Optimal Mechanism Design (without Priors) Jason D. Hartline - PowerPoint PPT Presentation

Step 3: Find Optimal Mechanism Step 3: Find Optimal Mechanism from class for distribution. Maximize Auction’s Profit: E b [ � i p i ( b ) − c ( x ( b ))] . Subject to truthfulness: 1. bidder i wins if b i > t i ⇔ x i ( b i ) is a step function. � b i 2. bidder i pays t i x i ( b i ) ⇔ p i ( b i ) = x i ( b i ) b i − 0 x i ( b ) db. Definition: The virtual valuation of a bidder i with value v i ∼ F i is ψ i ( v i ) = v i − 1 − F i ( v i ) f i ( v i ) . Lemma: For x i ( b ) and bids b with joint densify function f : � E b [ p i ( b )] = ψ i ( b i ) x i ( b ) f ( b ) d b . b 12 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Proof of Lemma ˆ pi ( b ) ˜ = Z pi ( bi ) f ( b ) d b E b b Z Z = pi ( bi ) fi ( bi ) f ( b i ) dbid b i b i bi Z bi » – Z Z = xi ( bi ) bi − xi ( b ) db fi ( bi )( b i ) dbid b i b i bi 0 Z h Z bi "Z # Z = xi ( bi ) bifi ( bi ) dbi − xi ( b ) fi ( bi ) dbdbi f ( b i ) d b i bi =0 b =0 b i bi Z h Z h "Z # Z = xi ( bi ) bifi ( bi ) dbi − xi ( b ) fi ( bi ) dbidb f ( b i ) d b i b i bi b =0 bi = b Z h Z h "Z # Z = xi ( bi ) bifi ( bi ) dbi − xi ( b ) fi ( bi ) dbidb f ( b i ) d b i b =0 bi = b b i bi Z h "Z # Z ` ´ = xi ( bi ) bifi ( bi ) dbi − xi ( b ) 1 − Fi ( b ) db f ( b i ) d b i b i bi b =0 " # 1 − Fi ( bi ) Z Z = xi ( bi ) fi ( bi ) f ( b i ) dbid b i bi − b i bi fi ( bi ) Z = ψi ( bi ) xi ( bi ) f ( b ) d b b 13 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Myerson Step 3: Find optimal mechanism. 21 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Myerson Step 3: Find optimal mechanism. Theorem: [Mye-81] Given allocation rule x and bids b with density function f the expected profit is � �� i ψ i ( b i ) x i ( b ) − c ( x ( b )) f ( b ) d b . b Definition: Myerson’s optimal mechanism for distribution F = F 1 × . . . × F n , is Myersion F ( b ) with � i ψ i ( b i ) x ′ i − c ( x ′ ) . x ( b ) = argmax x ′ Theorem: Myersion’s mechanism is optimal and truthful when the ψ i ( · ) s are monotone. Note 1: This applies to any cost function c ( x ) (not just for single-item auction). 21 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Myerson Note 2: For some c ( x ) non-monotone ψ i ( · ) can be ironed to be monotone. 22 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Example: Basic Auction The Basic Auction Problem: Given: • n identical items for sale. • n bidders, bidder i willing to pay at most v i for an item. Design: auction with maximal profit. 23 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Example Recall Theorem: [Mye-81] Given allocation rule x and bids b with density function f the expected profit is � �� i ψ i ( b i ) x i ( b ) − c ( x ( b )) f ( b ) d b . b Recall Example: single-item auction � 0 if � i x i ≤ 1 c ( x ) = ∞ otherwise. Result: • Winner: the bidder with highest ψ i ( b i ) (such that ψ i ( b i ) ≥ 0 ). • Winner’s Payment: argmin b { ψ i ( b ) ≥ ψ j ( b j ) & ψ i ( b ) ≥ 0 } 24 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Example Recall Theorem: [Mye-81] Given allocation rule x and bids b with density function f the expected profit is � �� i ψ i ( b i ) x i ( b ) − c ( x ( b )) f ( b ) d b . b Recall Example: single-item auction � 0 if � i x i ≤ 1 c ( x ) = ∞ otherwise. Result: • Winner: the bidder with highest ψ i ( b i ) (such that ψ i ( b i ) ≥ 0 ). • Winner’s Payment: argmin b { ψ i ( b ) ≥ ψ j ( b j ) & ψ i ( b ) ≥ 0 } • Suppose bids are identical, F i = F j : ⇒ max { b j : j � = i } ∪ { ψ − 1 (0) } 24 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Example • Interpretation: Optimal Auction = Vickrey w/reserve price ψ − 1 (0) . 25 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Example • Interpretation: Optimal Auction = Vickrey w/reserve price ψ − 1 (0) . Definition: opt( F ) = ψ − 1 (0) 25 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Example • Interpretation: Optimal Auction = Vickrey w/reserve price ψ − 1 (0) . Definition: opt( F ) = ψ − 1 (0) = argmax b b (1 − F ( b )) 25 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Other Directions 1. General ironing procedure for arbitrary costs? 2. Agent’s with correlated values. [Ron-03]. 3. Deficits. [CHRSU-04] 4. Iterative Mechanisms. [DRJSK-05] 5. Optimal Mechanism for multi-parameter agents? (needs characterization like [SW-05], related to [RL-05]) 26 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Optimal Mechanism Design without Priors Part II The Market Analysis Metaphor

