Price of anarchy in auctions & the smoothness framework Faidra - - PowerPoint PPT Presentation

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 The strategy profile (stay, swerve) is a mutual best response, a Nash equilibrium.

• pure strategies : correspond directly to actions in the game
• mixed strategies : are randomizations over actions in the game

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

• f the other players as given by the strategy profile

stay swerve stay (-10,-10) (1,-1) swerve (-1,1) (0,0)

INCOMPLETE INFORMATION GAMES (AUCTIONS)

• Each agent has some private information (agent’s valuation

𝑤𝑗) and this information affects the payoff of this agent in the game.

• strategy in a incomplete information auction = a function 𝑐𝑗 ∙

that maps an agent’s type to any bid of the agent’s possible bidding actions in the game

str trategy

𝑤𝑗

𝑐𝑗(∙) 𝑐𝑗(𝑤𝑗)

valuation bid

Example: Second Price Auction A strategy of player 𝑗 maps valuation to bid bi(vi) = "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 𝑤𝑗 ~𝐺𝒋 for the item.
• distribution 𝑮 is known and valuations 𝑤𝑗 are drawn independently

First Price Auction

• 1. the auction winner is the maximum bidder
• 2. the winner pays his bid

Player’s goal: maximize utility = valuation−price paid

F is the product distribution 𝑮 ≡ 𝐺

1 × ⋯ × 𝐺 𝑜

Then, 𝑮−𝑗|𝑤𝑗 = 𝑮−i

FIRST PRICE AUCTION: Symmetric Two bidders, independent valuations with uniform distribution U([0,1]) If the cat bids half her value, how should you bid?

value 𝑤1, bid 𝑐1 value 𝑤2, bid 𝑐2

Your expected utility: 𝐅 𝑣1 = 𝑤1 − 𝑐1 ∙ 𝐐 𝑧𝑝𝑣 𝑥𝑗𝑜 𝐐 𝑧𝑝𝑣 𝑥𝑗𝑜 = 𝐐 𝑐2 ≤ 𝑐1 = 2𝑐1 ⇒ 𝐅 𝑣1 = 2𝑤1𝑐1 − 2𝑐1

2

Optimal bid:

𝑒 𝑒𝑐1 𝐅 𝑣1 = 0 ⇒ 𝑐1 = 𝑤1 2 BNE

BAYES-NASH EQUILIBRIUM (BNE) + PRICE OF ANARCHY (PoA) A Bayes-Nash equilibrium (BNE) is a strategy profile where if for all 𝑗 𝑐𝑗(𝑤𝑗) is a best response when other agents play 𝑐−𝑗(𝑤−𝑗) with 𝑤−𝑗 ∼ 𝐆−𝐣|𝑤𝑗 (conditioned on 𝑤𝑗) Price of Anarchy (PoA) = the worst-case ratio between the

• bjective function value of an equilibrium and of an optimal
• utcome

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 )]

• b1 𝑤1 =

𝑤1 2 , b2 𝑤2 = 𝑤2 2 is BNE

• the player with the highest valuation wins in BNE ⇒

first-price auction maximizes social welfare Non-Symmetric Distributions [two bidders 𝑤1~ 𝑉 0,1 , 𝑤2~𝑉( 0,2 )]

• b1 𝑤1 =

2 3𝑤1 2 −

4 − 3𝑤1

2 , b2 𝑤2 = 2 3𝑤2 −2 +

4 + 3𝑤2

2 is BNE

• player 1 may win in cases where 𝑤2 > 𝑤1 ⇒ PoA>1

The smoothness framework

MOTIVATION: Simple and... not-so-simple auctions Simple! Single item second price auction Simple?

How realistic is the assumption that mechanisms run in isolation, as traditional mechanism design has considered? Typical mechanisms used in practice (ex. online markets) are extremely simple and not truthful!

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

• r 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 simple setting X Conclusion for a complex setting Y

proved under restriction P

SMOOTHNESS Smooth auctions An auction game is 𝝁, 𝝂 -smooth if ∃ a bidding strategy 𝐜∗ s.t. ∀𝐜 𝑣𝑗 𝑐𝑗

∗, 𝑐−𝑗 ≥ 𝝁 ⋅ 𝑃𝑄𝑈 − 𝝂 𝑞𝑗(𝐜) 𝑗 𝑗

Smoothness is property of auction not equilibrium!

