Simple versus Optimal Mechanisms

1. Simple versus Optimal Mechanisms Jason D. Hartline and Tim Roughgarden ACM EC’09 呂亦書

2. Outline • Abstract • Introduction • Myerson’s Lemma • VCG with Monopoly reserves • Q & A

3. Abstract Vickrey-Clarke-Groves (VCG) Revenue低 Myerson Valuation分佈不一樣 Valuation分佈相同 Downard-closed Environment

4. Introduction A B C D E F Feasible sets 1-item : {A},{B},{C},…….. 2-item : {A,B},{B,C},{A,C},…….. Downard-closed Environment 不考慮item : {A,B,C},{A,B,C,D},…….

5. Introduction bidder i private valuation The profile of agent valuations is denoted by allocation rule ( x ) : 0 : loser, 1 : winner payment rule : p utility : >0 maximize the social surplus :

6. Introduction ( Single-item auction ) • VCG revenue < Myerson revenue Example : 2 bidders, uniform distributions on [0, 1], 2nd highest bid: 1/3 Myerson reserve price:1/2 VCG revenue : 1/3 Myerson : bidder A > 1/2 : O O × × bidder B > 1/2 : O × O × 期望值 :1/4*1/2*3 = 3/8 > 1/3 1/4 1/4 1/4 1/4

7. Introduction • Is 1/2 the optimal reserve price ? hazard rate : virtual value : = (reserve price)

8. Introduction • Myserson: 加上item底價 Example : 拍賣一個item , 底價 :50 bidder A : 1.30 2.60 bidder B : (a)70 (b) 20 1. (a) revenue : 50 ( A < 50 ) 1. (b) revenue:0( 流拍 A,B < 50 ) 2. (a) revenue : 60 ( sell B second price ) 2. (b) revenue:50(B < 50 )

9. Myerson’s Lemma virtual surplus :

10. VCG with Monopoly reserves

11. VCG with Monopoly reserves

12. VCG with Monopoly reserves

13. VCG with Monopoly reserves • Example : bidder 0 : big bidder 1,2,3…..n : small exponential distribution Myerson : big valuation : small valuation : 1 revenue :

14. VCG with Monopoly reserves • VCGr : big monopoly reserves : small monopoly reserves : 1 big monopoly reserves : small monopoly reserves : 1 revenue : VCGr比 Myserson差2倍

15. Q &A • Question : Myerson是否為VCG + reserve price ? • Answer : Yes