, 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.

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

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

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.

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

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

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)

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

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

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

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.

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.