Automated Mechanism Design: Complexity Results Stemming From

the Single-Agent Setting

Vincent Conitzer and Tuomas SandholmComputer Science Department

Carnegie Mellon University

Mechanism design

• Mechanism design = – designing the rules of the game…– so that a (globally) good outcome happens… – although each agent plays strategically to maximize own utility– E.g. auctions, elections

• Setting– Each agent has a type which specifies the agent’s preferences– The designer doesn’t know the types, but has a prior over them– The designer constructs a mechanism (mapping from the

agents’ type reports to outcomes)– The agents report their types to the mechanism, which then

determines the outcome

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

• Mechanism design trivializes each agent’s search for a strategy

Available tools

• Sometimes the designer can force the agents to make certain payments– E.g. auction

• Sometimes the mechanism is allowed to be randomized– Each type vector defines a probability distribution

over outcomes

Objective

• The designer has an objective to maximize– Social welfare– Total payments to the designer– Own agenda for the outcome– …

• In general, function of how the outcome relates to the agents’ preferences

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 (not social welfare); 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

– Inputs: players, outcomes, type space, prior over types, designer’s objective, types of mechanism allowed (payments? randomized?)

– Output: optimal mechanism– Creating a mechanism for the specific setting

(instance) at hand rather than a class of settings

Conitzer & Sandholm, Bayesian Modeling Applications Workshop (UAI-03)

Small example: Divorce arbitration

• Outcomes:

• Each agent is of type high w.p. .2 and type low w.p. .8– Preferences of the high type:

• u(get the painting) = 11,000• u(museum) = 6,000• u(other gets the painting) = 1,000• u(burn) = 0

– Preferences of the low type:• u(get the painting) = 1,200• u(museum) = 1,100• u(other gets the painting) = 1,000• u(burn) = 0

Optimal deterministic mechanism for maximizing the sum of the divorcees’ utilities

high

low

lowhigh

Expected sum of divorcees’ utilities = 5,248

Mechanism

Optimal randomized mechanism for maximizing the sum of the divorcees’ utilities

high

low

lowhigh

.57.43

.55 .45

Expected sum of divorcees’ utilities = 5,510

Optimal randomized mechanism with payments for maximizing the sum of the divorcees’ utilities

high

low

lowhigh

Expected sum of divorcees’ utilities = 5,688

Wife pays 1,000

Optimal randomized mechanism with payments for maximizing the arbitrator’s revenue

high

low

lowhigh

Expected sum of divorcees’ utilities = 0 Arbitrator expects 4,320

Both pay 250Wife pays 13,750

Husband pays 11,250

Conitzer & Sandholm, Bayesian Modeling Applications Workshop (UAI-03)

Automated mechanism design has already created important mechanisms

• Reinvented the revenue-maximizing 1-object auction– Celebrated result in auction theory [Myerson 81]

• Invented revenue-maximizing combinatorial auctions– Recognized tough open research problem

• Public goods problems– Allowed us to model money burning as a loss in social

welfare– Circumvented the Myerson-Satterthwaite impossibility

theorem (satisfied social welfare maximization, budget balance, and voluntary participation)

• Combinatorial public goods problems

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. social welfare

– 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 (if any)

• maximizes the expectation of the objective function

Laying out the computational complexity of AMD

• So: how hard is AMD?• Many different variants

– Social welfare/payment maximizing/designer’s agenda for outcome

– Payments allowed/not allowed

– Deterministic mechanisms/randomized mechanisms

– Ex interim IR/ex post IR/no IR

– Dominant strategies/Bayes-Nash equilibrium

– …

• The above already gives 3*2*2*3*2 = 72 variants• Trick: hardness results in specific settings imply hardness

in more general settings

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!

How hard is AMD withdeterministic mechanisms?

NP-complete (even with one player)

Solvable in polynomial time (multiple players)

• Maximizing social welfare (without payments)

• Designer’s own agenda (without payments) [EC03]

• General payment-independent objectives

(with payments)• Maximizing expected

revenue [EC03]

• Maximizing social welfare (with payments)

(VCG)

Sketch of NP-hardness proofs: MINSAT

• In the MINSAT problem we seek to minimize the number of satisfied clauses in a formula in conjunctive normal form

• For example, (X1X2) (X1X3) (X2X3)• Say X1=false, X2=false, X3=false

– (X1X2) (X1X3) (X2X3) (two satisfied)

• Better: X1=true, X2=true, X3=false– (X1X2) (X1X3) (X2X3) (one satisfied)

• This problem is NP-complete [Kohli et al. 94]– We independently derived a reduction from MAX2SAT

Sketch of NP-hardness proofs: the reduction

• Given a MINSAT instance (a CNF formula, e.g. (X1X2) (X1X3) (X2X3)), we construct a single-agent deterministic AMD instance where– For every variable v, there is a type v, and outcomes o+v, o-v

– For every clause c, there is a type c, and outcome oc

– A good mechanism will always select one of o+v, o-v for type v

– To maximize the objective, would like to select oc for type c, but only possible if the mechanism never selects ol for lc

• E.g. for incentive compatibility reasons; details omitted here

• If we consider picking o+v for type v “setting v to true”, and picking o-v for type v “setting v to false”, then– Satisfying a clause preventing the best outcome for the type corresponding to that

clause– Want to minimize the number of satisfied clauses!

How hard is AMD withrandomized mechanisms?

• Everything becomes doable in polynomial time!!!• Use linear programming

– Probabilities of outcomes for given type reports are variables– Players’ payments for given type reports are variables

• Objectives under discussion are linear• IR constraints are linear

• IC constraints are linear

• Forcing the probabilities to be 0 or 1 also gives a MIP formulation for AMD with deterministic mechanisms

Ex post

Ex interim

Dominantstrategies

Bayes-Nashequilibrium

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

• We have shown that:– Designing deterministic mechanisms is typically NP-complete even with only 1 agent– Designing randomized mechanisms is in P for any constant number of agents (using LP)

• Future research:– New algorithms for greater scalability– New applications/bringing to industry– Using AMD as a tool in classical mechanism design

Thank you for your attention!