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Published byKaiya Frothingham Modified about 1 year ago

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Blackbox Reductions from Mechanisms to Algorithms

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Feasibility constraints on outcome space Algorithm Design v1v1 v1v1 v2v2 v2v2 v3v3 v3v3 v4v4 v4v4 v5v5 v5v5 Input v Output x GOAL: maximize (or minimize) some function f(x,v)

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Feasibility constraints on outcome space Mechanism Design Allocation x Payment p GOAL: maximize (or minimize) some function f(x,v) v1v1 v1v1 v2v2 v2v2 v3v3 v3v3 v4v4 v4v4 v5v5 v5v5 Input v b1b1 b1b1 b2b2 b2b2 b3b3 b3b3 b4b4 b4b4 b5b5 b5b5 Input b b i chosen to maximize utility = v i x i (b)-p i (b)

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behind every great mechanism is a great algorithm computation

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Black-Box Transformations Transformation Algorithm Input b Allocation x Payment p GOAL: for every algorithm, transformation preserves quality of solution in equilibrium. and is incentive compatible. Input v

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Black-Box Transformations Transformation Algorithm Allocation x … and is incentive compatible (IC), i.e., monotone: Input v ex-post IC (truthful in expectation): allocation to agent i is increasing in i’s bid for all bid profiles of others Bayesian IC: allocation to agent i is increasing in i’s bid in expectation w.r.t. prior of over bid profiles of others

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VCG Transformation Optimal Algorithm Input vAllocation x EXAMPLE: Vickrey-Clark-Groves auction transforms any optimal algorithm into an optimal ex-post IC mechanism for any monotone objective function.

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Single Item Auction one item, agent i has value v i for item VCG Transformation Selection Algorithm Input vAllocation x Find agent w/max value (single-parameter)

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i2i2 i2i2 Combinatorial Auction many items, agent i has value v ij for subset S j i1i1 i1i1 i2i2 i2i2 i3i3 i3i3 (multi-parameter)

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Combinatorial Auction VCG Transformation ??? Input vAllocation x Find max value non-overlapping collection of sets many items, agent i has value v ij for subset S j (multi-parameter)

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^ approximation

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BIC Transformation Positive Result: Transform approximation algorithms into Bayesian IC mechs with small loss in social welfare. Single-parameter: (single private value for allocation) 1.Monotonization. For dist. F and algorithm A, there is a Bayesian IC transformation T A,F satisfying E[T A,F (v)] ≥ E[A(v)]. 2.Blackbox computation. T A,F can be computed in polytime with queries to A. 3.Payment computation. Payments can be computed with two queries to A.

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x i (v i ) = E[alloc. to i | v i ] Not BIC BIC Monotonization Fact. There’re payments that make an alg. Bayesian IC if and only if for all i, expected allocation is monotone non-decreasing in value v i. vivi

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Monotonization Goal: construct y i from x i s.t. 1.Monotonicity. y i (.) non-decreasing monotone 2.Surplus-preservation. E v i [v i y i (v i )] ≥ E v i [v i x i (v i )] 3.Distribution-preservation. (can apply construction independently to each j)

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Monotonization Idea 1: remap values.

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Monotonization Idea 2: resample values.

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Monotonization Idea 3: resample values in region where cumulative allocation is not monotone. allocation cumulative curve

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Monotonization Construction of y i (v i ) from x i (v i ) preserves: 1.Distribution-preservation. 2.Monotonicity. y i non-decreasing monotone x i (v i ) y i (v i )

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Monotonization Construction of y i (v i ) from x i (v i ) preserves: 3.Surplus-preservation. E v i [v i (y i - x i )] ≥ 0 x i (v i ) y i (v i ) 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.)(2 nd term non-pos.) ≥ 0 a b a b a b

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BIC Transformation for Welfare Positive Result: Transform approximation algorithms into Bayesian IC mechs with small loss in social welfare. Single-parameter: (single private value for allocation) 1.Monotonization. For dist. F and algorithm A, there is a Bayesian IC transformation T A,F satisfying E[A(v)] ≥ E[T A,F (v)]. 2.Blackbox computation. T A,F can be computed in polytime with queries to A. 3.Payment computation. Payments can be computed with two queries to A.

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Blackbox Computation

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BIC Transformation for Welfare Positive Result: Transform approximation algorithms into Bayesian IC mechs with small loss in social welfare. Single-parameter: (single private value for allocation) 1.Monotonization. For dist. F and algorithm A, there is a Bayesian IC transformation T A,F satisfying E[A(v)] ≥ E[T A,F (v)]. 2.Blackbox computation. T A,F can be computed in polytime with queries to A. 3.Payment computation. Payments can be computed with two queries to A.

