Price of anarchy in auctions & the smoothness framework Faidra Monachou Algorithmic Game Theory 2016 CoReLab, NTUA
Introduction: The price of anarchy in auctions
COMPLETE INFORMATION GAMES Example : Chicken game stay swerve stay (-10,-10) (1,-1) swerve (-1,1) (0,0) The strategy profile (stay, swerve) is a mutual best response, a Nash equilibrium. A Nash equilibrium in a game of complete information is a strategy profile where each player’s strategy is a best response to the strategies of the other players as given by the strategy profile pure strategies : correspond directly to actions in the game mixed strategies : are randomizations over actions in the game
INCOMPLETE INFORMATION GAMES (AUCTIONS) Each agent has some private information (agent’s valuation 𝑤 𝑗 ) and this information affects the payoff of this agent in the game. strategy in a incomplete information auction = a function 𝑐 𝑗 ∙ that maps an agent’s type to any bid of the agent’s possible bidding actions in the game str trategy 𝑐 𝑗 (∙) 𝑐 𝑗 (𝑤 𝑗 ) 𝑤 𝑗 valuation bid Example : Second Price Auction A strategy of player 𝑗 maps valuation to bid b i (v i ) = "bid vi ” *This strategy is also truthful.
FIRST PRICE AUCTION OF A SINGLE ITEM a single item to sell 𝑜 players - each player 𝑗 has a private valuation 𝑤 𝑗 ~𝐺 𝒋 for the item. distribution 𝑮 is known and valuations 𝑤 𝑗 are drawn independently F is the product distribution First Price Auction 𝑮 ≡ 𝐺 1 × ⋯ × 𝐺 𝑜 1. the auction winner is the maximum bidder 2. the winner pays his bid Then, 𝑮 −𝑗 |𝑤 𝑗 = 𝑮 −i Player’s goal : maximize utility = valuation−price paid
FIRST PRICE AUCTION: Symmetric Two bidders, independent valuations with uniform distribution U([0,1]) value 𝑤 1 , bid 𝑐 1 value 𝑤 2 , bid 𝑐 2 If the cat bids half her value , how should you bid? Your expected utility: 𝐅 𝑣 1 = 𝑤 1 − 𝑐 1 ∙ 𝐐 𝑧𝑝𝑣 𝑥𝑗𝑜 2 𝐐 𝑧𝑝𝑣 𝑥𝑗𝑜 = 𝐐 𝑐 2 ≤ 𝑐 1 = 2𝑐 1 ⇒ 𝐅 𝑣 1 = 2𝑤 1 𝑐 1 − 2𝑐 1 𝑒 𝑤 1 𝑒𝑐 1 𝐅 𝑣 1 = 0 ⇒ 𝑐 1 = Optimal bid: 2 BNE
BAYES-NASH EQUILIBRIUM ( BNE ) + PRICE OF ANARCHY ( PoA ) A Bayes-Nash equilibrium (BNE) is a strategy profile where if for all 𝑗 𝑐 𝑗 (𝑤 𝑗 ) is a best response when other agents play 𝑐 −𝑗 (𝑤 −𝑗 ) with 𝑤 −𝑗 ∼ 𝐆 −𝐣 |𝑤 𝑗 (conditioned on 𝑤 𝑗 ) Price of Anarchy (PoA) = the worst-case ratio between the objective function value of an equilibrium and of an optimal outcome Example of an auction objective function: Social welfare = the valuation of the winner
FIRST PRICE AUCTION: Symmetric vs Non-Symmetric Symmetric Distributions [two bidders 𝑉( 0,1 ) ] 𝑤 1 𝑤 2 b 1 𝑤 1 = 2 , b 2 𝑤 2 = 2 is BNE the player with the highest valuation wins in BNE ⇒ first -price auction maximizes social welfare Non-Symmetric Distributions [two bidders 𝑤 1 ~ 𝑉 0,1 , 𝑤 2 ~𝑉( 0,2 ) ] 2 2 2 , b 2 𝑤 2 = 2 is BNE b 1 𝑤 1 = 3𝑤 1 2 − 4 − 3𝑤 1 3𝑤 2 −2 + 4 + 3𝑤 2 player 1 may win in cases where 𝑤 2 > 𝑤 1 ⇒ PoA>1
The smoothness framework
MOTIVATION: Simple and... not-so-simple auctions Simple! Single item second price auction Simple? Typical mechanisms used in practice (ex. online markets) are extremely simple and not truthful ! How realistic is the assumption that mechanisms run in isolation , as traditional mechanism design has considered?