Motivation Where does known prior come from? 1. previous sales. 2. market analysis. 28 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Motivation Where does known prior come from? 1. previous sales. 2. market analysis. Issues: 1. incentive properties. 2. accuracy. 28 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Motivation Where does known prior come from? 1. previous sales. 2. market analysis. Issues: 1. incentive properties. 2. accuracy. Argument 1: by assuming a known prior we ignore incentive and per- formance issues from obtaining the prior. 28 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Motivation Where does known prior come from? 1. previous sales. 2. market analysis. Issues: 1. incentive properties. 2. accuracy. Argument 1: by assuming a known prior we ignore incentive and per- formance issues from obtaining the prior. Argument 2: (Wilson Doctrine) Mechanisms should be independent of details. 28 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Market Analysis Market Analysis Approach: 1. Market Analysis ⇒ distributional knowledge F = ( F 1 , . . . , F n ) 2. Design mechanism for F : Myersion F Recall Incentive Compatibility: for all i , x i ( b i ) is monotone in b i . Can be arbitraty function of b i ! Insight: use b i for market analysis. 29 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Imperical Distributions Definition: The imperical distribution for b is F b ( x ) = |{ i : b i <x }| ˆ . n Recall: Myersion F ⇒ x F i ( b ) , p F i ( b ) Set x i ( b i ) be the allocation for bidder i in Myersion ˆ F b i 30 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Estimating Distributions Recall: Myerson’s Optimal Auction for bids i.i.d. from F : 1. optimal price = argmax p p (1 − F ( p )) . 2. offer all bidders the optimal price. 31 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Estimating Distributions Recall: Myerson’s Optimal Auction for bids i.i.d. from F : 1. optimal price = argmax p p (1 − F ( p )) . 2. offer all bidders the optimal price. Idea: For bidder i use empirical estimate of F from b i . 31 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Estimating Distributions Recall: Myerson’s Optimal Auction for bids i.i.d. from F : 1. optimal price = argmax p p (1 − F ( p )) . 2. offer all bidders the optimal price. Idea: For bidder i use empirical estimate of F from b i . Definition: The empirical distribution b i is ˆ 1 F b i ( p ) = “number of bids less than p ” × n − 1 . 31 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Deterministic Optimal Price Auction For basic auction problem: 32 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Deterministic Optimal Price Auction For basic auction problem: Deterministic Optimal Price Auction (DOP) [GHW-01,BV-03,Seg-03] On input b , for each bidder i : 1. p ← opt( b i ) . 2. If p ≤ b i , sell to bidder i at price p . 3. Otherwise, reject bidder i . 