Welfare Maximization with Production Costs: A Primal Dual Approach

Welfare Maximization with Production Costs: A Primal Dual Approach Zhiyi Huang Anthony Kim The University of Hong Kong Stanford University January 4, 2015 Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 1 / 22

Motivation General Mechanism Design Problem: Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 2 / 22

Motivation General Mechanism Design Problem: 1 A seller with items to allocate Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 2 / 22

Motivation General Mechanism Design Problem: 1 A seller with items to allocate 2 Buyers with private valuation functions for items Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 2 / 22

Motivation General Mechanism Design Problem: $*&A%% $X.XX :) 1 A seller with items to allocate 2 Buyers with private valuation functions for items 3 A mechanism that helps determine allocations and payments Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 2 / 22

Motivation General Mechanism Design Problem: $*&A%% $X.XX :) 1 A seller with items to allocate 2 Buyers with private valuation functions for items 3 A mechanism that helps determine allocations and payments Objective: design a mechanism that induces a (near-)optimal allocation Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 2 / 22

Model Online Comb. Auctions with Production Costs (Blum et al., 2011): Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 3 / 22

Model Online Comb. Auctions with Production Costs (Blum et al., 2011): 1 A seller with m items to allocate: Production cost function f : R + R + for each item: f (y) is the total cost to produce y units of an item f increasing (equiv., f convex) Ex) limited natural resources such as crude oil The set of available bundles S = {0, y} m : Bundle S represented by a vector (a 1S,..., a ms ) where a js {0, y} y is the unit-per-bundle constant Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 3 / 22

Model Online Comb. Auctions with Production Costs (Blum et al., 2011): 1 A seller with m items to allocate: Production cost function f : R + R + for each item: f (y) is the total cost to produce y units of an item f increasing (equiv., f convex) Ex) limited natural resources such as crude oil The set of available bundles S = {0, y} m : Bundle S represented by a vector (a 1S,..., a ms ) where a js {0, y} y is the unit-per-bundle constant 2 n buyers that arrive in the online fashion: Valuation function v i : S R + (Notation v is = v i (S)) Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 3 / 22

Model 3 Online mechanism: For i = 1, 2,...: Buyer i reports a valuation function ˆv i Seller allocates a bundle S i and charges a payment P i Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 4 / 22

Model 3 Online mechanism: For i = 1, 2,...: Buyer i reports a valuation function ˆv i Seller allocates a bundle S i and charges a payment P i Incentive compatible (IC) if each buyer i maximizes his expected utility, i.e., v i (S i ) P i, by reporting ˆv i = v i. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 4 / 22

Model 3 Online mechanism: For i = 1, 2,...: Buyer i reports a valuation function ˆv i Seller allocates a bundle S i and charges a payment P i Incentive compatible (IC) if each buyer i maximizes his expected utility, i.e., v i (S i ) P i, by reporting ˆv i = v i. Objective: design an online mechanism that maximizes the expected social welfare, v i (S i ) f (y j ): i j }{{}}{{} Tot. Value Prod. Cost Bundle S i is allocated to buyer i y j units of item j have been allocated in total at termination Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 4 / 22

Competitive Analysis Competitive Analysis: Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 5 / 22

Competitive Analysis Competitive Analysis: Let W (M) be the expected social welfare of an online mechanism M. Let OPT be the optimal social welfare in the offline setting. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 5 / 22

Competitive Analysis Competitive Analysis: Let W (M) be the expected social welfare of an online mechanism M. Let OPT be the optimal social welfare in the offline setting. An online mechanism M is α-competitive if there exists a constant β such that for all possible instances, W (M) 1 α OPT β. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 5 / 22

Overview of Results Via an online primal dual approach: Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 6 / 22

Overview of Results Via an online primal dual approach: For the fractional case ( y 0): We characterize a parameter α(f ) such that: 1 There are (α(f ) + ɛ)-competitive and IC mechanisms, and 2 There are no (α(f ) ɛ)-competitive online algorithms. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 6 / 22

Overview of Results Via an online primal dual approach: For the fractional case ( y 0): We characterize a parameter α(f ) such that: 1 There are (α(f ) + ɛ)-competitive and IC mechanisms, and 2 There are no (α(f ) ɛ)-competitive online algorithms. We compute bounds on α(f ) in some special cases. For the integral case ( y = 1), the fractional case results extend and improve previously obtained competitive ratios. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 6 / 22

Overview of Results Via an online primal dual approach: For the fractional case ( y 0): We characterize a parameter α(f ) such that: 1 There are (α(f ) + ɛ)-competitive and IC mechanisms, and 2 There are no (α(f ) ɛ)-competitive online algorithms. We compute bounds on α(f ) in some special cases. For the integral case ( y = 1), the fractional case results extend and improve previously obtained competitive ratios. For limited supply online comb. auctions, we obtain the same competitive ratio as previous work and show a new nearly matching lower bound. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 6 / 22

