Mechanism Design with E ! ciency and Equality Considerations (Wine - PowerPoint PPT Presentation

وا مان هب Mechanism Design with E ! ciency and Equality Considerations (Wine 2017) Mohamad Lati fi an Iman Jami Moghaddam

Outline • What’s an auction? • Equality, E ffi ciency, Truthfulness • Problem de fi nition • LP formulation • Proposed mechanism • Truthfulness • Computability • Conclusion and future works ! 2

What’s an auction? • Buying and selling items • Participants (bidders) call out their bids • Sell the good(s) w.r.t. bids ! 3

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Some issues • Who gets the good(s)? • How much should each bidder pay? • What’s the goal? • Did you bid your actual value? ! 5

What’s an auction? 
 (Closer look) • Single parameter • N bidders and K homogenous indivisible goods • Each bidder has a private value v i for a good • Bidders call out their bids b 1 , b 2 , …, b n • Allocation rule A(b) • Payment rule P(b) ! 6

What’s the goal? • Equality • E ffi ciency ! 7

The problem • An auction as de fi ned • Probabilistic allocation vector , • Total utility = or more general • The equality measure ! 8

LP Formulation ! 9

Equality • Generalized Gini inequality index • ⇒ • • ! 10

Equality • Min probability • and • Max di f ference • 
 • Gini-coe fi cient • ! 11

E ! ciency • Social welfare: • Some other e ffi ciency functions: • Expected revenue • Long-term revenue ! 12

Truthful Mechanism ! 13

Truthful Mechanism Two more definitions ! 13

Truthful Mechanism Let’s make it short ! 13

Truthful Mechanism ! 13


 
 Truthful Mechanism • Incentive Compatible 
 • Ex-post Individual Rational ! 14

Truthful Mechanism (Cont’d) ! 15


 Truthful Mechanism (Cont’d) • Allocation 
 ! 15


 Truthful Mechanism (Cont’d) • Allocation 
 Use the LP ! 15


 Truthful Mechanism (Cont’d) • Allocation 
 Use the LP ! 15


 Truthful Mechanism (Cont’d) • Allocation 
 Use the LP • Payment rule ! 15


 Truthful Mechanism (Cont’d) • Allocation 
 Use the LP • Payment rule ! 15

Computation of opt. allocation ! 16

Computation of opt. allocation • f (1) ≥ f (2) ≥ … ≥ f (n) ! 16

Computation of opt. allocation • f (1) ≥ f (2) ≥ … ≥ f (n) • There must exist an optimal solution with ! 16

Computation of opt. allocation • f (1) ≥ f (2) ≥ … ≥ f (n) • There must exist an optimal solution with • Theorem: For any bid pro " le b, exist an optimal solution q(b) such that: ! 16

Computation of opt. allocation • f (1) ≥ f (2) ≥ … ≥ f (n) • There must exist an optimal solution with • Theorem: For any bid pro " le b, exist an optimal solution q(b) such that: • n 1 player with q = 1 , ! 16

Computation of opt. allocation • f (1) ≥ f (2) ≥ … ≥ f (n) • There must exist an optimal solution with • Theorem: For any bid pro " le b, exist an optimal solution q(b) such that: • n 1 player with q = 1 , • n 2 player with q = q’ , ! 16

Computation of opt. allocation • f (1) ≥ f (2) ≥ … ≥ f (n) • There must exist an optimal solution with • Theorem: For any bid pro " le b, exist an optimal solution q(b) such that: • n 1 player with q = 1 , • n 2 player with q = q’ , • N - n 1 - n 2 - n 4 player with q = q’’ ! 16

Computation of opt. allocation • f (1) ≥ f (2) ≥ … ≥ f (n) • There must exist an optimal solution with • Theorem: For any bid pro " le b, exist an optimal solution q(b) such that: • n 1 player with q = 1 , • n 2 player with q = q’ , • N - n 1 - n 2 - n 4 player with q = q’’ • n 4 player with q = 0 , ! 16


 Computation of opt. allocation (cont’d) • 
 • 
 • n* = • O(N 4 ) ! 17

Computation of opt. Payment • Can not compute • Lemma: As player k’s bid x increases from t(i+1) to t(i) , her winning probability in the optimal allocation 3 ) times. q(x,b − k ) can change at most O(N ! 18

Conclusion and Future Works • Maximizes the e ! ciency while ensuring the equality level • Compute allocation and correspond payments in polynomial time • The Equality measure can be non linear and generalized in future works ! 19

Any questions? � 20

Back to your bids ! 21

Back to your bids o.w ! 21

Back to your bids ⇒ q i = 1 o.w ! 21

Back to your bids ⇒ q i = 1 ⇒ q i = 2/n o.w ! 21

Back to your bids ⇒ q i = 1 ⇒ q i = 2/n o.w ! 21

Back to your bids ⇒ q i = 1 ⇒ p i = 0 ⇒ q i = 2/n o.w ! 21

Mechanism Design with E ! ciency and Equality Considerations (Wine 2017) Mohamad Latifian Iman Jami Moghaddam � 22