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

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

Jan 09, 2023 46 likes •470 views

Mechanism Design with E ! ciency and Equality Considerations (Wine 2017) Mohamad Lati fi an Iman Jami Moghaddam Outline Whats an auction? Equality, E ffi ciency, Truthfulness Problem de fi nition LP

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Mechanism Design with E!ciency and Equality Considerations

(Wine 2017) Iman Jami Moghaddam Mohamad Latifian

وا مان هب

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Outline

  • What’s an auction?
  • Equality, Efficiency, Truthfulness
  • Problem definition
  • LP formulation
  • Proposed mechanism
  • Truthfulness
  • Computability
  • Conclusion and future works

!2

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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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  • 4
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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

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What’s an auction?
 (Closer look)

  • Single parameter
  • N bidders and K homogenous indivisible goods
  • Each bidder has a private value vi for a good
  • Bidders call out their bids b1, b2, …, bn
  • Allocation rule A(b)
  • Payment rule P(b)

!6

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What’s the goal?

  • Equality
  • Efficiency

!7

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  • An auction as defined
  • Probabilistic allocation vector ,
  • Total utility = or more general
  • The equality measure

The problem

!8

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LP Formulation

!9

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Equality

  • Generalized Gini inequality index
  • ⇒
  • !10
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Equality

  • Min probability
  • and
  • Max difference
  • Gini-coeficient
  • !11
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E!ciency

  • Social welfare:
  • Some other efficiency functions:
  • Expected revenue
  • Long-term revenue

!12

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Truthful Mechanism

!13

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Truthful Mechanism

Two more definitions

!13

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Truthful Mechanism

!13

Let’s make it short

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Truthful Mechanism

!13

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Truthful Mechanism

!14

  • Incentive Compatible



 


  • Ex-post Individual Rational
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Truthful Mechanism (Cont’d)

!15

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Truthful Mechanism (Cont’d)

!15

  • Allocation





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Truthful Mechanism (Cont’d)

!15

  • Allocation





Use the LP

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Truthful Mechanism (Cont’d)

!15

  • Allocation





Use the LP

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Truthful Mechanism (Cont’d)

!15

  • Allocation





  • Payment rule

Use the LP

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Truthful Mechanism (Cont’d)

!15

  • Allocation





  • Payment rule

Use the LP

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Computation of opt. allocation

!16

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Computation of opt. allocation

!16

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

!16

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

!16

  • 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:

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Computation of opt. allocation

!16

  • 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:

  • n1 player with q = 1,
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Computation of opt. allocation

!16

  • 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:

  • n1 player with q = 1,
  • n2 player with q = q’,
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Computation of opt. allocation

!16

  • 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:

  • n1 player with q = 1,
  • n2 player with q = q’,
  • N - n1 - n2 - n4 player with q = q’’
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Computation of opt. allocation

!16

  • 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:

  • n1 player with q = 1,
  • n2 player with q = q’,
  • N - n1 - n2 - n4 player with q = q’’
  • n4 player with q = 0,
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Computation of opt. allocation (cont’d)

!17




  • n* =
  • O(N

4)

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Computation of opt. Payment

!18

  • 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 q(x,b−k ) can change at most O(N

3 ) times.

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Conclusion and Future Works

!19

  • 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

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Any questions?

  • 20
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Back to your bids

!21

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Back to your bids

!21

  • .w
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Back to your bids

!21

  • .w

⇒ qi = 1

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Back to your bids

!21

  • .w

⇒ qi = 1 ⇒ qi = 2/n

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Back to your bids

!21

  • .w

⇒ qi = 1 ⇒ qi = 2/n

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Back to your bids

!21

  • .w

⇒ qi = 1 ⇒ qi = 2/n

⇒ pi = 0

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Mechanism Design with E!ciency and Equality Considerations

  • 22

(Wine 2017)

Iman Jami Moghaddam

Mohamad Latifian