ( b t + 1 ) − u i ( b t ) . Every increase in utility is lower-bounded by some ǫ > 0. Thomas Kesselheim 25/38