Class 17: Algorithmic Mechanism Design

Class 17: Algorithmic Mechanism Design
https://uvammm.github.io/class17

Markets, Mechanisms, and Machines
University of Virginia
cs4501/econ4559 Spring 2019
David Evans and Denis Nekipelov
https://uvammm.github.io/

Transcript

1. MARKETS, MECHANISMS, MACHINES University of Virginia, Spring 2019
   Class 17:
   Algorithmic
   Mechanism
   Design
   19 March 2019
   cs4501/econ4559 Spring 2019
   David Evans and Denis Nekipelov
   https://uvammm.github.io
   

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2. Reminder: Trial Auction Thursday!
   Your team should be prepared to participate in trial auction in-class
   Thursday:
   • Bring laptop (at least one for your team): on the network, with your
   client code ready to run
   • Client code ready to compete in trial auction
   limited number of rounds (will run throughout the class)
   • Winning team gets $1000 bonus for final auction
   • Teams that are not able to compete penalty $1000
   1
   

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3. Plan for Today
   Algorithmic Mechanism Design
   Crash Course Computational Complexity
   Complexity of Multi-Unit Allocation
   2
   

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4. Algorithmic Mechanism Design
   3
   (Classical)
   Mechanism Design
   Algorithms
   (Computer Science)
   Algorithmic
   Mechanism Design
   Goal
   Design mechanism that
   achieves economic goal
   (e.g., maximize social
   welfare)
   Design solution to problem
   that is computationally-
   efficient
   Design computationally-
   efficient mechanism that
   achieves economic goal
   Participants Models of rational agents Models of abstract machines Models of rational agents
   History
   1928: von Neumann,
   Theory of Games of
   Strategy
   1961: Vickey
   1673: Leibniz
   1930s: Turing, Church
   1999: Nissan and Ronen
   (STOC)
   

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5. Multi-Unit Auction
   ! identical items, " bidders
   Each bidder has private valuation function:
   #$
   : 0, … , ! → ℝ+
   #$
   , = value to bidder 9 for receiving , units
   #$
   0 = 0, ∀, ∈ 0, … , ! − 1 : #$
   , ≤ #$
   (, + 1)
   Goal: maximize social welfare = ∑
   $
   #$
   !$
   where ∑
   $
   !$
   ≤ !
   4
   Noam Nisan, Algorithmic Mechanism Design, 2014.
   

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6. Multi-Unit Allocation Problem
   Input:
   Output:
   5
   Noam Nisan, Algorithmic Mechanism Design, 2014.
   

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7. Multi-Unit Allocation Problem
   Input: Valuations for the players: !"
   , … , !%
   .
   Output: Allocation, '"
   , … , '%
   where ∑
   )
   ')
   ≤ ' that maximizes ∑
   )
   !)
   ')
   .
   6
   Noam Nisan, Algorithmic Mechanism Design, 2014.
   What does it mean to have a computationally-efficient solution?
   

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8. Measuring Cost (Recap from Class 4)
   Computation:
   number of steps needed by a model computer
   Communication:
   total amount of data communicated
   number of rounds of messages
   7
   Actual cost usually dominated by communication cost: bandwidth is expensive
   But, computation cost determines if your program will actually finish
   

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9. Complexity Classes
   Complexity Class: a set of problems that can be solved with the
   defined resource constraint
   Typically: given machine model (e.g., deterministic Turing
   Machine) within a limited amount (usually function of input size)
   of time or space
   8
   

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10. How many complexity classes are there?
    9
    

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11. How many complexity classes are there?
    10
    535 “interesting” classes (and counting)
    

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12. Most Important Complexity Class
    11
    P
    “Tractable”
    

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13. Polynomial Time
    12
    ! = #
    $ ∈ ℕ
    TIME(,$)
    

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14. Problems in Class P
    13
    

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15. How can we prove a problem is in P?
    14
    

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16. How can we prove a problem is not in P?
    15
    

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17. How can we prove a problem is not in P?
    16
    

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18. Is Multi-Unit Auction in P?
    17
    Input: Valuations for the players: !"
    , … , !%
    .
    Output: Allocation, '"
    , … , '%
    where ∑
    )
    ')
    ≤ ' that maximizes ∑
    )
    !)
    ')
    .
    

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19. Is Multi-Unit Auction in P?
    18
    Input: Valuations for the players: !"
    , … , !%
    .
    Output: Allocation, '"
    , … , '%
    where ∑
    )
    ')
    ≤ ' that maximizes ∑
    )
    !)
    ')
    .
    Polynomial in what?
    Length of input ' = log '
    How is the input represented?
    Query model:
    !)
    is an arbitrary function with !)
    0 = 0, ∀1 ∈ 0, … , ' − 1 : !)
    1 ≤ !)
    (1 + 1)
    Cost = number of queries to !)
    

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20. Is Multi-Unit Auction in P?
    19
    Input: Valuations for the players: !"
    , … , !%
    .
    Output: Allocation, '"
    , … , '%
    where ∑
    )
    ')
    ≤ ' that maximizes ∑
    )
    !)
    ')
    .
    Is it possible to find optimal allocation with +(log0 ') queries to !)
    ?
    