SMOOTHNESS IMPLIES PoA [PNE]

• Proof. Let 𝐜 : a Nash strategy profile,

𝐜∗: a strategy profile that satisfies smoothness 𝐜 Nash strategy profile ⇒ 𝑣𝑗(𝐜) ≥ 𝑣𝑗(𝑐𝑗

∗ , 𝑐−𝑗)

Summing over all players: 𝑣𝑗 𝐜 ≥

𝑗

𝑣𝑗(𝑐𝑗

∗ , 𝑐−𝑗) 𝑗

By auction smoothness: 𝑣𝑗 𝐜 ≥

𝑗

𝝁 ⋅ 𝑃𝑄𝑈 − 𝝂 𝑞𝑗(𝐜)

𝑗

⇒ 𝑣𝑗 𝐜

𝑗

+ 𝝂 𝑞𝑗(𝐜)

𝑗

≥ 𝝁 ⋅ 𝑃𝑄𝑈 ⇒ max 1, 𝜈 𝑇𝑋 𝐜 ≥ 𝜇 ⋅ 𝑃𝑄𝑈 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

(λ,μ)-smoothness ⇒ 𝑸𝒑𝑩 ≤ max(1, μ) λ

𝑄𝑝𝐵 = 𝑃𝑄𝑈(𝐰) 𝑇𝑋(𝐜)

An auction game is 𝝁, 𝝂 -smooth if ∃ a bidding strategy 𝐜∗ s.t. ∀𝐜 𝑣𝑗 𝑐𝑗

∗, 𝑐−𝑗 ≥ 𝝁 ⋅ 𝑃𝑄𝑈 − 𝝂 𝑞𝑗(𝐜) 𝑗 𝑗

A vector of strategies s is said to be a Nash equilibrium if for each player i and each strategy 𝑡𝑗

′:

𝑣𝑗 𝐭 ≥ 𝑣𝑗 (𝑡𝑗

′, 𝑡−𝑗)

SMOOTHNESS IMPLIES PoA [BNE!] THEOREM : Generalization to Bayesian settings The (𝜇, 𝜈)-smoothness property of an auction (with an 𝐜∗ such that 𝑐𝑗

∗depends only on the value of player i) implies that a Bayes-Nash

equilibrium strategy profile 𝐜 satisfies max 1, 𝜈 𝐅,𝑇𝑋 𝐜 - ≥ 𝜇 ⋅ 𝐅,𝑃𝑄𝑈-

(λ,μ)-smoothness ⇒ 𝑪𝑶𝑭 𝑸𝒑𝑩 ≤ max(1, μ) λ

𝑄𝑝𝐵 = 𝐹 𝑃𝑄𝑈 𝐰 𝐹 𝑇𝑋 𝐜 𝐰

A vector of strategies s is said to be a Bayes-Nash equilibrium if for each player i and each strategy 𝑡𝑗

′, maximizes utility

(conditional on valuation 𝑤𝑗) E𝑤 𝑣𝑗 𝐭 𝑤𝑗 ≥ E𝑤 ,𝑣𝑗 (𝑡𝑗

′ , 𝑡−𝑗 )|𝑤𝑗 -

Complete information PNE to BNE with correlated values:

Extension Theorem 1

“

Conclusion for a simple setting X Conclusion for a complex setting Y

POA extension theorem Complete information Pure Nash Equilibrium

𝐰 = (𝑤1, … , 𝑤𝑜) : common knowledge

Incomplete information Bayes-Nash Equilibrium with asymmetric correlated valuations 𝑄𝑝𝐵 = 𝑃𝑄𝑈(𝐰) 𝑇𝑋(𝐜) 𝑄𝑝𝐵 = 𝐹 𝑃𝑄𝑈 𝐰 𝐹 𝑇𝑋 𝐜 𝐰

FPA AND SMOOTHNESS

• Proof. We’ll prove that 𝑣𝑗

𝑗

𝑐𝑗

∗, 𝑐−𝑗 ≥ 1 2 𝑃𝑄𝑈 − 𝑞𝑗(𝐜) 𝑗

. Let’s try the bidding strategy 𝑐𝑗

∗ = 𝑤𝑗 2.

Maximum valuation bidder: 𝑘 = arg max

𝑗

𝑤𝑗

• If 𝑘 wins, 𝑣𝑘 = 𝑤𝑘 − 𝑐

𝑘 ∗ 𝑤𝑘 = 𝑤𝑘 2 ≥ 1 2 𝑤𝑘 − 𝑞𝑗(𝐜) 𝑗

• If 𝑘 loses, 𝑣𝑘 = 0, and 𝑞𝑗(𝐜)

𝑗

= max

𝑗

𝑐𝑗 >

1 2 𝑤𝑘

⇒ 𝑣𝑘 = 0 >

1 2 𝑤𝑘 − 𝑞𝑗(𝐜) 𝑗

. For all other bidders 𝑗 ≠ 𝑘 : 𝑣𝑗 𝑐𝑗

∗, 𝑐−𝑗 ≥ 0.

