CPS 296.1Mechanism designMechanism design: settingWhat should the center do?Quasilinear utility functionsDefinition of a (direct-revelation) mechanismIncentive compatibilityIndividual rationalityThe Clarke (aka. VCG) mechanism [Clarke 71]The Clarke mechanism is strategy-proofGeneralized Vickrey Auction (GVA) (= VCG applied to combinatorial auctions)Additional nice properties of the Clarke mechanismClarke mechanism is not perfectWhy restrict attention to truthful direct-revelation mechanisms?The revelation principleMyerson-Satterthwaite impossibility [1983]A few computational issues in mechanism designCPS 296.1Mechanism designVincent Conitzer [email protected] design: setting• The center has a set of outcomes O that she can choose from– Allocations of tasks/resources, joint plans, …• Each agent i draws a type θi from Θi– usually, but not necessarily, according to some probability distribution• Each agent has a (commonly known) valuation function vi: Θi xO → ℜ– Note: depends on θi, which is not commonly known• The center has some objective function g: Θ x O → ℜ– Θ = Θ1 x ... x Θn– E.g., efficiency (Σi vi(θi, o))– May also depend on payments (more on those later)– The center does not know the typesWhat should the center do?• She would like to know the agents’ types to make the best decision• Why not just ask them for their types?• Problem: agents might lie• E.g., an agent that slightly prefers outcome 1 may say that outcome 1 will give him a value of 1,000,000 and everything else will give him a value of 0, to force the decision in his favor• But maybe, if the center is clever about choosing outcomes and/or requires the agents to make some payments depending on the types they report, the incentive to lie disappears…Quasilinear utility functions• For the purposes of mechanism design, we will assume that an agent’s utility for – his type being θi,– outcome o being chosen, – and having to pay πi,can be written as vi(θi, o) - πi• Such utility functions are called quasilinear• Some of the results that we will see can be generalized beyond such utility functions, but we will not do soDefinition of a (direct-revelation) mechanism•A deterministic mechanism without payments is a mapping o: Θ→O•A randomized mechanism without payments is a mapping o: Θ→∆(O)– ∆(O) is the set of all probability distributions over O• Mechanisms with payments additionally specify, for each agent i, a payment function πi : Θ→ℜ(specifying the payment that that agent must make)• Each mechanism specifies a Bayesian game for the agents, where i’s set of actions Ai = Θi– We would like agents to use the truth-telling strategy defined by s(θi)=θiIncentive compatibility• Incentive compatibility (aka. truthfulness) = there is never an incentive to lie about one’s type• A mechanism is dominant-strategies incentive compatible (aka. strategy-proof) if for any i, for any type vector θ1, θ2, …, θi, …, θn, and for any alternative type θi’, we havevi(θi, o(θ1, θ2, …, θi, …, θn)) - πi(θ1, θ2, …, θi, …, θn) ≥vi(θi, o(θ1, θ2, …, θi’, …, θn)) - πi(θ1, θ2, …, θi’, …, θn)• A mechanism is Bayes-Nash equilibrium (BNE) incentive compatible if telling the truth is a BNE, that is, for any i, for any types θi, θi’, Σθ-iP(θ-i) [vi(θi, o(θ1, θ2, …, θi, …, θn)) - πi(θ1, θ2, …, θi, …, θn)] ≥Σθ-iP(θ-i) [vi(θi, o(θ1, θ2, …, θi’, …, θn)) - πi(θ1, θ2, …, θi’, …, θn)]Individual rationality• A selfish center: “All agents must give me all their money.” – but the agents would simply not participate– If an agent would not participate, we say that the mechanism is not individually rational• A mechanism is ex-post individually rational if for any i, for any type vector θ1, θ2, …, θi, …, θn, we havevi(θi, o(θ1, θ2, …, θi, …, θn)) - πi(θ1, θ2, …, θi, …, θn) ≥0• A mechanism is ex-interim individually rational if for any i, for any type θi, Σθ-iP(θ-i) [vi(θi, o(θ1, θ2, …, θi, …, θn)) - πi(θ1, θ2, …, θi, …, θn)] ≥ 0– i.e., an agent will want to participate given that he is uncertain about others’ types (not used as often)The Clarke (aka. VCG) mechanism [Clarke 71]• The Clarke mechanism chooses some outcome o that maximizes Σi vi(θi’, o)– θi’ = the type that i reports• To determine the payment that agent j must make:– Pretend j does not exist, and choose o-jthat maximizes Σi≠jvi(θi’, o-j)–j pays Σi≠jvi(θi’, o-j) - Σi≠jvi(θi’, o) = Σi≠j(vi(θi’, o-j) - vi(θi’, o)) • We say that each agent pays the externality that she imposes on the other agents• (VCG = Vickrey, Clarke, Groves)The Clarke mechanism is strategy-proof• Total utility for agent j is vj(θj, o) - Σi≠j(vi(θi’, o-j) - vi(θi’, o)) =vj(θj, o) + Σi≠jvi(θi’, o) - Σi≠jvi(θi’, o-j) • But agent j cannot affect the choice of o-j• Hence, j can focus on maximizing vj(θj, o) + Σi≠jvi(θi’, o)• But mechanism chooses o to maximize Σi vi(θi’, o)• Hence, if θj’= θj, j’s utility will be maximized!• Extension of idea: add any term to agent j’spayment that does not depend on j’s reported type• This is the family of Groves mechanisms [Groves 73]Generalized Vickrey Auction (GVA) (= VCG applied to combinatorial auctions)• Example:– Bidder 1 bids ({A, B}, 5)– Bidder 2 bids ({B, C}, 7)– Bidder 3 bids ({C}, 3)• Bidders 1 and 3 win, total value is 8• Without bidder 1, bidder 2 would have won– Bidder 1 pays 7 - 3 = 4• Without bidder 3, bidder 2 would have won– Bidder 3 pays 7 - 5 = 2• Strategy-proof, ex-post IR, weakly budget balanced• Vulnerable to collusion (more so than 1-item Vickrey auction)– E.g., add two bidders ({B}, 100), ({A, C}, 100)– What happens?– More on collusion in GVA in [Ausubel & Milgrom 06, Conitzer & Sandholm 06]Additional nice properties of the Clarke mechanism• Ex-post individually rational, assuming:– An agent’s presence never makes it impossible to choose an outcome that could have been chosen if the agent had not been present, and– No agent ever has a negative value for an outcome that would be selected if that agent were not present• Weak budget balanced - that is, the sum of the payments is always nonnegative - assuming:– If an agent leaves, this never makes the
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