AEA Continuing Education in Game Theory Avinash Dixit

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. • To handle this, we may want to create a combinatorial version of the Vickrey auction. • Example: Item A Item B Package AB Bidder 1 1 4 9 Bidder 2 3 2 5 Bidder 3 0 5 6 • 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. • Example: Item A Item B Package AB Bidder 1 1 4 9 Bidder 2 3 2 5 Bidder 3 0 5 6 • 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 A Item B Package AB Bidder 1 1 4 7 Bidder 2 3 2 5 Bidder 3 0 5 6

What happens if we change the values slightly in the example? • Now what are the VCG allocation and payments? Item A Item B Package AB Bidder 1 1 4 7 Bidder 2 3 2 5 Bidder 3 0 5 6 • 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? Item A Item B Package AB Bidder 1 0 3 0 Bidder 2 3 0 0 Bidder 3 1 1 5 • What’s undesirable about the outcome?

One more example. Item A Item B Package AB Bidder 1 0 3 0 Bidder 2 3 0 0 Bidder 3 1 1 5 • 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.

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 Mc. Afee & Mc. Millan (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 individuallicense bidders may tend to free-ride, fail to displace a bidder on a package.

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

What are 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 – B 2 B 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. Grooms: A: YXZ B: ZYX C: XZY • 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} 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.