Summing up over all players we get 𝑣𝑗(𝑐𝑗

∗, 𝑐−𝑗) 𝑗

≥ 1 2 𝑤𝑘 − 𝑞𝑗 𝐜

𝑗

= 1 2 𝑃𝑄 𝑈 − 𝑞𝑗

𝑗

𝐜

LEMMA

First Price Auction (complete information) of a single item is (

𝟐 𝟑 , 𝟐)-smooth

An auction game is 𝝁, 𝝂 -smooth if ∃ a bidding strategy 𝐜∗ s.t. ∀𝐜 𝑣𝑗 𝑐𝑗

∗, 𝑐−𝑗 ≥ 𝝁 ⋅ 𝑃𝑄𝑈 − 𝝂 𝑞𝑗(𝐜) 𝑗 𝑗

COMPLETE INFORMATION FIRST PRICE AUCTION : PNE & Complete Information Proof. Each bidder 𝑗 can deviate to 𝑐𝑗 =

𝑤𝑗 2 .

We prove that 𝑇𝑋(𝐜) ≥

1 2 𝑃𝑄𝑈(𝐰).

LEMMA

Complete Information First Price Auction of a single item has PoA ≤ 2 𝑄𝑝𝐵 = 𝑃𝑄𝑈(𝐰) 𝑇𝑋(𝐜) Complete Information First Price Auction of a single item has PoA =1. But…

“

First Extension Theorem

• 1. Prove smoothness property of simple setting
• 2. Prove PoA of simple setting via own-based

deviations

• 3. Use Extension Theorem to prove PoA bound
• f target setting

FPA (complete info) is (𝟐 −

𝟐 𝒇 , 𝟐)-smooth

FPA (complete info) has PoA ≤ 2

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

𝑸𝒑𝑩 ≤ 𝒇 𝒇 − 𝟐 ≈ 𝟐. 𝟔𝟗

The Composition Framework:

Extension Theorem 2

𝒘𝟑 = $𝟔 𝒘𝟐 = $𝟐𝟏𝟏 𝒘𝟓 = $𝟘 𝒘𝟒 = $𝟖 Simple setting. Single-item first price auction (complete information PNE). Target setting. Simultaneous single-item first price auctions with unit-demand bidders (complete information PNE). Uni Unit-Demand Val aluation 𝒘𝒋 𝑻 = 𝐧𝐛𝐲

𝒌∈𝑻 𝒘𝒋 𝒌

𝒘𝒋

𝟒

𝒘𝒋

𝟐

𝒘𝒋

𝟑

𝒄𝒋

𝟒

𝒄𝒋

𝟐

𝒄𝒋

𝟑

Can we derive global efficiency guarantees from local (1/2, 1)-smoothness of each first price auction?

FROM SIMPLE LOCAL SETTING TO TARGET GLOBAL SETTING EXTENSION THEOREM 2

PNE PoA bound of 1-item auction ⇒PNE PoA bound of simultaneous auctions based on proving smoothness

Proof sketch. Prove smoothness of the global mechanism!  Global deviation: Pick your item in the optimal allocation and perform the smoothness deviation for your local value 𝑤𝑗

𝑘, i.e. bi ∗ = 𝑤𝑗 𝑘/2.

 Smoothness locally: 𝑣𝑗 𝑐𝑗

∗, 𝑐−𝑗 ≥ 𝑤𝑗

𝑘

2 − 𝑞𝑘𝑗

∗

 Sum over players: 𝑣𝑗 𝑐𝑗

∗, 𝐜−𝐣 𝑗

≥

𝟐 𝟑 ⋅ 𝑃𝑄𝑈 𝐰 − 𝑆𝐹𝑊 𝐜

 (1/2, 1)-smoothness property globally

𝟏 𝑤𝑗

𝑘/2

𝟏

𝑘𝑗

∗

The Composition Framework:

Extension Theorem 3

FROM SIMPLE LOCAL SETTING TO TARGET GLOBAL SETTING EXTENSION THEOREM 3 If PNE PoA of single-item auction proved via (𝜇, 𝜈)-smoothness via valuation profile dependent deviation,

⇒ then BNE PoA bound of simultaneous auctions with submodular and

independent valuations also max*𝜈, 1+/𝜇 BNE PoA of simultaneous first price auctions with submodular and independent bidders ≤

𝑓 𝑓−1

Let f be a set function. f is submodular iff 𝑔(𝑇) + 𝑔(𝑈) ≥ 𝑔(𝑇 ∪ 𝑈) + 𝑔(𝑇 ∩ 𝑈)

SUMMARY  X: complete information PNE ⇒ Y: incomplete information BNE  X: single auction ⇒ Y: composition of auctions

• Applies to "any" auction, not only first price auction.
• Also true for sequential auctions.

