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Class 13: Auctioning Air

David Evans
February 26, 2019

Class 13: Auctioning Air

Class 13: Auctioning Air
https://uvammm.github.io/class13

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

David Evans

February 26, 2019
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  1. MARKETS, MECHANISMS, MACHINES University of Virginia, Spring 2019
    Class 13:
    Auctioning Air
    26 February 2019
    cs4501/econ4559 Spring 2019
    David Evans and Denis Nekipelov
    https://uvammm.github.io

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  2. Plan
    Recap and Finish: A-K-Q Poker
    Equilibrium Strategy
    Auctions
    Ascending Price Auctions
    Spectrum Auctions
    1

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  3. Recap: A-K-Q Poker
    2
    State: "
    ← A, K, Q , c)
    ← A, K, Q − { "
    }
    "
    = bet
    "
    = check
    5
    +1 if c"
    > c)
    −1 if )
    > "
    )
    = bet
    )
    = fold
    +1
    5
    +2 if c"
    > c2
    −2 if )
    > "
    Payoff for Player 1 shown; Player 2 payoff is negation.

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  4. Player 1: Ace Queen
    Bet Bet Check
    Player 2
    Ace Call -2 -1
    King
    Call +2 -2 -1
    Fold +1 +1
    The Tough Decisions
    What if Player 1 always bluffs?

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  5. Always Bluff
    Strategy
    Player 1: A K Q
    Bet Check Bet
    Player 2
    A Call -1 -2
    K
    Call +2 -2
    Fold +1 +1 +1
    Q Fold +1 +1

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  6. Always Bluff
    Strategy
    Player 1: A K Q
    Bet Check Bet
    Player 2
    A Call -1 -2
    K
    Call +2 -2
    Fold +1 +1 +1
    Q Fold +1 +1
    ",A) BCDDE FGHI
    =
    1
    3
    1
    2
    +
    2
    2
    +
    1
    3

    1
    2
    +
    1
    2
    +
    1
    3
    −2 = 0
    ",A) MNDOE FGHI
    =
    1
    3
    1
    2
    +
    1
    2
    +
    1
    3

    1
    2
    +
    1
    2
    +
    1
    3
    −2
    2
    +
    1
    2
    = −
    1
    6

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  7. Recap
    If player 1 never bluffs:
    If player 1 always bluffs:
    Looks like a break-even game for Player 1:
    is there a better strategy?
    "
    = 0, )
    = 0
    "
    = −
    1
    6
    , )
    =
    1
    6

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  8. Mixed Strategy
    Player 1: A K Q
    Bet Check Bet
    Player 2
    A Call -1 -2
    K Call +2 -2
    Q Fold +1 +1
    Always Bluff
    Player 1: A K Q
    Bet Check Check
    Player 2
    A Call -1 -1
    K Fold/Call +1 -1
    Q Fold +1 +1
    Never Bluff
    Pure strategy: always do the same action for a given input state.
    Mixed strategy: probabilistically select from a set of pure strategies.
    "
    = 0, )
    = 0 "
    = −
    1
    6
    , )
    =
    1
    6

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  9. Strategies
    Player 1
    Bluff with Queen
    Check with Queen
    Find the best strategy for Player 1 assuming Player 2 plays optimally.
    "
    "
    Q = bet, )
    = call = −
    1
    6
    "
    "
    Q = check, )
    = call =
    1
    6
    "
    "
    Q = bet, )
    = fold =
    1
    6
    "
    "
    Q = check, )
    = fold = 0
    Player 2
    Call with King Fold with King

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  10. Nash Equilibrium
    Player 1 is making the best decision she can, taking into account Player
    2’s decisions.
    Player 2 is making the best decision he can, taking into about Player 1’s
    decisions.
    Neither player can improve its expected value by deviating from the
    strategy.
    Hence, to find the best strategy for Player 1, we need to find a strategy
    that makes Player 2 indifferent between his options.