COMPOSITION OF MECHANISMS Simultaneous Composition of 𝒏 Mechanisms The player reports a bid at each mechanism 𝑁 𝑘 Sequential Composition of 𝒏 Mechanisms The player can base the bid he submits at mechanism 𝑁 𝑘 on the history of the submitted bids in previous mechanisms.
“ Reducing analysis of complex setting to simple setting. How to design mechanisms so that the efficiency guarantees for a single mechanism (when studied in isolation) carry over to the same or approximately the same guarantees for a market composed of such mechanisms? Key question What properties of local mechanisms guarantee global efficiency in a market composed of such mechanisms? Conclusion for a Conclusion for a proved under restriction P simple setting X complex setting Y
SMOOTHNESS Smooth auctions An auction game is 𝝁, 𝝂 -smooth if ∃ a bidding strategy 𝐜 ∗ s.t. ∀𝐜 ∗ , 𝑐 −𝑗 ≥ 𝝁 ⋅ 𝑃𝑄𝑈 − 𝝂 𝑞 𝑗 (𝐜) 𝑣 𝑗 𝑐 𝑗 𝑗 𝑗 Smoothness is property of auction not equilibrium!
𝑄𝑝𝐵 = 𝑃𝑄𝑈(𝐰) 𝑇𝑋(𝐜) SMOOTHNESS IMPLIES PoA [ PNE ] ( λ,μ) -smoothness ⇒ 𝑸𝒑𝑩 ≤ max(1, μ) λ THEOREM The (𝜇, 𝜈) -smoothness property of an auction implies that a Nash equilibrium strategy profile 𝐜 satisfies max 1, 𝜈 𝑇𝑋 𝐜 ≥ 𝜇 ⋅ 𝑃𝑄𝑈 Proof. Let 𝐜 : a Nash strategy profile, 𝐜 ∗ : a strategy profile that satisfies smoothness ∗ , 𝑐 −𝑗 ) 𝐜 Nash strategy profile ⇒ 𝑣 𝑗 (𝐜) ≥ 𝑣 𝑗 (𝑐 𝑗 ∗ , 𝑐 −𝑗 ) Summing over all players: 𝑣 𝑗 𝐜 ≥ 𝑣 𝑗 (𝑐 𝑗 𝑗 𝑗 By auction smoothness: 𝑣 𝑗 𝐜 ≥ 𝝁 ⋅ 𝑃𝑄𝑈 − 𝝂 𝑞 𝑗 (𝐜) 𝑗 𝑗 ⇒ 𝑣 𝑗 𝐜 + 𝝂 𝑞 𝑗 (𝐜) ≥ 𝝁 ⋅ 𝑃𝑄𝑈 ⇒ max 1, 𝜈 𝑇𝑋 𝐜 ≥ 𝜇 ⋅ 𝑃𝑄𝑈 𝑗 𝑗 An auction game is 𝝁, 𝝂 -smooth if ∃ A vector of strategies s is said to a bidding strategy 𝐜 ∗ s.t. ∀𝐜 be a Nash equilibrium if for each ′ : player i and each strategy 𝑡 𝑗 ∗ , 𝑐 −𝑗 ≥ 𝝁 ⋅ 𝑃𝑄𝑈 − 𝝂 𝑞 𝑗 (𝐜) 𝑣 𝑗 𝑐 𝑗 ′ , 𝑡 −𝑗 ) 𝑣 𝑗 𝐭 ≥ 𝑣 𝑗 (𝑡 𝑗 𝑗 𝑗
SMOOTHNESS IMPLIES PoA [ BNE! ] 𝐹 𝑃𝑄𝑈 𝐰 𝑄𝑝𝐵 = 𝐹 𝑇𝑋 𝐜 𝐰 ( λ,μ) -smoothness ⇒ 𝑪𝑶𝑭 𝑸𝒑𝑩 ≤ max(1, μ) λ THEOREM : Generalization to Bayesian settings The (𝜇, 𝜈) -smoothness property of an auction (with an 𝐜 ∗ such that ∗ depends only on the value of player i) implies that a Bayes-Nash 𝑐 𝑗 equilibrium strategy profile 𝐜 satisfies max 1, 𝜈 𝐅,𝑇𝑋 𝐜 - ≥ 𝜇 ⋅ 𝐅,𝑃𝑄𝑈- A vector of strategies s is said to be a Bayes-Nash equilibrium ′ , maximizes utility if for each player i and each strategy 𝑡 𝑗 (conditional on valuation 𝑤 𝑗 ) ′ , 𝑡 −𝑗 )|𝑤 𝑗 - E 𝑤 𝑣 𝑗 𝐭 𝑤 𝑗 ≥ E 𝑤 ,𝑣 𝑗 (𝑡 𝑗
