A Review of Auction Theory: Sequential Auctions and Vickrey Auctions

Transcription

1 A Review of Auction Theory: and Vickrey Daniel R. 1 1 Department of Economics University of Maryland, College Park. September 2017 / Econ415

2 . Vickrey s. Vickrey.

3 Example Two goods, one per bidder Suppose we have the standard IPV situation with n >2 bidders and the seller has 2 objects to sell and each bidder desires a single object. The seller could modify the to a Third Price Auction (highest losing bid for two objects) and all our previous results concerning dominant strategies would carry over. Alternatively, she could conduct two in sequence, each selling a single object. How would a bidder bid in the second of the two auctions? What would be the expected price? Knowing this, how would bidders bid in the first of the two auctions? Although the actual equilibrium prices might differ from the TPA, in expectation, the prices will be the same.. Why?

4 Example Two goods, many per bidder Suppose we have n = 3 bidders and the seller has 2 objects to sell. Two bidders desire a single object A for 100 and B for 50 or 100. The remaining bidder only wants both objects AB for 180. (Example, telecom footprint.) Suppose with two s, A is sold then B. Suppose large bidder buys A for x(= 100) > 0. What should it bid for B? If the remaining single object bidder bids 50, then large bidder is ok. But what if it bids $100? Large bidder is exposed and risks losing a lot at the auction. Problem arises because for small bidders, objects are substitutes, while for large bidder, they are complements.

5 General Setup Suppose that there are M objects and n bidders have positive value for any subset of them. Show that there are 2 M possible subsets. For any subset C, the bidder s willingness to pay is v C. What are conditions on {v C } that create exposure issues? C, D, C D =, v C D > v C + v D. How to sell all subsets to many potential buyers? Let D be the set of all possible allocations of subsets to n bidders. Computing D is a complicated fitting problem.

6 Set-up allocation The type θ j of a bidder is the collection of values v j C the bidder places on each possible subset C it could acquire. If the bidder acquires C j and pays t j it gets v j C j t j. Bidders report to the auction, a claim of θ j (it does not have to be the truth). For any report, θ = (θ 1, θ 2,..., θ n ), the auctioneer computes W (θ) = max (C1,...,C n) D and selects this allocation, C(θ). n j=1 v j C j

7 Set-up payment For any report, θ, and bidder i the auctioneer computes W i (θ i ) = max C D Bidder i must pay the difference n v j C j j i t i (θ) = W i (θ i ) j i v j C j (θ). Observe that if the report of bidder i does not change the allocation to the other bidders, bidder i pays zero. Otherwise, bidder i is termed a pivotal bidder.

8 Main Result It is a weakly dominant strategy to report your true values to the mechanism. As long as bidders follow their dominant strategy, the mechanism is efficient it maximizes total bidder value. With just one object, the mechanism collapses to a. Show. Compute the equilibrium outcome for the exposure example. Compute.

9 Example 1 3 objects and 3 bidders. Note that in general, each bidder will have 8 = 2 3 values to report. They are reported in the following table. B 1 B 2 B a b c ab ac bc abc 5 5 5

10 Example 2: Low Revenues (From Ausubel/Milgrom) 2 objects and 3 bidders. The values (in Billions of dollars) are reported in the following table. B 1 B 2 B a b ab Suppose the licenses were sold as a bundle only. Revenue and efficiency. What would occur with only bidder B1 and B3? Compare it to having all three bidders.

11 Criticisms Complexity. Size of bid space rises rapidly. Lack of price discovery. Related to complexity. Also, need to learn about market. Failure of monotonicity in number of bidders. Examples Vulnerability to collusive behavior. Examples. Potential for very low revenues. Examples.