Bayesian Algorithmic Mechanism Design

Blackbox Reductions fromMechanisms to AlgorithmsNicole Immorlica, Northwestern U. & MSR1MACHINESCHEDULINGASSIGNMENTFeasibility constraintson outcome spaceAlgorithm Designv1v2v3v4v5Input vOutput xSET COVERGOAL: maximize (or minimize) some function f(x,v)ASSIGNMENTFeasibility constraintson outcome spaceMACHINESCHEDULINGMechanism DesignAllocation xSET COVERPayment pGOAL: maximize (or minimize) some function f(x,v)v1v2v3v4v5Input vb1b2b3b4b5Input bbi chosen to maximize utility = vixi(b)-pi(b)Algorithmic Mechanism Design:behind every great mechanism is a great algorithmcomputationincentives4HOLY GRAIL:general technique to convert algorithms into mechanisms5Black-Box TransformationsTransformationAlgorithmInput bAllocation xPayment pGOAL: for every algorithm, transformation preserves quality of solution in equilibrium.and is incentive compatible.Input vBlack-Box TransformationsTransformationAlgorithmAllocation x and is incentive compatible (IC), i.e., monotone:Input vex-post IC (truthful in expectation): allocation to agent i is increasing in is bid for all bid profiles of othersBayesian IC: allocation to agent i is increasing in is bid in expectation w.r.t. prior of over bid profiles of othersVCG TransformationOptimal AlgorithmInput vAllocation xEXAMPLE: Vickrey-Clark-Groves auction transforms any optimal algorithm into an optimal ex-post IC mechanism for any monotone objective function.Requires only black-box access to an optimal allocation rule.Does not depend on the social welfare problem being solved.Requires optimality.8Single Item Auctionone item, agent i has value vi for itemVCG TransformationSelectionAlgorithmInput vAllocation xFind agentw/max value(single-parameter)i2Combinatorial Auctionmany items, agent i has value vij for subset Sj i1i2i3agentsitems$10$12(multi-parameter)Combinatorial AuctionVCG Transformation???Input vAllocation xFind max valuenon-overlappingcollection of setsmany items, agent i has value vij for subset Sj (multi-parameter)HOLY GRAIL:general technique to convert algorithms into mechanisms^approximation12polynomial time?Bayesian IC transformations for welfaretruthful transformations for welfare(single- or multi-parameter!)Bayesian IC transformations for non-linear objectives(see Shuchis talk)= max i xivi s.t. feasibility constraints on alloc. x13BIC TransformationPositive Result: Transform approximation algorithms into Bayesian IC mechs with small loss in social welfare.Single-parameter: (single private value for allocation)Monotonization. For dist. F and algorithm A, there is a Bayesian IC transformation TA,F satisfying E[TA,F(v)] E[A(v)].Blackbox computation.TA,F can be computed in polytime with queries to A.Payment computation.Payments can be computed with two queries to A.xi(vi) = E[alloc. to i | vi]Not BICBICMonotonizationFact. Therere payments that make an alg. Bayesian IC if and only if for all i, expected allocation is monotone non-decreasing in value vi.viDescribe difference between BIC and TIE15MonotonizationGoal: construct yi from xi s.t.Monotonicity. yi(.) non-decreasing monotoneSurplus-preservation. Evi[viyi(vi)] Evi[vixi(vi)]Distribution-preservation. (can apply construction independently to each j)MonotonizationIdea 1: remap values.Monotone and surplus preserving, but not dist-pres.

17MonotonizationIdea 2: resample values.Dist-pres, but not monotone and surplus preserving.18MonotonizationIdea 3: resample values in region wherecumulative allocation is not monotone.allocationcumulativecurve19MonotonizationConstruction of yi(vi) from xi(vi) preserves:Distribution-preservation. Monotonicity. yi non-decreasing monotonexi(vi)yi(vi)MonotonizationConstruction of yi(vi) from xi(vi) preserves:Surplus-preservation. Evi[vi(yi - xi)] 0xi(vi)yi(vi)E[v(y-x)] = v(y-x) d f(v)(integration by parts)

= v(Y-X)| v(Y-X) d f(v)(v , X dominates Y)

= 0 (non-neg.) x (non-pos.)(2nd term non-pos.)

0 ababab21BIC Transformation for WelfarePositive Result: Transform approximation algorithms into Bayesian IC mechs with small loss in social welfare.Single-parameter: (single private value for allocation)Monotonization. For dist. F and algorithm A, there is a Bayesian IC transformation TA,F satisfying E[A(v)] E[TA,F(v)].Blackbox computation.TA,F can be computed in polytime with queries to A.Payment computation.Payments can be computed with two queries to A.Blackbox ComputationBIC Transformation for WelfarePositive Result: Transform approximation algorithms into Bayesian IC mechs with small loss in social welfare.Single-parameter: (single private value for allocation)Monotonization. For dist. F and algorithm A, there is a Bayesian IC transformation TA,F satisfying E[A(v)] E[TA,F(v)].Blackbox computation.TA,F can be computed in polytime with queries to A.Payment computation.Payments can be computed with two queries to A.Payment Computationpayment identityp(v) = v y(v) y(z) dzv0Idea: compute random variable P with E[P] = p(v)Payment Computationpayment identityp(v) = v y(v) y(z) dzv0Idea: compute random variable P with E[P] = p(v) Y indicator random variable for whether agent wins in A (with y(v)) z drawn uniformly from [0,v] Yz indicator random variable for whether agent wins in A (with y(z)) P = v (Y Yz)1st call to A2nd call to Aconst. # calls per agentfor a particular agent, do above26Payment Computationgoal: given A, find an alg. A that computesallocation and payments with just 1 call to A Pick agent k uniformly at random and draw wk from Fk Calculate outcome y for A(wk, v-k) For each agent i k, set pi = viyi For agent k, set pk = 0 if wk > vk and pk = -(n 1)yk/fk(wk) otherwise Output (y, p)only call to Augly formulaPayment ComputationThm. Algorithm A is Bayesian IC.Proof. Monotone. y linear transformation of y. y(v) = (1 - 1/n) y(v) + 1/n E[y(w)]Payment Identity.p(v) = v y(v) y(z) dzv0p(v) = (1 - 1/n) vy(v) (1/n)(n - 1) y(z) dzv0payment for i kpayment for i = k(see ugly formula)28Payment ComputationThm. Welfare is E[A(v)] E[A(v) max(v)]Proof. Each buyer has welfare (1 - 1/n) vy(v)Since y(v) is a probability, vy(v) max(v)Lose at most max(v) in total buyer welfareExpected payments are the same, so lose nothing in seller welfare Finds (alloc, payments) with 1 call to monotone alg.