32 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Deterministic Optimal Price Auction For basic auction problem: Deterministic Optimal Price Auction (DOP) [GHW-01,BV-03,Seg-03] On input b , for each bidder i : 1. p ← opt( b i ) . 2. If p ≤ b i , sell to bidder i at price p . 3. Otherwise, reject bidder i . Theorem: For b i.i.d. from F on range [1 , h ] , profit of DOP approaches optimal profit as n → ∞ . [BV-03,Seg-03] 32 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Deterministic Optimal Price Auction For basic auction problem: Deterministic Optimal Price Auction (DOP) [GHW-01,BV-03,Seg-03] On input b , for each bidder i : 1. p ← opt( b i ) . 2. If p ≤ b i , sell to bidder i at price p . 3. Otherwise, reject bidder i . Theorem: For b i.i.d. from F on range [1 , h ] , profit of DOP approaches optimal profit as n → ∞ . [BV-03,Seg-03] Lemma: Worst-case profit is bad. [GHW-01] 32 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case Analysis of DOP 10 bidders � �� Example: for DOP and b = ( 10 , 10 , . . . , 10 , 1 , 1 , . . . , 1 ) � �� 100 bidders Profit: 10 × + 90 × 33 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case Analysis of DOP 10 bidders � �� Example: for DOP and b = ( 10 , 10 , . . . , 10 , 1 , 1 , . . . , 1 ) � �� 100 bidders Profit: 10 × Revenue from 10 bid + 90 × Revenue from 1 bid 33 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case Analysis of DOP 10 bidders � �� Example: for DOP and b = ( 10 , 10 , . . . , 10 , 1 , 1 , . . . , 1 ) � �� 100 bidders Profit: 10 × Revenue from 10 bid + 90 × Revenue from 1 bid Revenue from 10 bid What does DOP do for b i = 10 ? 9 bidders z }| { opt( b i ) = ( 10 , . . . , 10 , 1 , 1 , . . . , 1 ) | {z } 99 bidders 33 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case Analysis of DOP 10 bidders � �� Example: for DOP and b = ( 10 , 10 , . . . , 10 , 1 , 1 , . . . , 1 ) � �� 100 bidders Profit: 10 × Revenue from 10 bid + 90 × Revenue from 1 bid Is opt( b i ) = 1 or 10 ? Revenue from 10 bid What does DOP do for b i = 10 ? 9 bidders z }| { opt( b i ) = ( 10 , . . . , 10 , 1 , 1 , . . . , 1 ) | {z } 99 bidders 33 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case Analysis of DOP 10 bidders � �� Example: for DOP and b = ( 10 , 10 , . . . , 10 , 1 , 1 , . . . , 1 ) � �� 100 bidders Profit: 10 × Revenue from 10 bid + 90 × Revenue from 1 bid Is opt( b i ) = 1 or 10 ? Revenue from 10 bid • Revenue 10 What does DOP do for b i = 10 ? = 10 × 9 = 90 . 9 bidders z }| { • Revenue 1 opt( b i ) = ( 10 , . . . , 10 , 1 , 1 , . . . , 1 ) | {z } = 1 × 99 = 99 . 99 bidders 33 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case Analysis of DOP 10 bidders � �� Example: for DOP and b = ( 10 , 10 , . . . , 10 , 1 , 1 , . . . , 1 ) � �� 100 bidders Profit: 10 × Revenue from 10 bid + 90 × Revenue from 1 bid � �� 1 Is opt( b i ) = 1 or 10 ? Revenue from 10 bid • Revenue 10 What does DOP do for b i = 10 ? = 10 × 9 = 90 . 9 bidders z }| { • Revenue 1 opt( b i ) = ( 10 , . . . , 10 , 1 , 1 , . . . , 1 ) | {z } = 1 × 99 = 99 . 99 bidders Result: Bidder i buys item at price 1! • Thus, opt( b i ) = 1 . 