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  11. Winning the AKQ Game
    Bluff Check
    Call -1 +1
    Fold +1 0
    Player 1 wants to make Player 2 indifferent between )
    = call and )
    = fold
    "
    "
    Q = bet, )
    = call = −
    1
    6
    "
    "
    Q = check, )
    = call =
    1
    6
    "
    "
    Q = bet, )
    = fold =
    1
    6
    "
    "
    Q = check, )
    = fold = 0

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  12. Winning the AKQ Game
    Bluff Check
    Call -1 +1
    Fold +1 0
    Player 1 wants to make Player 2 indifferent between )
    = call and )
    = fold

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  13. Winning the AKQ Game
    Bluff Check
    Call -1 +1
    Fold +1 0
    Player 1 wants to make Player 2 indifferent between )
    = call and )
    = fold
    )
    , )
    = call = )
    , )
    = fold
    )
    , )
    = call
    = −2 "
    = A ) + 2("
    = | )
    = −2 1 − + 2
    )
    , )
    = fold = −1
    = "
    = Q "
    = bet, )
    = K)

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  14. Winning the AKQ Game
    Player 1 wants to make Player 2 indifferent between )
    = call and )
    = fold
    )
    , )
    = call = )
    , )
    = fold
    )
    , )
    = fold = −1
    = ("
    = Q | "
    = bet, c)
    = K)
    −2 1 − + 2 = −1
    = 1/4
    Given experimental observations, how do you determine the probability of an event?
    )
    , )
    = call
    = −2 "
    = A ) + 2("
    = | )
    = −2 1 − + 2

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  15. Thomas Bayes, 1702-1761
    Essay Towards
    Solving a Problem in
    the Doctrine of
    Chances
    (presented to Royal
    Society in 1763 after
    Bayes’ death)

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  16. Bayes’ Theorem
    Prior Probability: likelihood of
    regardless of other event
    Conditional Probability: likelihood of
    given that you observed .
    ) =
    ()
    ()
    ()
    )

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  17. AKQ Example
    Given that I have a King, what is the probability that
    you have an Ace?
    ) =
    ()
    ()

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  18. AKQ Example
    Given that I have a King, what is the probability that
    you have an Ace?
    "
    = A )
    = K) =
    )
    = K "
    = A) ("
    = A)
    ()
    = K)
    ) =
    ()
    ()

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  19. AKQ Example
    Given that I have a King and you bet, what (should be)
    the probability that you have a Queen (assuming you
    are playing optimally)?
    = ("
    = Q | "
    = bet, )
    = K)
    For Player 2 to be indifferent:
    = 1/4

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  20. When to Bluff?
    How often should Player 1 bluff with a Queen?
    = ("
    = Q | "
    = bet, )
    = K) = "
    _

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  21. Value of Game for Player 1
    20
    "
    , =
    1
    3
    1 +
    1
    3
    +
    1
    3
    0 +
    1
    3
    (
    2
    3
    −1 +
    1
    3
    (
    1
    2
    −2 +
    1
    2
    (
    1
    3
    −2 +
    2
    3
    1 )

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  22. Auctions
    21

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  23. “English Auction”
    Open, ascending first price auction
    22
    https://youtu.be/iiO_1XRnMt4?t=18

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  24. Open Ascending Auction
    Initialization:
    , = None, starting bid
    while new bid (bidder, ) where > :
    , = bidder,
    announce winner, bid
    outcome = (, )
    23
    How does this compare to a
    second price auction?

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  25. Simultaneous Ascending Auction
    Initialization:
    , = Bidders, starting bid
    while | | > 1:
    = , +
    announce remaining active bidders
    outcome = (. ℎ(), )
    24
    How does this compare to a
    second price auction?

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  26. Top-Level Domains
    25

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  27. Top-Level Domains
    26

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  28. Top-Level Domains
    27

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  29. Why is mobile
    service so bad
    in the US?
    28

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  30. 29

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  31. 30

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  32. 31

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  33. 32

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  34. 33

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  35. 34

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  36. Multiple Simultaneous Auctions
    35
    Multiple items (spectrum licenses) up for sale
    Each round:
    bidders submit bids for items they want
    must increase previous high bid for that item
    Rounds continue as long as price increases for at least one item
    (alternate proposal: item is sold after one round with no increase)
    Bidders cannot withdraw bids:
    if they are the current high bid for a license, committed to buying
    High bidder pays final price