[Babaioff, Kleinberg, Slivkins10]TransformationApprox. AlgorithmInput vPOSSIBILITY: can transform any approximation algorithm into a Bayesian IC mech. with small loss for f(x,v) = ixivi.

[Hartline-Lucier10]Dist. of values(drawn fromknown dist.)Allocation xPayment p30Multi-parameter TransformationGoal: construct allocation from algorithm s.t.Monotonicity.Surplus-preservation. Distribution-preservation. By mapping types of an agent to surrogates in a way that preserves above properties.Replicas and Surrogatesreplicas(drawn from F)surrogates(drawn from F)surrogate allocationsv(t,x(t))max-weightmatchingoriginal type tsurrogatetype tx(t)Set payment equal to VCG payment for type t.Set allocation equal to output on surrogate type profile.imagine replica/surrogates are lists of everything

32Replicas and SurrogatesThm. Transformation is distribution-preserving.Thm. Transformation is Bayesian IC.

Thm. Transformation doesnt lose much welfare.Prf. Because replicas are close to matched surrogates in values for outcomes.[Hartline, Kleinberg, Malekian11][Bei, Huang11]first imagine imagine replica/surrogates are lists of everything. next note same as experiment in which we select k things on left, k on right, and one random one on left as our original type, which gives us random on right due to perfect matching property.in fact follows from VCG payments being truthful and using dist.-pres. of other agents to argue values of replicas are accurate, so BIC conditioned on replicas/surrogates, now remove conditioning (note important replicas are diff from surrogates)follows from transportation cost analysis, pretend both sides are the same (but since we cant for BIC we need enough samples)

problems: assumed could calc expected value of replica for surrogate (can circumvent); poly in type space (can be large)33Strengthening the ResultSolution concept: black-box transformations for social welfare that preserve approximation and are truthful in expectation?

Social objective: black-box transformations that preserve approximation, are Bayesian IC, and work for other social objectives?

34GOAL:Find a general technique to convert approximation algorithms into truthful mech. for social welfare.IMPOSSIBLE

35Multi-parameter TransformationsThm. Theres no truthful in expectation mech. for combinatorial auctions with submodular valuations that guarantees a sub-linear approx.

Note: there is a (1-1/e)-approximation alg.[Dughmi, Vondrak11]Single-parameter TransformationsTruthful in Expectation. For all algorithms A, TA is truthful in expectation, i.e., expected allocation is monotone for all i.

Worst-case approximation preserving. For all values vectors v and algorithms A, expected welfare of transformation is close to expected welfare of algorithm.BAD NEWS:For any polytime truthful in expectation transformation, there is a welfare problem and alg. such that worst-case welfare of transformation is polynomially larger than the alg.s.38Proof OutlineDefine welfare instance (feasible allocations, values of agents).Find algorithm with high welfare.Use monotonicity to show any ex-post transformation has low worst-case welfare.Intuition(.5,.5)v1v2(x1,x2)Bayesian ICcolumn ave. of x2 increasingrow ave. of x1 increasingIntuition(.7,.9)(.6,.2)(.6,.3)(.3,.4)(.5,.5)(.1,.6)(.7,.7)(.4,.1)(.3,.9)(.2,.7)v1v2Ex-post ICNote here diff btw TIE and BIC

41IntuitionTransformation must fix non-monotonicitiesin every row and column.QueryQueryQueryQueryInput vector(.1,.3)(.2,.2)(.3,.4)(.8,.7)(.5,.5)v1v2note MIR as global fix, not efficient. Must be local fixes.42Intuition(.6,.2)(.2,.6)Idea: hide non-monotonicity on high-dim. diagonal.Make all allocations constant on these agents.(.5,.5)(.3,.3)21Easy local fix43Truthful in ExpectationThm. Any truthful-in-expectation transformation loses a polynomial factor in welfare approximation.

[Chawla, Immorlica, Lucier12]downward closed, with hu fu, lucier44Blackbox Transformations.Bayesian IC transformations for welfareex-post IC transformations for welfare(single- or multi-parameter!)Bayesian IC transformations for non-linear objectives(see Shuchis talk)Constructions for certain problem and algorithm classes.LP-based construction for packing problems [Lavi-Swamy05] Downward-closed, binary allocation, continuous valuations [Babaioff-Lavi-Pavlov09] Extensions via convex optimization [Dugmi-Roughgarden-Yan11]FPTAS algorithms [Dughmi-Roughgarden10]

45Thank YouBeethoven's Fifth Symphony (Excerpt)London Philharmonic OrchestraWith Love... From Beethoven, track 11/112011Classical3186.9426 - 000002A1 00000218 00001C1E 00001348 0003CEA4 00010608 000079C0 00007A0E 0004CA05 00026B48