33 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case Analysis of DOP 10 bidders � �� Example: for DOP and b = ( 10 , 10 , . . . , 10 , 1 , 1 , . . . , 1 ) � �� 100 bidders Profit: 10 × Revenue from 10 bid + 90 × Revenue from 1 bid � �� 1 Is opt( b i ) = 1 or 10 ? Revenue from 10 bid Revenue from 1 bid • Revenue 10 What does DOP do for b i = 1 ? What does DOP do for b i = 10 ? = 10 × 9 = 90 . 9 bidders 10 bidders z z }| }| { { • Revenue 1 opt( b i ) = ( opt( b i ) = ( 10 , 10 , . . . , 10 , 1 , . . . , 1 10 , . . . , 10 , 1 , 1 , . . . , 1 ) ) | | {z {z } } = 1 × 99 = 99 . 99 bidders 99 bidders Result: Bidder i buys item at price 1! • Thus, opt( b i ) = 1 . 33 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case Analysis of DOP 10 bidders � �� Example: for DOP and b = ( 10 , 10 , . . . , 10 , 1 , 1 , . . . , 1 ) � �� 100 bidders Profit: 10 × Revenue from 10 bid + 90 × Revenue from 1 bid � �� 1 Is opt( b i ) = 1 or 10 ? Is opt( b i ) = 1 or 10 ? Revenue from 10 bid Revenue from 1 bid • Revenue 10 What does DOP do for b i = 10 ? What does DOP do for b i = 1 ? = 10 × 9 = 90 . 9 bidders 10 bidders z z }| }| { { • Revenue 1 opt( b i ) = ( opt( b i ) = ( 10 , . . . , 10 , 1 , 1 , . . . , 1 10 , 10 , . . . , 10 , 1 , . . . , 1 ) ) | | {z {z } } = 1 × 99 = 99 . 99 bidders 99 bidders Result: Bidder i buys item at price 1! • Thus, opt( b i ) = 1 . 33 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case Analysis of DOP 10 bidders � �� Example: for DOP and b = ( 10 , 10 , . . . , 10 , 1 , 1 , . . . , 1 ) � �� 100 bidders Profit: 10 × Revenue from 10 bid + 90 × Revenue from 1 bid � �� 1 Is opt( b i ) = 1 or 10 ? Is opt( b i ) = 1 or 10 ? Revenue from 1 bid Revenue from 10 bid • Revenue 10 • Revenue 10 What does DOP do for b i = 10 ? What does DOP do for b i = 1 ? = 10 × 10 = 100 . = 10 × 9 = 90 . 9 bidders 10 bidders z z }| }| { { • Revenue 1 • Revenue 1 opt( b i ) = ( opt( b i ) = ( 10 , 10 , . . . , 10 , 1 , . . . , 1 10 , . . . , 10 , 1 , 1 , . . . , 1 ) ) | | {z {z } } = 1 × 99 = 99 . = 1 × 99 = 99 . 99 bidders 99 bidders Result: Bidder i buys item at price 1! • Thus, opt( b i ) = 1 . 33 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case Analysis of DOP 10 bidders � �� Example: for DOP and b = ( 10 , 10 , . . . , 10 , 1 , 1 , . . . , 1 ) � �� 100 bidders Profit: 10 × Revenue from 10 bid + 90 × Revenue from 1 bid = 10 � �� 1 0 Is opt( b i ) = 1 or 10 ? Is opt( b i ) = 1 or 10 ? Revenue from 1 bid Revenue from 10 bid • Revenue 10 • Revenue 10 What does DOP do for b i = 1 ? What does DOP do for b i = 10 ? = 10 × 10 = 100 . = 10 × 9 = 90 . 9 bidders 10 bidders z z }| }| { { • Revenue 1 • Revenue 1 opt( b i ) = ( opt( b i ) = ( 10 , . . . , 10 , 1 , 1 , . . . , 1 10 , 10 , . . . , 10 , 1 , . . . , 1 ) ) | | {z {z } } = 1 × 99 = 99 . = 1 × 99 = 99 . 99 bidders 99 bidders Result: Bidder i is rejected! Result: Bidder i buys item at price 1! • Thus, opt( b i ) = 1 . • Thus, opt( b i ) = 10 . 33 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