Primal Dual Approach The convex program relaxation in the offline setting: (P) max x,y i i : S x is 1 j : i S v isx is j f (y j) S a jsx is y j x, y 0 Variable x is indicates whether or not buyer i purchases bundle S. (Constraint Type 1) Each buyer is allocated at most one bundle. (Constraint Type 2) Each item is allocated (at most) y j units in total. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 7 / 22

Primal Dual Approach The dual program: (D) min u,p i u i + j f (p j ) i, S : u i + j a jsp j v is u, p 0 Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 8 / 22

Primal Dual Approach The dual program: (D) min u,p i u i + j f (p j ) i, S : u i + j a jsp j v is u, p 0 Variable u i for each buyer i and variable p j for each item j. Interpret u i as the utility of buyer i and p j as the price per unit of item j. f is the convex conjugate of f, f (p) = sup y 0 {py f (y)}. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 8 / 22

Primal Dual Approach Consider the online setting: max x,y i i : j : S v isx is j f (y j) min u,p i u i + j f (p j ) S x is 1 i, S : u i + j a jsp j v is i S a jsx is y j u, p 0 x, y 0 Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 9 / 22

Primal Dual Approach Consider the online setting: max x,y i i : j : S v isx is j f (y j) min u,p i u i + j f (p j ) S x is 1 i, S : u i + j a jsp j v is i S a jsx is y j u, p 0 x, y 0 An online mechanism M leads to feasible primal and dual solutions, (P n, D n ), at termination. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 9 / 22

Primal Dual Approach Consider the online setting: max x,y i i : j : S v isx is j f (y j) min u,p i u i + j f (p j ) S x is 1 i, S : u i + j a jsp j v is i S a jsx is y j u, p 0 x, y 0 An online mechanism M leads to feasible primal and dual solutions, (P n, D n ), at termination. Note that if then M is α-competitive. P n 1 α Dn β, Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 9 / 22

Primal Dual Approach Consider the online setting: max x,y i i : j : S v isx is j f (y j) min u,p i u i + j f (p j ) S x is 1 i, S : u i + j a jsp j v is i S a jsx is y j u, p 0 x, y 0 An online mechanism M leads to feasible primal and dual solutions, (P n, D n ), at termination. Note that if then M is α-competitive. P n 1 α Dn β, Design M so that the above condition holds! Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 9 / 22

Posted Pricing Mechanisms Assume pricing function p : R + R + for the y-th unit of an item. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 10 / 22

Posted Pricing Mechanisms Assume pricing function p : R + R + for the y-th unit of an item. Define posted pricing mechanism M p : Algorithm M p 1: Initialize y j = 0 for all j 2: for i = 1,..., n do 3: Offer item j at price p j = p(y j ) for all j 4: Buyer i chooses bundle S and pays j S p j 5: Update y j = y j + y for all j S 6: end for Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 10 / 22

Posted Pricing Mechanisms Assume pricing function p : R + R + for the y-th unit of an item. Define posted pricing mechanism M p : Algorithm M p 1: Initialize y j = 0 for all j 2: for i = 1,..., n do 3: Offer item j at price p j = p(y j ) for all j 4: Buyer i chooses bundle S and pays j S p j 5: Update y j = y j + y for all j S 6: end for M p is incentive compatible. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 10 / 22

Fractional Case: Characterization The central differential equation is: y 0 p(ȳ)dȳ f (y) 1 α f (p(y)) β, for all y 0. (1) Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 11 / 22

Fractional Case: Characterization The central differential equation is: y 0 p(ȳ)dȳ f (y) 1 α f (p(y)) β, for all y 0. (1) Theorem (Upper Bound) If a monotonically increasing pricing function p satisfies (1) for some constant β, then M p is α-competitive. Theorem (Lower Bound) If there is an α-competitive algorithm, then there exists a monotonically increasing pricing function p that satisfies (1) for some constant β. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 11 / 22

Fractional Case: Characterization The central differential equation is: y 0 p(ȳ)dȳ f (y) 1 α f (p(y)) β, for all y 0. (1) Theorem (Upper Bound) If a monotonically increasing pricing function p satisfies (1) for some constant β, then M p is α-competitive. Theorem (Lower Bound) If there is an α-competitive algorithm, then there exists a monotonically increasing pricing function p that satisfies (1) for some constant β. Both reduce to the same equation! Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 11 / 22

Fractional Case: Characterization The central differential equation is: y 0 p(ȳ)dȳ f (y) 1 α f (p(y)) β, for all y 0. (1) We define: α(f ) = inf { α : there exist a constant β and a monotonically increasing p so that (1) holds }. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 12 / 22

