Self-interested Automated Mechanism Design and Implications for Optimal Combinatorial Auctions

1. Self-interested Automated Mechanism Design and Implications for Optimal Combinatorial Auctions Vincent Conitzer and Tuomas Sandholm Computer Science Department Carnegie Mellon University

2. Self-interested designers • There are three types of self-interested designers: • Payment maximizing designers that care only about the (expected) sum of payments by the agents, ii • For example, optimal (revenue-maximizing) auctions • Designers that care only about their own agenda for the outcome, g(o) • In contrast with benevolent (e.g., social welfare maximizing) designers that care about how the outcome relates to the agents’ types, g(,o) • Designers that care about both about payments and their own agenda, g(o) + ii • Designer has control over outcome and payments, but: The designer cannot make the agents worse off than they would have been without the mechanism • This is an individual rationality (IR) constraint

3. Constraints on the mechanism • Incentive compatibility constraints: Each agent (for each type) best off reporting truthfully • Dominant strategies: for any type reports by the other agents, each agent is best of reporting truthfully • Bayes-Nash equilibrium (weaker): each agent is best off reporting truthfully when not aware of other agents’ types • Participation constraints: Each agent (for each type) benefits from participating in the mechanism • Ex post: beneficial for any type reports by the other agents • Ex interim: beneficial when not aware of other agents’ types

4. General preferences Quasilinear prefs Classical mechanism design • Classical mechanism design has created a number of canonical mechanisms • Vickrey, Clarke, Groves mechanisms; Myerson auction; … • These obtain a particular goal over a range of settings • It has also created impossibility results • Gibbard-Satterthwaite; Myerson-Satterthwaite; … • Show that no mechanism obtains a goal over a range of settings

5. Difficulties with canonical mechanisms • A single preference aggregation instance comes along • A particular set of outcomes, players, sets of possible preferences (types), priors over preferences, … • What if no canonical mechanism covers this instance? • Unusual objective; payments not possible; … • Impossibility results may exist for the general type of setting • But the particular instance may have additional structure so that good mechanisms do exist => can circumvent impossibility result • What if a canonical mechanism does cover the setting? • Can we use instance’s structure to get higher objective value? • Can we get stronger nonmanipulability/participation properties? • Dominant strategies instead of Bayes-Nash equilibrium • Ex-post IR instead of ex-interim • SOLUTION: hire a mechanism designer for every instance!

6. A cheaper, faster solution: Automated mechanism design Solve mechanism design as an optimization problem automatically for the instance at hand

7. Defining the computational problem: Input • An instance is given by • Set of possible outcomes • Set of agents • For each agent • set of possible types • probability distribution over these types • utility function converting type/outcome pairs to utilities • Objective function • Gives a value for each outcome for each combination of agents’ types • E.g. payment maximization • Restrictions on the mechanism • Are side payments allowed? • Is randomization over outcomes allowed? • What concept of nonmanipulability is used? • What participation constraint (if any) is used?

8. Defining the computational problem: Output • The algorithm should produce • a mechanism • A mechanism maps combinations of agents’ revealed types to outcomes • Randomized mechanism maps to probability distributions over outcomes • Also specifies payments by agents (if side payments are allowed) • … which • is nonmanipulable (according to the given concept) • By revelation principle, we can focus on truth-revealing direct-revelation mechanisms w.l.o.g. • satisfies the given participation constraint • maximizes the expectation of the objective function

9. Incentive compatibility constraints coincide with 1 (reporting) agent Bayes-Nash equilibrium: Reporting truthfully is optimal in expectation over the other agents’ (true) types Dominant strategies: Reporting truthfully is optimal for any types the others report P(21)u1(11,o5) + P(22)u1(11,o9) P(21)u1(11,o3) + P(22)u1(11,o2) u1(11,o5)u1(11,o3) AND u1(11,o9) u1(11,o2) With only 1 reporting agent, the constraints are the same! u1(11,o5)u1(11,o3)  P(21)u1(11,o5) P(21)u1(11,o3)

10. Individual rationality constraints coincide with 1 (reporting) agent Ex interim: Participating does not hurt in expectation over the other agents’ (true) types Ex post: Participating never hurts (for any reported types for the other agents) u1(11,o5) 0 AND u1(11,o9) 0 P(21)u1(11,o5) + P(22)u1(11,o9) 0 With only 1 reporting agent, the constraints are the same! u1(11,o5) 0  P(21)u1(11,o5) 0

11. Results • Theorem. Designing the payment maximizing deterministic mechanism is NP-complete. • Holds even in the single agent setting • This implies it holds for any combination of IC and IR constraints • Theorem. When payments are not possible (and hence the designer is only interested in her own agenda), designing the optimal deterministic mechanism is NP-complete. • Holds even in the single agent setting • This implies it holds for any combination of IC and IR constraints • Theorem. Designing the optimal randomized mechanism is solvable in polynomial time by linear programming • For any (self-interested) objective • For any constant number of agents • For any combination of IC and IR constraints

12. Optimal combinatorial auctions • Optimal auction = revenue maximizing auction • For one item: Myerson auction [1981] • For more than one item, open problem (even with only two items) • Two items case with special structure solved [Armstrong 00] • Can solve optimal auction design problems directly using AMD (for setting at hand) … • … but typically leads to exponential blowup in outcome space

13. Best-only preferences • Suppose that the bidders only care about the best item in their bundle relative to their type • E.g. because each bidder will discard the other ones • Formally, this means we can express the utility function as ui(, S) = maxsSvi(, s) for some function vi

14. Mechanism design with best-only preferences • Observation: with BO preferences, it never makes sense to award a bidder more than one item • Proof: • Take a mechanism that sometimes awards bundles • Replace each (nonempty) bundle awarded to an agent with the best item in it (for that agent for her reported type) • If the agent was truthful, then she is no worse off • If she was lying, she is no better off

15. BO preferences and AMD (1) • Number of allocations where each bidder receives at most 1 item is at most (|I|+1)kwhere k is the number of bidders • Exponential only in number of bidders! • So: Theorem. When the number of bidders is a constant, we can find the optimal randomized mechanism in polynomial time using LP

16. BO preferences and AMD (2) • What about hardness results for deterministic mechanisms? • If there is only one bidder, outcome = which item does the bidder win (if any) • Thus, the bidder can have an arbitrary utility function over the outcomes • Except for the “default” outcome of getting nothing • So: Theorem. The one-agent BO-preferences optimal-auction problem can capture the full generality of one-agent revenue maximizing AMD => it is NP-complete

17. Conclusions • In automated mechanism design, mechanisms are designed on the fly for the setting at hand • Applicable in settings not covered by classical mechanisms • Can outperform classical mechanisms • Circumvents impossibility results about general mechanisms • Here the focus was on self-interesteddesigners • Payment maximization • Own agenda for the outcome • Self-interested AMD is NP-complete even in the simplest settings for deterministic mechanisms • Even with 1 agent, so for any IR and IC notions • But for randomized mechanisms, it is in P (linear programming) even in complicated settings • Any (constant) number of agents, any combination of IR and IC notions • These results transfer to the optimal combinatorial auction design problem with best-only preferences

18. Thank you for your attention!