General Consistency Issue Emperical Myerson Auction may be inconsistent Double Auction Problem. 34 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Approximation via Random Sampling Random Sampling Optimal Price Auction, RSOP 1. Randomly partition bids into two sets: b ′ and b ′′ . 2. Use p ′ = opt( b ′ ) as price for b ′′ . b 35 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Approximation via Random Sampling Random Sampling Optimal Price Auction, RSOP 1. Randomly partition bids into two sets: b ′ and b ′′ . 2. Use p ′ = opt( b ′ ) as price for b ′′ . b b ′ b ′′ 35 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Approximation via Random Sampling Random Sampling Optimal Price Auction, RSOP 1. Randomly partition bids into two sets: b ′ and b ′′ . 2. Use p ′ = opt( b ′ ) as price for b ′′ . b b ′ p ′ = opt( b ′ ) b ′′ 35 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Approximation via Random Sampling Random Sampling Optimal Price Auction, RSOP 1. Randomly partition bids into two sets: b ′ and b ′′ . 2. Use p ′ = opt( b ′ ) as price for b ′′ . 3. Use p ′′ = opt( b ′′ ) as price for b ′ (optional). b b ′ p ′ = opt( b ′ ) p ′′ = opt( b ′′ ) b ′′ 35 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Approximation via Random Sampling Random Sampling Optimal Price Auction, RSOP 1. Randomly partition bids into two sets: b ′ and b ′′ . 2. Use p ′ = opt( b ′ ) as price for b ′′ . 3. Use p ′′ = opt( b ′′ ) as price for b ′ (optional). b b ′ p ′ = opt( b ′ ) p ′′ = opt( b ′′ ) b ′′ Theorem: For b on range [1 , h ] , profit of RSOP approaches optimal profit as n → ∞ . 35 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case with Assumption Recall Theorem: For b on range [1 , h ] , profit of RSOP approaches optimal profit as n → ∞ . 36 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case with Assumption Recall Theorem: For b on range [1 , h ] , profit of RSOP approaches optimal profit as n → ∞ . Implicit Assumption: optimal profit ≫ h . 36 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case with Assumption Recall Theorem: For b on range [1 , h ] , profit of RSOP approaches optimal profit as n → ∞ . Implicit Assumption: optimal profit ≫ h . Implicit Definition: optimal profit = “optimal profit from single price sale with bidders’ valuations.” 36 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Worst Case with Assumption Recall Theorem: For b on range [1 , h ] , profit of RSOP approaches optimal profit as n → ∞ . Implicit Assumption: optimal profit ≫ h . Implicit Definition: optimal profit = “optimal profit from single price sale with bidders’ valuations.” Fact: impossible to approximate optimal profit when it is optimal to sell only one unit. E.g., b = (1 , 1 , 1 , 1 , h, 1 , 1) 36 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Consistency Concern: lack of consistency? (bidders offered optimal prices from different empirical distributions) 37 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Consistency Concern: lack of consistency? (bidders offered optimal prices from different empirical distributions) Result: DOP generalization via Myerson-VCG construction fails. 37 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Consistency Concern: lack of consistency? (bidders offered optimal prices from different empirical distributions) Result: DOP generalization via Myerson-VCG construction fails. Recall: Myerson-VCG Construction: 1. Compute each player’s virtual valuation φ ( v i ) = v i − 1 − F ( v i ) . f ( v i ) 2. Run VCG on virtual valuations. 37 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Consistency Concern: lack of consistency? (bidders offered optimal prices from different empirical distributions) Result: DOP generalization via Myerson-VCG construction fails. Recall: Myerson-VCG Construction: 1. Compute each player’s virtual valuation φ ( v i ) = v i − 1 − F ( v i ) . f ( v i ) 2. Run VCG on virtual valuations. Generalized DOP Technique: for each bidder i , 1. Compute virtual valuations using ˆ F b i . 2. Compute outcome of VCG on virtual valuations for bidder i . 37 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Consistency Concern: lack of consistency? (bidders offered optimal prices from different empirical distributions) Result: DOP generalization via Myerson-VCG construction fails. Recall: Myerson-VCG Construction: 1. Compute each player’s virtual valuation φ ( v i ) = v i − 1 − F ( v i ) . f ( v i ) 2. Run VCG on virtual valuations. Generalized DOP Technique: for each bidder i , 1. Compute virtual valuations using ˆ F b i . 2. Compute outcome of VCG on virtual valuations for bidder i . Different empirical distributions ⇒ inconsistency. 37 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

