370

# 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

## 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

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

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.

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?

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

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

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

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

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

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.

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

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

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)

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

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

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

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

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

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

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

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 )

22. Auctions
21

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

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?

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?

26. Top-Level Domains
25

27. Top-Level Domains
26

28. Top-Level Domains
27

29. Why is mobile
in the US?
28

30. 29

31. 30

32. 31

33. 32

34. 33

35. 34

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

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)

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

39. Simple Spectrum Auction Model
38
= 1, … , licenses for sale
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

40. Simple Spectrum Auction Model
39
= 1, … , licenses for sale
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

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

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.

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.

44. Really Mutual Substitutes?
43

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

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

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

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

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

50. What could possibly go wrong?
49

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.

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.

53. 52

54. 53

55. 54