Automated Mechanism Design Tuomas Sandholm Presented by Dimitri Mostinski November 17, 2004

Mechanism Design Art of designing the rules of the game (aka. mechanism) so that a desirable outcome (according to a given objective) is reached despite the fact that each agent acts in his own selfinterest Art of designing the rules of the game (aka. mechanism) so that a desirable outcome (according to a given objective) is reached despite the fact that each agent acts in his own selfinterest Some examples of applications Some examples of applications Auctions Auctions Voting protocols Voting protocols Divorce settlement procedures Divorce settlement procedures Collaborative rating systems Collaborative rating systems

Manual Mechanism Design Traditional approach to mechanism design Traditional approach to mechanism design Good design is hypothesized based on designers experience and intuition and then desirable properties are proven formally Good design is hypothesized based on designers experience and intuition and then desirable properties are proven formally Over last 40 years a small number of canonical mechanisms were created, each designed for a class of settings and a specific objective Over last 40 years a small number of canonical mechanisms were created, each designed for a class of settings and a specific objective

Problems with Manual MD The most famous and most broadly applicable general mechanisms, VCG and dAGVA, only maximize social welfare The most famous and most broadly applicable general mechanisms, VCG and dAGVA, only maximize social welfare The general mechanisms that do focus on a self-interested designer are only applicable in very restricted settings The general mechanisms that do focus on a self-interested designer are only applicable in very restricted settings The designer may also be interested in the outcome per se The designer may also be interested in the outcome per se It is often assumed that side payments can be used to tailor the agents' incentives, but this is not always practical It is often assumed that side payments can be used to tailor the agents' incentives, but this is not always practical The most common mechanisms assume that the agents have quasilinear preferences u i (o;  1,..,  N ) = v i (o)−  i The most common mechanisms assume that the agents have quasilinear preferences u i (o;  1,..,  N ) = v i (o)−  i

Impossibility Results Traditional research has yielded a number of impossibility results of the form “no mechanism works across a class of settings” for different definitions of “works” and different classes of settings. Traditional research has yielded a number of impossibility results of the form “no mechanism works across a class of settings” for different definitions of “works” and different classes of settings. E.g. Gibbard-Satterthwaite theorem states that for the class of general preferences, no mechanism exists where E.g. Gibbard-Satterthwaite theorem states that for the class of general preferences, no mechanism exists where an outcome outcome can be any one of at least three candidates an outcome outcome can be any one of at least three candidates the mechanism is nondictatorial the mechanism is nondictatorial truth telling is a dominant strategy for all agents truth telling is a dominant strategy for all agents

Automatic Mechanism Design (AMD) A novel approach to mechanism design proposed by Conitzer and Sandholm in 2002 A novel approach to mechanism design proposed by Conitzer and Sandholm in 2002 Mechanism is computationally created for the specic problem instance at hand Mechanism is computationally created for the specic problem instance at hand

Advantages of AMD It can be used in settings beyond the classes of problems that have been successfully studied in manual mechanism design to date It can be used in settings beyond the classes of problems that have been successfully studied in manual mechanism design to date It can allow one to circumvent the impossibility results by considering an instance of the class not the class itself It can allow one to circumvent the impossibility results by considering an instance of the class not the class itself It can yield mechanisms that produce better results and are harder to manipulate by using the information that the mechanism designer has about the agents‘ preferences It can yield mechanisms that produce better results and are harder to manipulate by using the information that the mechanism designer has about the agents‘ preferences It shifts the burden of mechanism design from humans to a machine. It shifts the burden of mechanism design from humans to a machine.