The Double Auction Problem The Double Auction Problem : Given: • n sellers, seller i willing to sell a unit for at least s i . • n buyers, buyer i willing to buy a unit for at most b i . Design: Double auction maximize profit of broker. [BV-03,DGHK-02] 38 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

The Double Auction Problem The Double Auction Problem : Given: • n sellers, seller i willing to sell a unit for at least s i . • n buyers, buyer i willing to buy a unit for at most b i . Design: Double auction maximize profit of broker. [BV-03,DGHK-02] Consistency Constraint: number of winning buyers = number of winning sellers. 38 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

The Double Auction Problem The Double Auction Problem : Given: • n sellers, seller i willing to sell a unit for at least s i . • n buyers, buyer i willing to buy a unit for at most b i . Design: Double auction maximize profit of broker. [BV-03,DGHK-02] Consistency Constraint: number of winning buyers = number of winning sellers. Generalized DOP ⇒ inconsistent. Generalized RSOP ⇒ consistent. 38 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Generalizing RSOP Random Sampling Optimal Price Double Auction, RSOP 1. Randomly partition bids into two sets: b ′ , s ′ and b ′′ , s ′′ 2. Compute virtual valuations for b ′ and s ′ using ˆ F b ′′ and ˆ F s ′′ . 3. Run VCG on virtual valuations of b ′ and s ′ . 39 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Generalizing RSOP Random Sampling Optimal Price Double Auction, RSOP 1. Randomly partition bids into two sets: b ′ , s ′ and b ′′ , s ′′ 2. Compute virtual valuations for b ′ and s ′ using ˆ F b ′′ and ˆ F s ′′ . 3. Run VCG on virtual valuations of b ′ and s ′ . 4. Vice versa. 39 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Generalizing RSOP Random Sampling Optimal Price Double Auction, RSOP 1. Randomly partition bids into two sets: b ′ , s ′ and b ′′ , s ′′ 2. Compute virtual valuations for b ′ and s ′ using ˆ F b ′′ and ˆ F s ′′ . 3. Run VCG on virtual valuations of b ′ and s ′ . 4. Vice versa. Consistency: because both partitions are consistent. 39 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Generalizing RSOP Random Sampling Optimal Price Double Auction, RSOP 1. Randomly partition bids into two sets: b ′ , s ′ and b ′′ , s ′′ 2. Compute virtual valuations for b ′ and s ′ using ˆ F b ′′ and ˆ F s ′′ . 3. Run VCG on virtual valuations of b ′ and s ′ . 4. Vice versa. Consistency: because both partitions are consistent. Theorem: [BV-03] The RSOP double auction approaches optimal profit as n → ∞ . 39 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Generalizing RSOP Random Sampling Optimal Price Double Auction, RSOP 1. Randomly partition bids into two sets: b ′ , s ′ and b ′′ , s ′′ 2. Compute virtual valuations for b ′ and s ′ using ˆ F b ′′ and ˆ F s ′′ . 3. Run VCG on virtual valuations of b ′ and s ′ . 4. Vice versa. Consistency: because both partitions are consistent. Theorem: [BV-03] The RSOP double auction approaches optimal profit as n → ∞ . Subtlety: Must iron emperical distribution when it fails the monotone hazard rate condition. 39 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Is consistency feasible? Difficulty: Consistency, Truthfulness, and Profit Maximization. Example: • Basic Auction problem ( n bidders, n units). • Envy-freedom: all bidders are offered the same price. 40 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Is consistency feasible? Difficulty: Consistency, Truthfulness, and Profit Maximization. Example: • Basic Auction problem ( n bidders, n units). • Envy-freedom: all bidders are offered the same price. Theorem: [GH-03] No auction is truthful, envy-free, and approximates the optimal profit better than o (log n/ log log n ) . But. . . 40 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Is consistency feasible? Difficulty: Consistency, Truthfulness, and Profit Maximization. Example: • Basic Auction problem ( n bidders, n units). • Envy-freedom: all bidders are offered the same price. Theorem: [GH-03] No auction is truthful, envy-free, and approximates the optimal profit better than o (log n/ log log n ) . But. . . Theorem: Exists approximately optimal auctions that are • truthful with high probability and envy-free, or • envy-free with high probability and truthful. 40 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Optimal Mechanism Design without Priors Part III The Worst Case