Complete information PNE to BNE with correlated values: Extension Theorem 1
“ POA extension Conclusion for a Conclusion for a simple setting X theorem complex setting Y Complete information Incomplete information Pure Nash Equilibrium Bayes-Nash Equilibrium 𝐰 = (𝑤 1 , … , 𝑤 𝑜 ) : common knowledge with asymmetric correlated 𝑄𝑝𝐵 = 𝑃𝑄𝑈(𝐰) valuations 𝑇𝑋(𝐜) 𝐹 𝑃𝑄𝑈 𝐰 𝑄𝑝𝐵 = 𝐹 𝑇𝑋 𝐜 𝐰
An auction game is 𝝁, 𝝂 -smooth if ∃ a bidding strategy 𝐜 ∗ s.t. ∀𝐜 ∗ , 𝑐 −𝑗 ≥ 𝝁 ⋅ 𝑃𝑄𝑈 − 𝝂 𝑞 𝑗 (𝐜) 𝑣 𝑗 𝑐 𝑗 FPA AND SMOOTHNESS 𝑗 𝑗 LEMMA 𝟐 First Price Auction (complete information) of a single item is ( 𝟑 , 𝟐) -smooth 1 ∗ , 𝑐 −𝑗 ≥ Proof. We’ll prove that 𝑣 𝑗 2 𝑃𝑄𝑈 − 𝑞 𝑗 (𝐜) 𝑐 𝑗 . 𝑗 𝑗 ∗ = 𝑤 𝑗 Let’s try the bidding strategy 𝑐 𝑗 2 . Maximum valuation bidder: 𝑘 = arg max 𝑤 𝑗 𝑗 ∗ 𝑤 𝑘 = 𝑤 𝑘 1 2 𝑤 𝑘 − 𝑞 𝑗 (𝐜) If 𝑘 wins, 𝑣 𝑘 = 𝑤 𝑘 − 𝑐 2 ≥ 𝑘 𝑗 1 If 𝑘 loses, 𝑣 𝑘 = 0 , and 𝑞 𝑗 (𝐜) = max 𝑐 𝑗 > 2 𝑤 𝑘 𝑗 𝑗 1 ⇒ 𝑣 𝑘 = 0 > 2 𝑤 𝑘 − 𝑞 𝑗 (𝐜) . 𝑗 ∗ , 𝑐 −𝑗 ≥ 0 . For all other bidders 𝑗 ≠ 𝑘 : 𝑣 𝑗 𝑐 𝑗 Summing up over all players we get ≥ 1 = 1 ∗ , 𝑐 −𝑗 ) 𝑣 𝑗 (𝑐 𝑗 2 𝑤 𝑘 − 𝑞 𝑗 𝐜 2 𝑃𝑄 𝑈 − 𝑞 𝑗 𝐜 𝑗 𝑗 𝑗
COMPLETE INFORMATION FIRST PRICE AUCTION : PNE & Complete Information LEMMA Complete Information First Price Auction of a single item has PoA ≤ 2 𝑄𝑝𝐵 = 𝑃𝑄𝑈(𝐰) Proof. 𝑤 𝑗 𝑇𝑋(𝐜) Each bidder 𝑗 can deviate to 𝑐 𝑗 = 2 . 1 We prove that 𝑇𝑋(𝐜) ≥ 2 𝑃𝑄𝑈(𝐰) . Complete Information First Price Auction of a single item has PoA =1. But…
“ First Extension Theorem 𝟐 FPA (complete info) is (𝟐 − 𝒇 , 𝟐) -smooth 1. Prove smoothness property of simple setting 2. Prove PoA of simple setting via own-based deviations FPA (complete info) has PoA ≤ 2 3. Use Extension Theorem to prove PoA bound 𝒇 𝑸𝒑𝑩 ≤ 𝒇 − 𝟐 ≈ 𝟐. 𝟔𝟗 of target setting EXTENSION THEOREM 1 PNE PoA proved by showing 𝜇, 𝜈 − smoothness property via own-value deviations ⇒ PoA bound of BNE with correlated values max*𝜈,1+ λ
The Composition Framework: Extension Theorem 2
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