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

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.

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

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

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

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.

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

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

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

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?

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?

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

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)

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

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

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

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.

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.

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

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