AMD formalism Am automatic mechanism design setting is Am automatic mechanism design setting is A finite set of outcomes O A finite set of outcomes O A finite set of N agents A finite set of N agents For each agent I For each agent I A finite set of types  i A finite set of types  i A probability distribution  i over  i A probability distribution  i over  i A utility function u i :  i x O  R A utility function u i :  i x O  R An objective function whose expectation the designer wishes to maximize g(o;  1,..,  N ) An objective function whose expectation the designer wishes to maximize g(o;  1,..,  N )

More AMD formalism A mechanism consists of A mechanism consists of An outcome selection function An outcome selection function o :  1 x.. x  N  O if it is deterministic A distribution selection function A distribution selection function p :  1 x.. x  N  P(O) if it is randomized For each agent i a payment selection function For each agent i a payment selection function  i  :  1 x.. x  N  R if it involves payments  i  :  1 x.. x  N  R if it involves payments

Individual Rationality An agent must never be worse off by participating in the mechanism An agent must never be worse off by participating in the mechanism Types of Individual Rationality Types of Individual Rationality Ex ante the agent would participate if it knew nothing at all (not even its own type) Ex ante the agent would participate if it knew nothing at all (not even its own type) Ex interim the agent would always participate if it knew only its own type Ex interim the agent would always participate if it knew only its own type Ex post the agent would always participate even if it knew everybody's type Ex post the agent would always participate even if it knew everybody's type In an AMD setting with an IR constraint there exists a fallback outcome o 0 such that for every agent i u i (  i,o 0 ) = 0 In an AMD setting with an IR constraint there exists a fallback outcome o 0 such that for every agent i u i (  i,o 0 ) = 0

Incentive Compatibility The agents should never have an incentive to misreport their type The agents should never have an incentive to misreport their type Two most common solution concepts are Two most common solution concepts are implementation in dominant strategies implementation in dominant strategies Truth telling is the optimal strategy even if all other agents’ types are known Truth telling is the optimal strategy even if all other agents’ types are known implementation in Bayesian Nash equilibrium implementation in Bayesian Nash equilibrium Truth telling is the optimal strategy if other agents’ types are not yet known, but they are assumed to be truthful Truth telling is the optimal strategy if other agents’ types are not yet known, but they are assumed to be truthful

Formally the AMD problem Given Given Automated mechanism design setting Automated 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 Nash equilibrium) A solution concept (dominant strategies or Bayesian Nash equilibrium) Possibility of payments and randomization Possibility of payments and randomization A target value G A target value G Determine Determine If there exists a mechanism of the specified type that satisfies both the IR notion and the solution concept, and gives an expected value of at least G for the objective. If there exists a mechanism of the specified type that satisfies both the IR notion and the solution concept, and gives an expected value of at least G for the objective.

Complexity results Consider a case of only one agent Consider a case of only one agent The two discussed IR options coincide here The two discussed IR options coincide here The two solution concepts coincide as well The two solution concepts coincide as well Proving hardness for this case would imply lower bound on the general problem Proving hardness for this case would imply lower bound on the general problem AMD is NP-hard (by reduction to MINSAT) if AMD is NP-hard (by reduction to MINSAT) if Payments are not allowed Payments are not allowed Payments are allowed but the designer is looking for something other than social welfare maximization Payments are allowed but the designer is looking for something other than social welfare maximization AMD can be solved in (expected) polynomial time using randomized algorithm for LP problems AMD can be solved in (expected) polynomial time using randomized algorithm for LP problems If the input is structured in a way that it can be concisely communicated it can also be faster processed If the input is structured in a way that it can be concisely communicated it can also be faster processed

An example 1LowHighLowHusbandHusband HighHusbandHusband 2LowHighLowJointHusband HighWifeBurned 3LowHighLow.57 Husband.43 Wife Husband HighWife.45 Burned.55 Husband

Some results of AMD It reinvented the Myerson auction which maximizes the seller's expected revenue in a 1- object auction It reinvented the Myerson auction which maximizes the seller's expected revenue in a 1- object auction It created expected revenue maximizing combinatorial auctions It created expected revenue maximizing combinatorial auctions It created optimal mechanisms for a public good problem (deciding whether or not to build a bridge) It created optimal mechanisms for a public good problem (deciding whether or not to build a bridge) It created optimal mechanisms for public goods problems with multiple goods It created optimal mechanisms for public goods problems with multiple goods

Conclusion Automated mechanism design is a brand new area of research Automated mechanism design is a brand new area of research The problems that were long studied for manual mechanism design can all be applied to AMD The problems that were long studied for manual mechanism design can all be applied to AMD