Fractional Case: Characterization The central differential equation is: y 0 p(ȳ)dȳ f (y) 1 α f (p(y)) β, for all y 0. (1) We define: α(f ) = inf { α : there exist a constant β and Corollary a monotonically increasing p so that (1) holds }. There is an (α(f ) + ɛ)-competitive and incentive compatible mechanism. Corollary There are no (α(f ) ɛ)-competitive algorithms. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 12 / 22

Fractional Case: Characterization The central differential equation is: y 0 p(ȳ)dȳ f (y) 1 α f (p(y)) β, for all y 0. (1) We define: α(f ) = inf { α : there exist a constant β and Corollary a monotonically increasing p so that (1) holds }. There is an (α(f ) + ɛ)-competitive and incentive compatible mechanism. Corollary There are no (α(f ) ɛ)-competitive algorithms. α(f ) is the optimal competitive ratio achievable! Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 12 / 22

Fractional Case: Characterization y 0 p(ȳ)dȳ f (y) 1 α f (p(y)) β, for all y 0 (1) Theorem (Upper Bound) If a monotonically increasing pricing function p satisfies (1) for some constant β, then the pricing mechanism M p is α-competitive. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 13 / 22

Fractional Case: Characterization y 0 p(ȳ)dȳ f (y) 1 α f (p(y)) β, for all y 0 (1) Theorem (Upper Bound) If a monotonically increasing pricing function p satisfies (1) for some constant β, then the pricing mechanism M p is α-competitive. Proof. For any M p, P n = i v i j f (y j) and D n = i u i + j f (p j ). Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 13 / 22

Fractional Case: Characterization y 0 p(ȳ)dȳ f (y) 1 α f (p(y)) β, for all y 0 (1) Theorem (Upper Bound) If a monotonically increasing pricing function p satisfies (1) for some constant β, then the pricing mechanism M p is α-competitive. Proof. For any M p, P n = i v i j f (y j) and D n = i u i + j f (p j ). Equivalently, P n = i u i + yj j 0 p(y)dy j f (y j). Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 13 / 22

Fractional Case: Characterization y 0 p(ȳ)dȳ f (y) 1 α f (p(y)) β, for all y 0 (1) Theorem (Upper Bound) If a monotonically increasing pricing function p satisfies (1) for some constant β, then the pricing mechanism M p is α-competitive. Proof. For any M p, P n = i v i j f (y j) and D n = i u i + j f (p j ). Equivalently, P n = i u i + yj j 0 p(y)dy j f (y j). The condition P n 1 α Dn β follows from yj j 0 p(y)dy j f (y j) 1 α j f (p j ) mβ, which follows from (1). Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 13 / 22

Fractional Case: Characterization Theorem (Lower Bound) If there is an α-competitive algorithm, then there exists a monotonically increasing p that satisfies (1). Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 14 / 22

Fractional Case: Characterization Theorem (Lower Bound) If there is an α-competitive algorithm, then there exists a monotonically increasing p that satisfies (1). Proof. Consider a family of single-item instances, {I v } v 0: An infinite continuum of stages: At stage v, a continuum of buyers with value v per unit of the item and a total demand of f (v)(= f 1 (v)). I v is the continuum of stages from 0 to v. Consider an α-competitive algorithm: y(v) be the expected number of units sold up to stage v. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 14 / 22

Fractional Case: Characterization Theorem (Lower Bound) If there is an α-competitive algorithm, then there exists a monotonically increasing p that satisfies (1). Proof. Consider a family of single-item instances, {I v } v 0: An infinite continuum of stages: At stage v, a continuum of buyers with value v per unit of the item and a total demand of f (v)(= f 1 (v)). I v is the continuum of stages from 0 to v. Consider an α-competitive algorithm: y(v) be the expected number of units sold up to stage v. Then, v 0 vdy(v) f (y(v )) 1 α f (v ) β, for v 0. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 14 / 22

Fractional Case: Characterization Theorem (Lower Bound) If there is an α-competitive algorithm, then there exists a monotonically increasing p that satisfies (1). Proof. Consider a family of single-item instances, {I v } v 0: An infinite continuum of stages: At stage v, a continuum of buyers with value v per unit of the item and a total demand of f (v)(= f 1 (v)). I v is the continuum of stages from 0 to v. Consider an α-competitive algorithm: y(v) be the expected number of units sold up to stage v. Then, v 0 vdy(v) f (y(v )) 1 α f (v ) β, for v 0. Which, in turn, implies that there exists a function p satisfying (1). Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 14 / 22