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21. Is Multi-Unit Auction in P?
    20
    Input: Valuations for the two players: !"
    , !$
    Output: Allocation, %"
    , %$
    where ∑
    '
    %'
    ≤ % that maximizes ∑
    '
    !'
    %'
    .
    Is it possible to find optimal allocation with )(log. %) queries to !'
    ?
    

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22. Proof: Multi-Unit Auction ∉ "
    21
    Input: Valuations for the two players: #$
    , #&
    Output: Allocation, '$
    , '& where ∑
    )
    ')
    ≤ ' that maximizes ∑
    )
    #)
    ') .
    If the algorithm makes fewer than 2' – 2 queries, this means there is some value
    of either #$ or #& below ' that wasn’t queried.
    Without loss of generality lets assume it is #$.
    Suppose #$
    - = - and #&
    - = - for all queried inputs.
    The output is ('$
    , '&
    ). Value = #$
    '$
    + #&
    '&
    = '$
    + '&
    = '.
    For the unqueried point, 3 ∉ '$
    , '&
    : #$
    3 = 3 + 1.
    The allocation (3, ' − 3) has value:
    #$
    3 + #&
    ' − 3 = 3 + 1 + ' − 3 = ' + 1 > '.
    

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23. Multi-Unit Auction: Input Representation
    22
    Input: Valuations for the players: !"
    , … , !%
    .
    Output: Allocation, '"
    , … , '%
    where ∑
    )
    ')
    ≤ ' that maximizes ∑
    )
    !)
    ')
    .
    Polynomial in what?
    Length of input ' = log '
    How is the input represented?
    Query model:
    !)
    is an arbitrary function with !)
    0 = 0, ∀1 ∈ 0, … , ' − 1 : !)
    1 ≤ !)
    (1 + 1)
    Representing input as arbitrary function requires '9 bits for each valuation function,
    so solution that requires querying each value once is polynomial in input size number of queries.
    

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24. Multi-Unit Auction: Input Representation
    23
    Input: Valuations for the players: !"
    , … , !%
    .
    Output: Allocation, '"
    , … , '%
    where ∑
    )
    ')
    ≤ ' that maximizes ∑
    )
    !)
    ')
    .
    Query model:
    !)
    0 = 0, ∀. ∈ 0, … , ' − 1 : !)
    . ≤ !)
    (. + 1)
    Single-Minded Step Bids:
    !)
    . = 0 for . < .)
    ∗; !)
    . = <)
    ∗ for . ≥ .)
    ∗.
    Downward Sloping:
    ∀. ∈ 0, … , ' − 1 : !)
    . + 1 − !)
    . ≤ !)
    . − !)
    (. − 1)
    

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25. Downward Sloping Valuations
    24
    Input: Valuations for the players: !"
    , … , !%
    .
    Output: Allocation, '"
    , … , '%
    where ∑
    )
    ')
    ≤ ' that maximizes ∑
    )
    !)
    ')
    .
    Downward Sloping:
    ∀, ∈ 0, … , ' − 1 : !)
    , + 1 − !)
    , ≤ !)
    , − !)
    (, − 1) “convex economy”
    '
    0
    !(')
    

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26. Downward Sloping Valuations
    25
    Input: Valuations for the players: !"
    , … , !%
    .
    Output: Allocation, '"
    , … , '%
    where ∑
    )
    ')
    ≤ ' that maximizes ∑
    )
    !)
    ')
    .
    Downward Sloping:
    ∀, ∈ 0, … , ' − 1 : !)
    , + 1 − !)
    , ≤ !)
    , − !)
    (, − 1) “convex economy”
    '
    0
    !(')
    Some clearing price 5 such that: for all 6,
    !)
    ')
    − !)
    ')
    − 1 ≥ 5 > !)
    ')
    + 1 − !)
    ')
    and ∑
    )
    ')
    = '
    Use binary search (: log ') to find 5 and allocation.
    

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27. Multi-Unit Auction: Input Representation
    26
    Input: Valuations for the players: !"
    , … , !%
    .
    Output: Allocation, '"
    , … , '%
    where ∑
    )
    ')
    ≤ ' that maximizes ∑
    )
    !)
    ')
    .
    Query model:
    !)
    0 = 0, ∀. ∈ 0, … , ' − 1 : !)
    . ≤ !)
    (. + 1)
    Single-Minded Step Bids:
    !)
    . = 0 for . < .)
    ∗; !)
    . = <)
    ∗ for . ≥ .)
    ∗.
    Downward Sloping:
    ∀. ∈ 0, … , ' − 1 : !)
    . + 1 − !)
    . ≤ !)
    . − !)
    (. − 1)
    No polynomial (in log ') time solution
    Know polynomial (in log ') time solution
    ???
    

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28. Next class:
    Computational complexity of Multi-Unit
    Auctions with Step Bids
    P = NP?
    Charge
    Be ready for trial
    auction at
    beginning of
    Thursday’s class!
    

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