Analysis Framework Recall Goal: Truthful profit maximizing basic auction. 42 P RIOR - FREE M ECHANISM D ESIGN – J UNE 5, 2005

Optimal Mechanism Design (without Priors) Jason D. Hartline - PowerPoint PPT Presentation

Optimal Mechanism Design (without Priors) Jason D. Hartline Microsoft Research Silicon Valley June 5, 2005 Also at EC Sunday 2:00: G. Aggarwal, J. Hartline, Knapsack Auctions . Sunday 2:30: M.-F . Balcan, A. Blum, J. Hartline, Y.

Mechanism Design with Unknown Correlated Distributions: Can We Learn Optimal Mechanisms? Michael

Mechanism Design and Auctions Game Theory MohammadAmin Fazli Algorithmic Game Theory 1 TOC

Linear Programming & Mechanism Design Rakesh Vohra August 2012 Rakesh V. Vohra ()

Mechanism Feasibility Design Dr. James Gopsill Design & Manufacture 2 Mechanism

A Truthful (1-)-Optimal Mechanism for On-demand

Recent Advances and Techniques in Algorithmic Mechanism Design Part 2: Bayesian Mechanism Design

Voting and Mechanism Design Jos e M Vidal Department of Computer Science and Engineering,

Mechanism Feasibility Design Task Dr. James Gopsill Design & Manufacture 2 Mechanism

Mechanism Feasibility Design Task Dr. James Gopsill Design & Manufacture 2 Mechanism

CMU 15-896 Mechanism design 1: Without money Teacher: Ariel Procaccia Approximate MD wo money

Near Optimal Compressed Sensing without Priors: Parametric SURE Approximate Message Passing Chunli

MECHANISM DESIGN Game Theory: Interaction of rational, competing, strategic agents

Mechanism Feasibility Design Task Dr. James Gopsill Design & Manufacture 2 Mechanism

Mechanism Design: A Highly Selective Review E. Maskin May 7, 2005 1 Mechanism Design part

Mechanism Design and Auctions Jun Shu EECS228a, Fall 2002 UC Berkeley Class Objectives To

Introduction to Mechanism Design Mehdi Dastani BBL-521 M.M.Dastani@cs.uu.nl What is Mechanism

A Mathematical Solution to Power Optimal Pipeline Design Power Optimal Pipeline Design by

Truth or Envy? (a theory for prior-free mechanism design) Jason D. Hartline Northwestern

Introduction to Mechanism Design for Single Parameter Environments Based on slides by V. Markakis

Approximation in Mechanism Design Jason D. Hartline Northwestern University August 8 and 10,

