Upgrade to Pro — share decks privately, control downloads, hide ads and more …

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
Tweet

More Decks by David Evans

Other Decks in Technology

Transcript

  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

    View full-size slide

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

    View full-size slide

  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.

    View full-size slide

  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?

    View full-size slide

  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

    View full-size slide

  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

    View full-size slide

  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

    View full-size slide

  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

    View full-size slide

  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

    View full-size slide

  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.

    View full-size slide

  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

    View full-size slide

  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

    View full-size slide

  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)

    View full-size slide

  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

    View full-size slide

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

    View full-size slide

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

    View full-size slide

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

    View full-size slide

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

    View full-size slide

  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

    View full-size slide

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

    View full-size slide

  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 )

    View full-size slide

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

    View full-size slide

  23. 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?

    View full-size slide

  24. 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?

    View full-size slide

  25. Top-Level Domains
    25

    View full-size slide

  26. Top-Level Domains
    26

    View full-size slide

  27. Top-Level Domains
    27

    View full-size slide

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

    View full-size slide

  29. 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

    View full-size slide

  30. 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)

    View full-size slide

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

    View full-size slide

  32. 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

    View full-size slide

  33. 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

    View full-size slide

  34. 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

    View full-size slide

  35. 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.

    View full-size slide

  36. 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.

    View full-size slide

  37. Really Mutual Substitutes?
    43

    View full-size slide

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

    View full-size slide

  39. 45
    What is actual value of 30
    MHz of Charlottesville’s
    spectrum?

    View full-size slide

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

    View full-size slide

  41. 47
    https://www.fcc.gov/sites/default/files/wireless/auctions/05/charts/5markets.xls

    View full-size slide

  42. 48
    https://www.fcc.gov/sites/default/files/wireless/auctions/05/charts/5markets.xls
    $7,415,250 (won by Virginia Wireless)

    View full-size slide

  43. What could possibly go wrong?
    49

    View full-size slide

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

    View full-size slide

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

    View full-size slide

  46. 54
    Get your 5G
    bandwidth
    bids ready!

    View full-size slide

  47. 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

    View full-size slide