Automated Mechanism Automated Mechanism DesignDesign

Tuomas SandholmTuomas Sandholm

Presented by Dimitri MostinskiPresented by Dimitri MostinskiNovember 17, 2004November 17, 2004

Mechanism DesignMechanism Design

AArt of designing the rules of thert of designing the rules of the game game (aka. (aka. mechanismmechanism) so that a desirable ) so that a desirable outcome (according to a givenoutcome (according to a given objective) objective) is reached despite the fact that each agent is reached despite the fact that each agent acts in his own selfinterestacts in his own selfinterest

Some examples of applicationsSome examples of applications AAuctionsuctions VVoting protocolsoting protocols DDivorce settlement proceduresivorce settlement procedures CCollaborative rating systemsollaborative rating systems

Manual Mechanism DesignManual Mechanism Design

Traditional approach to mechanism designTraditional approach to mechanism design Good design is hypothesized based on Good design is hypothesized based on

designers experience and intuition and designers experience and intuition and then desirable properties are proven then desirable properties are proven formallyformally

Over last 40 years a small number of Over last 40 years a small number of canonical mechanisms were created, each canonical mechanisms were created, each designed for a class of settings and a designed for a class of settings and a specific objectivespecific objective

Problems with Manual MDProblems with Manual MD

The most famous and most broadly applicable general The most famous and most broadly applicable general mechanisms, VCGmechanisms, VCG and dAGVA, only maximize social welfareand dAGVA, only maximize social welfare

The general mechanisms that do focus on a self-interested The general mechanisms that do focus on a self-interested designer are onlydesigner are only applicable in very restricted settingsapplicable in very restricted settings

TThe designer mayhe designer may also be interested in the outcomealso be interested in the outcome per seper se It is often assumed that side payments can be used to tailor It is often assumed that side payments can be used to tailor

the agents' incentives,the agents' incentives, but this is not always practicalbut this is not always practical The most common mechanismsThe most common mechanisms assume that the agents assume that the agents

have quasilinear preferenceshave quasilinear preferences uuii((o; o; 11, .., .. ,, NN) = ) = vvii((oo))− − ii

Impossibility ResultsImpossibility Results

Traditional research has yielded a number Traditional research has yielded a number of impossibility results of the form “of impossibility results of the form “no no mechanism mechanism worksworks across a across a classclass of settings of settings” ” for different definitions of “works” and for different definitions of “works” and different classes of settings.different classes of settings. E.g. E.g. Gibbard-Satterthwaite theoremGibbard-Satterthwaite theorem states states

that that for the class of for the class of general preferencesgeneral preferences,, no no mechanism mechanism exists where exists where

an outcomean outcome outcome can outcome can be any one of at least three be any one of at least three candidatescandidates

the mechanism is nondictatorialthe mechanism is nondictatorial truth telling is a dominant strategy for all agentstruth telling is a dominant strategy for all agents

Automatic Mechanism Design Automatic Mechanism Design (AMD)(AMD)

A novel approach to mechanism A novel approach to mechanism design proposed by design proposed by Conitzer and Conitzer and SandholmSandholm in 2002 in 2002

MMechanismechanism is computationally is computationally created for the specic problem created for the specic problem instance atinstance at handhand

Advantages of AMDAdvantages of AMD

It can be used in settings beyond the classes of It can be used in settings beyond the classes of problems that have beenproblems that have been successfully studied in successfully studied in manual mechanism design to datemanual mechanism design to date

It can allow one to circumvent the impossibility It can allow one to circumvent the impossibility resultsresults by considering an instance of the class not by considering an instance of the class not the class itselfthe class itself

It can yield It can yield mechanisms that produce better mechanisms that produce better results and are harder to manipulate by using the results and are harder to manipulate by using the information that the mechanism designer has information that the mechanism designer has about the agents‘about the agents‘ preferencespreferences

It shifts the burden of mechanism design from It shifts the burden of mechanism design from humans to a machine.humans to a machine.

AMD formalismAMD formalism

Am automatic mechanism design Am automatic mechanism design setting issetting is A finite set of outcomes OA finite set of outcomes O A finite set of N agentsA finite set of N agents For each agent IFor each agent I

A finite set of types A finite set of types ii

A probability distribution A probability distribution ii over over ii

A utility function uA utility function uii : : ii x O x O RR An objective function whose An objective function whose expectation the expectation the

designer wishes to maximizedesigner wishes to maximize gg((o; o; 11, .., .. ,, NN))

