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

Vincent Conitzer and Tuomas SandholmComputer Science Department

Carnegie Mellon University

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, ii

• 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) + ii

• 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

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

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

General preferences

Quasilinear prefs

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!

A cheaper, faster solution: Automated mechanism design

Solve mechanism design as an optimization problem automatically for the instance at hand

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?

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

Incentive compatibility constraints coincide with 1 (reporting) agent

Dominant strategies:

Reporting truthfully is optimal for any types the others report

Bayes-Nash equilibrium:

Reporting truthfully is optimal

in expectation over the other agents’ (true) types

21 22

11 o5 o9

12 o3 o2

21 22

11 o5 o9

12 o3 o2

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)

21

11 o5

11 o3

u1(11,o5) u1(11,o3)

P(21)u1(11,o5) P(21)u1(11,o3)

With only 1 reporting agent, the constraints

are the same!

Individual rationality constraints coincide with 1 (reporting) agent

Ex post:

Participating never hurts (for any reported types for the other

agents)

Ex interim:

Participating does not hurt in expectation over the other

agents’ (true) types

21 22

11 o5 o9

12 o3 o2

21 22

11 o5 o9

12 o3 o2

P(21)u1(11,o5) +

P(22)u1(11,o9) 0

u1(11,o5) 0

AND

u1(11,o9) 0

21

11 o5

11 o3

u1(11,o5) 0

P(21)u1(11,o5) 0

With only 1 reporting agent, the constraints

are the same!

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

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

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) = maxsSvi(, s)

for some function vi

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 bundlesReplace 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 offIf she was lying, she is no better off

BO preferences and AMD (1)

• Number of allocations where each bidder receives at most 1 item is at most (|I|+1)k where 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

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

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-interested designers– 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

Thank you for your attention!