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Payment Computation payment identity p(v) = v y(v) – ∫ y(z) dz v 0 Idea: compute random variable P with E[P] = p(v)

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Payment Computation payment identity p(v) = v y(v) – ∫ y(z) dz v 0 Idea: compute random variable P with E[P] = p(v) 1. Y indicator random variable for whether agent wins in A (with y(v)) 2. z drawn uniformly from [0,v] 3. Y z indicator random variable for whether agent wins in A (with y(z)) 4. P = v (Y – Y z ) 1 st call to A 2 nd call to A const. # calls per agent

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Payment Computation goal: given A, find an alg. A’ that computes allocation and payments with just 1 call to A 1. Pick agent k uniformly at random and draw w k from F k 2. Calculate outcome y’ for A(w k, v -k ) 3. For each agent i ≠ k, set p’ i = v i y’ i 4. For agent k, set p’ k = 0 if w k > v k and p’ k = -(n – 1)y’ k /f k (w k ) otherwise 5. Output (y’, p’) only call to A

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Payment Computation Thm. Algorithm A’ is Bayesian IC. Proof. 1.Monotone. y’ linear transformation of y. y’(v) = (1 - 1/n) y(v) + 1/n E[y(w)] 2.Payment Identity. p’(v) = v y’(v) – ∫ y’(z) dz v 0 p’(v) = (1 - 1/n) vy(v) – (1/n)(n - 1)∫ y(z) dz v 0 payment for i ≠ k payment for i = k (see ugly formula)

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Payment Computation Thm. Welfare is E[A’(v)] ≥ E[A(v) – max(v)] Proof. 1.Each buyer has welfare ≥ (1 - 1/n) vy(v) 2.Since y(v) is a probability, vy(v) ≤ max(v) 3.Lose at most max(v) in total buyer welfare 4.Expected payments are the same, so lose nothing in seller welfare Finds (alloc, payments) with 1 call to monotone alg. [Babaioff, Kleinberg, Slivkins’10]

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Transformation Approx. Algorithm Input v POSSIBILITY: can transform any approximation algorithm into a Bayesian IC mech. with small loss for f(x,v) = Σ i x i v i. [Hartline-Lucier’10] Dist. of values (drawn from known dist.) Allocation x Payment p

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Multi-parameter Transformation Goal: construct allocation from algorithm s.t. 1.“Monotonicity”. 2.Surplus-preservation. 3.Distribution-preservation. By mapping types of an agent to surrogates in a way that preserves above properties.

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Replicas and Surrogates replicas (drawn from F) surrogates (drawn from F) surrogate allocations v(t,x(t’)) max-weight matching original type t surrogate type t’ x(t’) Set payment equal to VCG payment for type t. Set allocation equal to output on surrogate type profile.

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Replicas and Surrogates Thm. Transformation is distribution-preserving. Thm. Transformation is Bayesian IC. Thm. Transformation doesn’t lose much welfare. Prf. Because replicas are “close” to matched surrogates in values for outcomes. [Hartline, Kleinberg, Malekian’11] [Bei, Huang’11]

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Strengthening the Result Solution 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?

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Multi-parameter Transformations Thm. There’s 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, Vondrak’11]

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Single-parameter Transformations Truthful in Expectation. For all algorithms A, T A 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.

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Proof Outline 1.Define welfare instance (feasible allocations, values of agents). 2.Find algorithm with high welfare. 3.Use monotonicity to show any ex-post transformation has low worst-case welfare.

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Intuition (.5,.5) v1v1 v2v2 (x 1,x 2 ) Bayesian IC column ave. of x 2 increasing row ave. of x 1 increasing

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Intuition (.7,.9) (.6,.2) (.6,.3)(.3,.4)(.5,.5)(.1,.6)(.7,.7) (.4,.1) (.3,.9) (.2,.7) v1v1 v2v2 Ex-post IC

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Intuition Transformation must fix non-monotonicities in every row and column. Query Input vector (.1,.3)(.2,.2)(.3,.4)(.8,.7)(.5,.5) v1v1 v2v2

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Intuition (.6,.2) (.2,.6) Idea: hide non-monotonicity on high-dim. diagonal. Make all allocations constant on these agents. (.5,.5) (.3,.3) 2 1

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Truthful in Expectation Thm. Any truthful-in-expectation transformation loses a polynomial factor in welfare approximation. [Chawla, Immorlica, Lucier’12]

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