Approximation in Mechanism Design Jason Hartline Northwestern University August 8 and 10, 2012

AN INTRODUCTION TO OPTIMAL DESIGN G. Allaire, Ecole Polytechnique Optimal design of structures

Signal detection theory z p[r|-] p[r|+] <r> + <r> - Role of priors : Find z by

Sequential mechanism design Krzysztof R. Apt (so not Krzystof and definitely not Krystof) CWI,

Optimal Mechanism Design (without Priors) Jason D. Hartline - PowerPoint PPT Presentation

Optimal Mechanism Design (without Priors) Jason D. Hartline Microsoft Research Silicon Valley June 5, 2005 Also at EC Sunday 2:00: G. Aggarwal, J. Hartline, Knapsack Auctions . Sunday 2:30: M.-F . Balcan, A. Blum, J. Hartline, Y.

Mechanism Design with Unknown Correlated Distributions: Can We Learn Optimal Mechanisms? Michael

Mechanism Design and Auctions Game Theory MohammadAmin Fazli Algorithmic Game Theory 1 TOC

Linear Programming &amp; Mechanism Design Rakesh Vohra August 2012 Rakesh V. Vohra ()

Mechanism Feasibility Design Dr. James Gopsill Design &amp; Manufacture 2 Mechanism

A Truthful (1-)-Optimal Mechanism for On-demand

Recent Advances and Techniques in Algorithmic Mechanism Design Part 2: Bayesian Mechanism Design

Voting and Mechanism Design Jos e M Vidal Department of Computer Science and Engineering,

Mechanism Feasibility Design Task Dr. James Gopsill Design &amp; Manufacture 2 Mechanism

Mechanism Feasibility Design Task Dr. James Gopsill Design &amp; Manufacture 2 Mechanism

CMU 15-896 Mechanism design 1: Without money Teacher: Ariel Procaccia Approximate MD wo money

Near Optimal Compressed Sensing without Priors: Parametric SURE Approximate Message Passing Chunli

MECHANISM DESIGN Game Theory: Interaction of rational, competing, strategic agents

Mechanism Feasibility Design Task Dr. James Gopsill Design &amp; Manufacture 2 Mechanism

Mechanism Design: A Highly Selective Review E. Maskin May 7, 2005 1 Mechanism Design part

Mechanism Design and Auctions Jun Shu EECS228a, Fall 2002 UC Berkeley Class Objectives To

Introduction to Mechanism Design Mehdi Dastani BBL-521 M.M.Dastani@cs.uu.nl What is Mechanism

A Mathematical Solution to Power Optimal Pipeline Design Power Optimal Pipeline Design by

Truth or Envy? (a theory for prior-free mechanism design) Jason D. Hartline Northwestern

Introduction to Mechanism Design for Single Parameter Environments Based on slides by V. Markakis

Approximation in Mechanism Design Jason D. Hartline Northwestern University August 8 and 10,

Approximation in Mechanism Design Jason Hartline Northwestern University August 8 and 10, 2012

AN INTRODUCTION TO OPTIMAL DESIGN G. Allaire, Ecole Polytechnique Optimal design of structures

Signal detection theory z p[r|-] p[r|+] &lt;r&gt; + &lt;r&gt; - Role of priors : Find z by

Sequential mechanism design Krzysztof R. Apt (so not Krzystof and definitely not Krystof) CWI,

Linear Programming & Mechanism Design Rakesh Vohra August 2012 Rakesh V. Vohra ()

Mechanism Feasibility Design Dr. James Gopsill Design & Manufacture 2 Mechanism

Mechanism Feasibility Design Task Dr. James Gopsill Design & Manufacture 2 Mechanism

Mechanism Feasibility Design Task Dr. James Gopsill Design & Manufacture 2 Mechanism

Mechanism Feasibility Design Task Dr. James Gopsill Design & Manufacture 2 Mechanism

Signal detection theory z p[r|-] p[r|+] <r> + <r> - Role of priors : Find z by