On the E ffi ciency of the Walrasian Mechanism Moshe Babaio ff Brendan Lucier (Microsoft Research) (Microsoft Research) Noam Nisan Renato Paes Leme (Microsoft Research (Google Research) and Hebrew University)
First Welfare Theorem • If there exist prices in the market such that no good is under- or over-demanded, then those prices implement an e ffi cient allocation. • Given some natural conditions ( gross substitutability ), such prices always exist. • Those prices can be found via very natural distributed greedy algorithms.
First Welfare Theorem • Set of agent and goods M = { 1 ...m } N = { 1 ...n } v i : 2 M → R + • Each agent has a valuation • Demands: for prices , each agent purchases his p ∈ R M + favorite bundle: D ( v i ; p ) = argmax S [ v i ( S ) − P j ∈ S p j ] p ∈ R M • Walrasian prices s.t. there exist S i ∈ D ( v i ; p ) + that clear the market. • E ffi ciency: in a WE, the welfare is maximized. P i v i ( S i )
First Welfare Theorem • [Kelso-Crawford] If the valuations are gross substitutes , a Walrasian equilibrium always exists and can be found via tatonnement (trial-and-error). p ∈ R M ‣ Fix arbitrary prices: + ‣ Compute demands S i ∈ D ( v i ; p ) ‣ For every if j ∈ N ‣ is not demanded p j ← p j · (1 − ✏ ) j ‣ is over-demanded p j ← p j · (1 + ✏ ) j
First Welfare Theorem • [Kelso-Crawford] If the valuations are gross substitutes , a Walrasian equilibrium always exists and can be found via tatonnement (trial-and-error). p ∈ R M ‣ Fix arbitrary prices: + truthfully reporting ‣ Compute demands S i ∈ D ( v i ; p ) preferences ‣ For every if j ∈ N ‣ is not demanded p j ← p j · (1 − ✏ ) j ‣ is over-demanded p j ← p j · (1 + ✏ ) j
First Welfare Theorem • [Kelso-Crawford] If the valuations are gross substitutes , a Walrasian equilibrium always exists and can be found via tatonnement (trial-and-error). p ∈ R M ‣ Fix arbitrary prices: + truthfully reporting ‣ Compute demands S i ∈ D ( v i ; p ) preferences ‣ For every if j ∈ N ‣ is not demanded p j ← p j · (1 − ✏ ) j ‣ is over-demanded p j ← p j · (1 + ✏ ) j
Large vs Small Markets Bargaining / Haggling Price taking behavior
Large vs Small Markets Bargaining / Haggling Price taking behavior Strategic demand
Our goal : prove welfare theorems with strategic agents
Related Work • [Hurwicz’72]: observes that market equilibrium is not strategy-proof and proposes a game-theoretic framework to analyze its equilibrium properties. • [Rutischi, Sattherwaite, Williams, Econometrica’94] [Sattherwaite, Williams, Econometrica’02]: observe that many markets use variations of market clearing, such as stock exchange opening price or call market for copper and gold and observe: “Such behavior, which is the essence of bargaining, may lead to an impasse that delays or lesses the gains of trade”
Related Work • [Jackson, Manelli], [Otani, Sicilian], [Roberts, Postlewaite], [Azevedo, Budish]: for large markets and suitable regularity conditions, the Walrasian mechanism is approximately strategyproof. • Here : Approximate version of the fi rst welfare theorem without any large market or regularity assumptions.