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  37. Spectrum Auction: Bidding Rules
    36
    Before auction, each bidder established initial eligibility
    - makes deposits for amount of spectrum it wants to buy
    Must maintain activity level each round:
    - submit bids (or hold highest bid from previous round) that
    cover fraction (varies by phases) of its eligibility (or have
    eligibility reduced)
    - 5 waivers (intended to prevent errors, but also could be
    used strategically)

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  38. Walrasian “Auction”
    37
    Léon Walras
    tâtonnement
    (“trial-and-error”)
    to reach equilibrium

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  39. Simple Spectrum Auction Model
    38
    = 1, … , licenses for sale
    ⊆ subset of licenses
    p
    value of to bidder
    ∈ ℝt
    u vector of prices (price for each license)
    Paul Milgrom, Putting Auction Theory to Work: The Simultaneous Ascending Auction. Journal of Political Economy, 2000.
    Total price for = v x

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  40. Simple Spectrum Auction Model
    39
    = 1, … , licenses for sale
    ⊆ subset of licenses
    p
    value of to bidder
    ∈ ℝt
    u vector of prices (price for each license)
    Paul Milgrom, Putting Auction Theory to Work: The Simultaneous Ascending Auction. Journal of Political Economy, 2000.
    Total price for = v x
    p
    = argmax
    x
    {p
    − v x
    } Demand Correspondence

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  41. Maximize Value Theorem
    Theorem: Assume that the licenses are mutual substitutes for all
    bidders, and all bidders bid “straightforwardly”. Then:
    1. The auction ends after a finite number of rounds
    2. Final assignment maximizes total value (to within a single bid
    increment).
    40

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  42. Maximize Value Theorem
    Theorem: Assume that the licenses are mutual substitutes for all
    bidders, and all bidders bid “straightforwardly”. Then:
    1. The auction ends after a finite number of rounds
    2. Final assignment maximizes total value (to within a single bid
    increment).
    41
    Proof of #1: At each round, each bidders bid on each item either stays the
    same or increases by some amount. Each bidder has some maximum bid
    (without exceeding }
    ()), so must eventually stop increasing bid.

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  43. Maximize Value Theorem
    Theorem: Assume that the licenses are mutual substitutes for all
    bidders, and all bidders bid “straightforwardly”. Then:
    1. The auction ends after a finite number of rounds
    2. Final assignment, ∗, ∗ , maximizes total value (to within a single
    bid increment).
    42
    Intuition: because of mutual substitutes assumption, each bidder
    will keep bidding until either they meet their demand, or another
    bidder is willing to pay more.

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  44. Really Mutual Substitutes?
    43

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  45. Scenario
    44
    North
    Charlottesville
    South
    Charlottesville
    Both
    Charlottesville
    Bidder 1 (AT&T) $130K $230K $420K
    Bidder 2 (Sprint) $140K $240K $380K

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  46. 45
    What is actual value of 30
    MHz of Charlottesville’s
    spectrum?

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  47. C Block (1995-1996) Auction
    https://www.fcc.gov/auction/5
    46

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  48. 47
    https://www.fcc.gov/sites/default/files/wireless/auctions/05/charts/5markets.xls

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  49. 48
    https://www.fcc.gov/sites/default/files/wireless/auctions/05/charts/5markets.xls
    $7,415,250 (won by Virginia Wireless)

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  50. What could possibly go wrong?
    49

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  51. Collusion in Practice
    50
    Peter Cramton and Jesse A. Schwartz. Collusive Bidding: Lessons from the
    FCC Spectrum Auctions. Journal of Regulatory Economics, May 2000.

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  52. Collusion in Practice
    51
    Peter Cramton and Jesse A. Schwartz. Collusive Bidding: Lessons from the
    FCC Spectrum Auctions. Journal of Regulatory Economics, May 2000.

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  53. 52

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  54. 53

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  55. 54
    Get your 5G
    bandwidth
    bids ready!

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  56. Project 4 Teaming
    Team must have 3 or 4 students, at least one in CS, one in Econ
    Cannot team with someone you have already worked with twice
    55

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