Fractional Case: Case Study The central differential equation is: y 0 p(ȳ)dȳ f (y) 1 α f (p(y)) β, for all y 0. (1) Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 15 / 22

Fractional Case: Case Study The central differential equation is: y 0 p(ȳ)dȳ f (y) 1 α f (p(y)) β, for all y 0. (1) For specific production costs, we can: 1 Plug in the cost function into (1); and 2 Find bounds on the competitive ratio α. Ex) A feasible p with a particular α An α-competitive M p. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 15 / 22

Fractional Case: Case Study For specific cases: Marginal Prod. Cost f Concave α(f ) 4 Power (f = y γ, γ 1) α(f ) = (γ + 1) (γ+1)/γ Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 16 / 22

Fractional Case: Case Study For specific cases: Marginal Prod. Cost f Concave α(f ) 4 Power (f = y γ, γ 1) α(f ) = (γ + 1) (γ+1)/γ In general: Theorem (A Unified Theorem) For a cost function f with f differentiable and strictly increasing, M p with p(y) = f λ (λy) is 2 λ 1 Γ f,λ-competitive for any λ > 1. Where, Γ f,λ = max { y>0 1, (λ 1)yf (λy)} f (λy) f (y) ( how fast f increases ). Hence, α(f ) min λ>1 { λ2 λ 1 Γ f,λ }. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 16 / 22

Integral Case ( y = 1) The integral case reduces to the fractional case when the seller has already sold many copies of the items: The contribution until then can be accounted for by an additive cost β Competitive ratios from the fractional case hold arbitrarily close (with a tradeoff in β) For specific cases, W (M p ) 1 OPT β : α Marg Prod f p in M p α, β Concave p(y) = f 4(1 + ɛ) (2(y + 1)) j ( 1 4(1+ɛ) f (f ( 2 ɛ )) + f ( 1 ɛ 1)) Power (y γ ) p(y) = (γ + 1) 2 y γ (1 + ɛ) γ (γ + 1) (γ+1)/γ j ( 1 α f (f ( 2 ɛ )) + f ( 1 ɛ 1)) Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 17 / 22

Integral Case ( y = 1) In general: Theorem (A Unified Theorem) For cost functions f with f differentiable and strictly increasing, M p with p(y) = f (λ(y + 1)) is α-competitive for α = (1+ɛ)λ2 λ 1 Γ + f,λ,λ/ɛ Γ f,λ and arbitrarily small ɛ > 0. Where, Γ f,λ = max y>0{1, (λ 1)yf (λy) f (λy) f (y) } Where, Γ + f,λ,τ = max y τ {1, f (y+λ) f (y) } Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 18 / 22

Integral Case ( y = 1) Can improve previous competitive ratios obtained by Blum et al.: Marginal Prod. Costs f Blum et al. This Paper Linear 6 4(1 + ɛ) Polynomial (1 + ɛ)4d (1 + ɛ)(1 + 1 d )d+1 d Logarithmic 4.93 4(1 + ɛ) Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 19 / 22

Limited k-supply Case Model: Items are allocated integrally and there are exactly k units of each item for sale. { 0, if y [0, k] In our framework, production cost f (y) =. +, if y > k Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 20 / 22

Limited k-supply Case Model: Items are allocated integrally and there are exactly k units of each item for sale. { 0, if y [0, k] In our framework, production cost f (y) =. +, if y > k Theorem The pricing mechanism M p with p(y) = p(0)r y, where p(0) = v min 2m r = (2mρ) 1/k, is Θ(k((2mρ) 1/k 1))-competitive and IC. and Where, ρ = v max /v min Where, v max = max i,s v is and v min = min i,s:vis >0 v is Competitive ratio is Θ(log(mρ)) when k = Ω(log m). Matches Bartal et al. and Buchbinder and Gonen. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 20 / 22

Limited k-supply Case Model: Items are allocated integrally and there are exactly k units of each item for sale. { 0, if y [0, k] In our framework, production cost f (y) =. +, if y > k Theorem No online algorithms are o( log m log log m + log ρ)-competitive. Where, ρ = v max /v min Where, v max = max i,s v is and v min = min i,s:vis >0 v is Resolves an open problem in Buchbinder and Gonen. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 21 / 22

Recap A characterization of the optimal competitive ratio achievable, α(f ), in the fractional case. For specific functions, bounds on α(f ) in both fractional and integral cases. Application of the primal dual framework in the limited supply case. Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 22 / 22

Recap A characterization of the optimal competitive ratio achievable, α(f ), in the fractional case. For specific functions, bounds on α(f ) in both fractional and integral cases. Application of the primal dual framework in the limited supply case. Thank you! Zhiyi Huang, Anthony Kim Welfare Maximization with Production Costs January 4, 2015 22 / 22