More AMD formalismMore AMD formalism

A mechanism consists ofA mechanism consists of An outcome selection function An outcome selection function

o : o : 11x .. x x .. x NN O if it is deterministic O if it is deterministic A distribution selection function A distribution selection function

p : p : 11x .. x x .. x NN P(O) if it is randomized P(O) if it is randomized For each agent i a payment selection For each agent i a payment selection

functionfunction

ii: : 11x .. x x .. x NN R if it involves payments R if it involves payments

Individual RationalityIndividual Rationality

An agent must never be worse off by participating An agent must never be worse off by participating in the mechanismin the mechanism

Types of Individual RationalityTypes of Individual Rationality Ex anteEx ante the agent would participate if it knewthe agent would participate if it knew nothing at nothing at

all (not even its own type)all (not even its own type) Ex interimEx interim the agent would always participate if it knewthe agent would always participate if it knew

only its own typeonly its own type Ex postEx post the agentthe agent would always participate even if it knew would always participate even if it knew

everybody's typeeverybody's type In an AMD setting with an IR constraint there In an AMD setting with an IR constraint there

exists a fallback outcome oexists a fallback outcome o0 0 such that for every such that for every agent iagent i uuii((ii,o,o00) = 0) = 0

Incentive CompatibilityIncentive Compatibility

TThe agents should never have an incentivehe agents should never have an incentive to misreport their typeto misreport their type

TTwo most commonwo most common solution conceptssolution concepts are are implementation in dominant strategiesimplementation in dominant strategies

Truth telling is the optimal strategy even if all other Truth telling is the optimal strategy even if all other agents’ types are knownagents’ types are known

implementation in Bayesian Nash equilibriumimplementation in Bayesian Nash equilibrium Truth telling is the optimal strategy if other agents’ Truth telling is the optimal strategy if other agents’

types are not yet known, but they are assumed to be types are not yet known, but they are assumed to be truthfultruthful

Formally the AMD problemFormally the AMD problem

GivenGiven Automated mechanism design settingAutomated mechanism design setting An IR notion (ex interim, ex post, or none)An IR notion (ex interim, ex post, or none) A solution concept (dominant strategies or Bayesian A solution concept (dominant strategies or Bayesian

Nash equilibrium)Nash equilibrium) Possibility of payments and randomizationPossibility of payments and randomization A target value GA target value G

DetermineDetermine If there exists a mechanism of the specified type that If there exists a mechanism of the specified type that

satisfies both the IR notion and the solution concept, and satisfies both the IR notion and the solution concept, and gives an expected value of at least G for the objective.gives an expected value of at least G for the objective.

Complexity resultsComplexity results

Consider a case of only one agentConsider a case of only one agent The two discussed IR options coincide hereThe two discussed IR options coincide here The two solution concepts coincide as wellThe two solution concepts coincide as well

Proving hardness for this case would imply lower bound on Proving hardness for this case would imply lower bound on the general problemthe general problem

AMD is NP-hard (by reduction to MINSAT) ifAMD is NP-hard (by reduction to MINSAT) if Payments are not allowedPayments are not allowed Payments are allowed but the designer is looking for Payments are allowed but the designer is looking for

something other than social welfare maximizationsomething other than social welfare maximization AMD can be solved in (expected) polynomial time using AMD can be solved in (expected) polynomial time using

randomized algorithm for LP problemsrandomized algorithm for LP problems If the input is structured in a way that it can be concisely If the input is structured in a way that it can be concisely

communicated it can also be faster processedcommunicated it can also be faster processed

An exampleAn example

11 LowLow HighHigh

LowLow HusbandHusband HusbandHusband

HighHigh HusbandHusband HusbandHusband

22 LowLow HighHigh

LowLow JointJoint HusbandHusband

HighHigh WifeWife BurnedBurned

33 LowLow HighHigh

LowLow .57 .57 HusbandHusband

.43 Wife.43 Wife

HusbandHusband

HighHigh WifeWife .45 .45 BurnedBurned

.55 .55 HusbandHusband

Some results of AMDSome results of AMD

It reinvented the Myerson auction which It reinvented the Myerson auction which maximizes the seller's expected revenue maximizes the seller's expected revenue in a 1-object auctionin a 1-object auction

It created expected revenue maximizing It created expected revenue maximizing combinatorial auctionscombinatorial auctions

It created optimal mechanisms for a public It created optimal mechanisms for a public good problem (deciding whether or not to good problem (deciding whether or not to build a bridge)build a bridge)

It created optimal mechanisms for public It created optimal mechanisms for public goods problems with multiple goodsgoods problems with multiple goods

ConclusionConclusion

Automated mechanism design is a Automated mechanism design is a brand new area of researchbrand new area of research

The problems that were long studied The problems that were long studied for manual mechanism design can all for manual mechanism design can all be applied to AMDbe applied to AMD