AEA Continuing Education in Game Theory Avinash Dixit and David Reiley Session 6: Market Design and Algorithms David Reiley Yahoo! Research January 2011

FCC spectrum auctions involve bidding on multiple licenses, possibly complementary. Item AItem BPackage AB Bidder 1149 Bidder 2325 Bidder 3056 To handle this, we may want to create a combinatorial version of the Vickrey auction. Example: What is the optimal allocation in this auction? What are the prices paid by bidders in the VCG auction mechanism?

FCC spectrum auctions involve bidding on multiple licenses, possibly complementary. Item AItem BPackage AB Bidder 1149 Bidder 2325 Bidder 3056 Example: Solution: Maximize by giving AB to Bidder 1. Price equals surplus if Bidder 1 were absent, which is 3+5=8.

What happens if we change the values slightly in the example? Now what are the VCG allocation and payments? Item AItem BPackage AB Bidder 1147 Bidder 2325 Bidder 3056

What happens if we change the values slightly in the example? Now what are the VCG allocation and payments? Item AItem BPackage AB Bidder 1147 Bidder 2325 Bidder 3056 Solution: 2 wins A, 3 wins B. Total surplus 8. Without 2, surplus would have been 7. So 2 pays 2. Without 3, surplus would have been 7. So 3 pays 4.

One more example. What are the VCG allocation and payments? What’s undesirable about the outcome? Item AItem BPackage AB Bidder 1030 Bidder 2300 Bidder 3115

One more example. Allocation: 1 gets B, 2 gets A. Surplus is 6. Without A, surplus would be 5. So A pays 2. Without B, surplus would be 5. So B pays 2. What’s undesirable? Not in the core. Item AItem BPackage AB Bidder 1030 Bidder 2300 Bidder 3115

Several problems for VCG auctions with complementarities: The revenues may be low, and the outcome may not be in the core. –Literature on Core-Selecting Auctions If there are winner’s-curse problems, ascending-bid auctions may be better. –SAA, plus work by Ausubel The problem can quickly get computationally complex. –100 items and all possible packages?

The Simultaneous Ascending Auction has been used in practice by the FCC. See McAfee & McMillan (1996). Two interesting strategic problems in market design: Exposure problem: Without package bidding, a package bidder may get “stuck” overpaying for a single license. Threshold problem: two individual- license bidders may tend to free-ride, fail to displace a bidder on a package.

What are the three lessons from Roth’s market-design work?

Provide market thickness. Overcome the congestion that thickness can bring, so that participants can to consider alternative transactions. Make it safe to participate in the market –Rather than staying out –Rather than behaving strategically in a way that distorts market efficiency

More questions on Roth What are the five markets that Roth has worked on? Can you think of any other examples of market design?

More questions on Roth What are the five markets that Roth has worked on? Can you think of any other examples of market design? –Financial markets –Sponsored-search auctions –B2B exchanges –Privatization auctions –College course selection

How does the Gale-Shapley algorithm work? Students and hospitals report complete preference orderings. Order the students (perhaps randomly). The first student proposes to her first-choice hospital. If the hospital finds her acceptable, make this tentative assignment. The next student does the same. If a student proposes to a hospital that already has a tentative match, replace that student if the match can be improved, otherwise go to the next-choice hospital and try again. If someone gets “bumped” from a tentative match, move them to a tentative match with their next available choice. Continue until all students either have been matched, or have no remaining options available from their preference list.

Exercise: use the Gale-Shapley algorithm to compute matchings in the following example. Three potential grooms: A, B, C. Three potential brides: X, Y, Z. Grooms are the proposers, brides are the receivers. Preferences are (best to worst): A: YXZ B: ZYX C: XZY X: BAC Y: CBA Z: ACB

In this example, the algorithm converges in one step. A proposes to Y. –Y tentatively accepts. B proposes to Z. –Z tentatively accepts. C proposes to X. –X tentatively accepts. Final matching: {AY,BZ,CX} Stable matching: No two would exchange places. Grooms: A: YXZ B: ZYX C: XZY Brides: X: BAC Y: CBA Z: ACB

Exercise: What is an example of an unstable matching? Grooms: A: YXZ B: ZYX C: XZY Brides: X: BAC Y: CBA Z: ACB

Exercise: What is an example of an unstable matching? There are six possible matchings. Only three are stable. {AX,BY,CZ} - stable {AX,BZ,CY} - C & Z prefer each other {AY,BX,CZ} - B & Y prefer each other {AY,BZ,CX} - stable {AZ,BX,CY} - stable {AZ,BY,CX} - A & X prefer each other Grooms: A: YXZ B: ZYX C: XZY Brides: X: BAC Y: CBA Z: ACB

Nice properties of Gale-Shapley: Always converges to a stable matching. Proposer side has a (weakly) dominant strategy to report truthfully.

A receiver can have a strategic incentive to shorten her list. With truthtelling and grooms as proposers,the final matching was: {AY,BZ,CX} Though this matching is stable, each bride is getting her last choice. What if Y reports just “CB,” indicating that A is unacceptable to her? Grooms: A: YXZ B: ZYX C: XZY Brides: X: BAC Y: CBA Z: ACB

Suppose Y reports CB instead of CBA. Then the algorithm proceeds as follows. A proposes to Y. –Y rejects. A proposes to X. –X tentatively accepts. B proposes to Z. –Z tentatively accepts. C proposes to X. –X rejects (because she prefers A). C proposes to Z. –Z tentatively accepts, rejecting B. B proposes to Y. –Y tentatively accepts. Final matching: {AX,BY,CZ} Grooms: A: YXZ B: ZYX C: XZY Brides: X: BAC Y: CBA Z: ACB Note that Y is better off than in the previous stable matching: {AY,BZ,CX}.

An example of game theory’s role in market design. Since truthtelling is a dominant strategy for the proposer side in the GS algorithm, we might assign that role to agents whose strategy we want to simplify. –Students in school choice –Doctors in residency matc Note that with larger markets, the incentives for strategic behavior are relatively small on the receiver side as well.