Related Work • Our perspective: PoA of Auctions [Christodoulou, Kovacs and Shapira] and follow up work… • Also on strategic aspects of markets: [Markakis, Telelis], [de Keijze, Markakis, Shafer, Telelis], [Adsul, Babu, Garg, Mehta, Sohoni], [Chen, Deng, Zhang, Zhang], [Zhang], …
Hurwicz Framework v i : 2 M → R + • Each agent has a valuation • … but reports (bid) b i : 2 M → R + • compute allocation and prices according to a Walrasian p ∈ R M + , { S i } i equilibrium of the reported market. u i = v i ( S i ) − P • Utilities: j ∈ S i p j • Welfare: W = P i v i ( S i )
Hurwicz Framework v i : 2 M → R + • Each agent has a valuation • … but reports (bid) b i : 2 M → R + • compute allocation and prices according to a Walrasian p ∈ R M + , { S i } i equilibrium of the reported market. u i = v i ( S i ) − P • Utilities: j ∈ S i p j • Welfare: W = P i v i ( S i )
Example 1.1 1.1 1.1 2 2 4
Example 1.1 1.1 1.1 2 2 4 u = 1.8 prices 1.1 1.1
Example 1.1 1.1 1.1 2 2 4 2 0 2
Example 1.1 1.1 1.1 2 2 4 2 0 2 u = 1.8 2 prices 0 0
Assumption : - exposure: Payment( b i , b − i ) ≤ (1 + γ ) v i ( S i ) , ∀ b − i γ Main Theorem : If agent values and bids are Gross substitutes and agents employ -exposure strategies, then for all Nash γ equilibria of (any fl avor of) the Walrasian mechanism: 1 P i v i ( S i ) ≥ P i v i ( S ∗ i ) 4+2 γ • guarantees also hold for the (correlated) Bayesian setting • existence of e ffi cient pure 0-exposure equilibria (PoS = 1) • lower bound of 2 for 0-exposure
Assumption : - exposure: Payment( b i , b − i ) ≤ (1 + γ ) v i ( S i ) , ∀ b − i γ Main Theorem : If agent values and bids are Gross substitutes and agents employ -exposure strategies, then for all Nash γ equilibria of (any fl avor of) the Walrasian mechanism any declared welfare maximizer mechanism : � 1 P i v i ( S i ) ≥ P i v i ( S ∗ i ) 4+2 γ
Assumption : - exposure: Payment( b i , b − i ) ≤ (1 + γ ) v i ( S i ) , ∀ b − i γ Main Theorem : If agent values and bids are Gross substitutes and agents employ -exposure strategies, then for all Nash γ equilibria of (any fl avor of) the Walrasian mechanism any declared welfare maximizer mechanism : � 1 P i v i ( S i ) ≥ P i v i ( S ∗ i ) 4+2 γ � � A mechanism is a declared welfare maximizer if it chooses P i b i ( S i ) { S i } an allocation maximizing and charges payments no larger then the declared value of that set.
Assumption : - exposure: Payment( b i , b − i ) ≤ (1 + γ ) v i ( S i ) , ∀ b − i γ Main Theorem : If agent values and bids are Gross substitutes and agents employ -exposure strategies, then for all Nash γ equilibria of (any fl avor of) the Walrasian mechanism any declared welfare maximizer mechanism : � 1 P i v i ( S i ) ≥ P i v i ( S ∗ i ) 4+2 γ � � A mechanism is a declared welfare maximizer if it chooses P i b i ( S i ) { S i } an allocation maximizing and charges payments no larger then the declared value of that set. � Examples: Walrasian mechanism, VCG, Pay-Your-Bid, …
Assumption : - exposure: Payment( b i , b − i ) ≤ (1 + γ ) v i ( S i ) , ∀ b − i γ Main Theorem : If agent values and bids are Gross substitutes and agents employ -exposure strategies, then for all Nash γ equilibria of (any fl avor of) the Walrasian mechanism any declared welfare maximizer mechanism : � 1 P i v i ( S i ) ≥ P i v i ( S ∗ i ) 4+2 γ � � A mechanism is a declared welfare maximizer if it chooses P i b i ( S i ) { S i } an allocation maximizing and charges payments no larger then the declared value of that set. � Examples: Walrasian mechanism, VCG, Pay-Your-Bid, …
General Theorem : If agent values are in , bids are in , and V B agents employ -exposure strategies, then the Price of Anarchy γ of mechanism is . M PoA M V B PoA declared welfare maximizers 4 + 2 γ GS GS maximizers
General Theorem : If agent values are in , bids are in , and V B agents employ -exposure strategies, then the Price of Anarchy γ of mechanism is . M PoA M V B PoA declared welfare maximizers 4 + 2 γ GS GS maximizers 6 + 4 γ declared welfare maximizers XOS XOS maximizers
General Theorem : If agent values are in , bids are in , and V B agents employ -exposure strategies, then the Price of Anarchy γ of mechanism is . M PoA M V B PoA declared welfare maximizers 4 + 2 γ GS GS maximizers 6 + 4 γ declared welfare maximizers XOS XOS maximizers So far, we considered . However, a simpler bidding B = V language can be useful for various reasons: � • representation / communication • computational e ffi ciency • auction simplicity
General Theorem : If agent values are in , bids are in , and V B agents employ -exposure strategies, then the Price of Anarchy γ of mechanism is . M PoA M V B PoA declared welfare maximizers 4 + 2 γ XOS Add ⊆ B ⊆ Gs maximizers 6 + 4 γ declared welfare maximizers XOS Add ⊆ B ⊆ Xos maximizers e.g. item bidding auctions [CKS], [BR], [FFGL]. We can allow for more expressive, yet still computationally e ffi cient mechanisms, i.e., run the Walrasian mechanism with GS bids, even if valuations are XOS.
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