Conclusion for a simple setting X Conclusion for a complex setting Y

proved under restrictions Smooth auctions An auction game is 𝝁, 𝝂 -smooth if ∃ a bidding strategy 𝐜∗ s.t. ∀𝐜 𝑣𝑗 𝑐𝑗

∗, 𝑐−𝑗 ≥ 𝝁 ⋅ 𝑃𝑄𝑈 − 𝝂 𝑞𝑗(𝐜) 𝑗 𝑗

“

The Composition Framework

Simultaneous Composition of 𝒏 Mechanisms Suppose that

• each mechanism 𝑁

𝑘 is (𝜇, 𝜈) -smooth

• the valuation of each player across mechanisms is XOS.

Then the global mechanism is (𝜇, 𝜈) -smooth. Sequential Composition of 𝒏 Mechanisms Suppose that

• each mechanism 𝑁

𝑘 is (𝜇, 𝜈) -smooth

• the valuation of each player comes from his best mechanism’s outcome

𝑤𝑗 𝑦𝑗 = max

𝑘

𝑤𝑗𝑘 (𝑦𝑗𝑘). Then the global mechanism is (𝜇, 𝜈 + 1) −smooth, independent of the information released to players during the sequential rounds.

We can combine these two theorems to prove efficiency guarantees when mechanisms are run in a sequence of rounds and at each round several mechanisms are run simultaneously.

Applications

APPLICATIONS  m simultaneous first price auctions and bidders have budgets and fractionally subadditive valuations ⇒ BNE achieves at least

𝑓−1 𝑓 ≈ 0.63 of the expected optimal effective welfare

 Generalized First-Price Auction: 𝑜 bidders, 𝑛 slots. We allocate slots by bid and charge bid per-click. Bidder’s utility: 𝑣𝑗 𝐜 = 𝑏𝜏 𝑗 𝑤𝑗 − 𝑐𝑗 BNE 𝑄𝑝𝐵 ≤ 2  Public Goods Auctions: 𝑜 bidders, 𝑛 public projects. Choose a single public project to implement . Each player 𝑗 has a value 𝑤𝑗𝑘 if project 𝑘 is implemented

Effective Welfare

𝐹𝑋(𝑦) = min *𝑤𝑗(𝑦𝑗), 𝐶𝑗+

𝑗

APPLICATIONS  m simultaneous with budgets/sequential bandwidth allocation mechanisms  Second Price Auction weakly smooth mechanism (λ, μ1, μ2) + willingness-to-pay  All-pay auction - proof similar to FPA

REFERENCES  WINE 2013 Tutorial: Price of Anarchy in Auctions, by Jason Hartline and Vasilis Syrgkanis http://wine13.seas.harvard.edu/tutorials/  Hartline, J.D., 2012. Approximation in economic

• design. Lecture Notes.

 Syrgkanis, V. and Tardos, E., 2013. Composable and efficient mechanisms. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing (pp. 211-220). ACM.  Roughgarden, T., 2012. The price of anarchy in games of incomplete information. In Proceedings of the 13th ACM Conference on Electronic Commerce (pp. 862-879). ACM.  Syrgkanis, V. and Tardos, E., 2012. Bayesian sequential

• auctions. In Proceedings of the 13th ACM Conference on

Electronic Commerce (pp. 929-944). ACM.

REFERENCES  Lucier, B. and Paes Leme, R., 2011. GSP auctions with correlated types. In Proceedings of the 12th ACM conference

• n Electronic commerce(pp. 71-80). ACM.

 Roughgarden, T., 2009. Intrinsic robustness of the price of

• anarchy. InProceedings of the forty-first annual ACM

symposium on Theory of computing (pp. 513-522). ACM.  Leme, R.P., Syrgkanis, V. and Tardos, É., 2012. The curse of

• simultaneity. In Proceedings of the 3rd Innovations in

Theoretical Computer Science Conference (pp. 60-67). ACM.  Krishna, V., 2009. Auction theory. Academic press.