Linear Programming & Mechanism Design Rakesh Vohra August 2012 Rakesh V. Vohra () LP&Mech August 2012 1 / 47
Mechanism Design ‘Optimizing’ the allocation of resources. Parameters (called type) needed to determine an optimal allocation are privately held by agents who will consume the resources to be allocated. Those parameters determine the utility an agent will enjoy from a particular allocation. Agents report of her type will influence the resulting allocation which in turn will affect the agents utility. How can planner simultaneously elicit the information that is privately held and choose the optimal allocation? Via allocation and monetary transfers. Rakesh V. Vohra () LP&Mech August 2012 2 / 47
Examples Many mechanism design problems are optimization problems. Auctions Matching (school choice, randomized rules) Models of Persuasion Team formation Rakesh V. Vohra () LP&Mech August 2012 3 / 47
Today & Tomorrow 1 Use of polymatroids in mechanism design. 2 Use of shortest path problems to analyze rationalizability and incentive compatibility. 3 Use of iterative rounding. Rakesh V. Vohra () LP&Mech August 2012 4 / 47
Submodularity & Polymatroids Let E = { 1 , 2 , . . . , n } be a ground set. Real valued function g defined on subsets of E is non-decreasing if S ⊆ T ⇒ g ( S ) ≤ g ( T ), and g is submodular if ∀ S , T ⊂ E g ( S ) + g ( T ) ≥ g ( S ∪ T ) + g ( S ∩ T ) . Equivalent: Suppose S ⊂ T and j �∈ T . Then, g ( T ∪ j ) − g ( T ) ≤ g ( S ∪ j ) − g ( S ) Rakesh V. Vohra () LP&Mech August 2012 5 / 47
Examples of Submodular Functions 1 E a finite set of vectors in ℜ m and g ( S ) is the rank of the subset S . 2 E a finite set of vectors in ℜ m and g ( S ) is log volume of set S . 3 E set of columns of a non-negative determinant matrix and g ( S ) is log of determinant of principal minor associated with S (also M matrices). 4 E the edge set of a graph and g ( S ) size of largest acyclic subset of S . 5 E the vertex set of an edge capacitated network with a distinguished source vertex and g ( S ) the maximum flow into S . 6 E the vertex set of a graph and g ( S ) the cardinality of the set of neighbors of S . 7 E a set of events and − g ( S ) is the probability that all events in S are realized. 8 Entropy of joint distribution. Rakesh V. Vohra () LP&Mech August 2012 6 / 47
Polymatroid Optimization Polymatroid: � P ( g ) = { x ∈ ℜ n + : x j ≤ g ( S ) ∀ S ⊆ E } j ∈ S max { cx : x ∈ P ( g ) } c 1 ≥ c 2 . . . ≥ c k ≥ 0 > c k +1 . . . ≥ c n . 1 S 0 = ∅ 2 S j = { 1 , 2 , . . . , j } for all j ∈ E . 3 x j = g ( S j ) − g ( S j − 1 ) for 1 ≤ j ≤ k 4 x j = 0 for j ≥ k + 1. Rakesh V. Vohra () LP&Mech August 2012 7 / 47
So What? For economic applications goal is not merely to solve the underlying optimization problem but identify qualitative properties of optimal solution and how it varies with changes in the parameters of the problem. Polymatroid optimizations problems are valuable because they admit a simple greedy solution. Reduction allows one to handle certain kinds of additional constraints like budget and quota constraints. Rakesh V. Vohra () LP&Mech August 2012 8 / 47
Allocation with Inspection Inspired by Dekel, Lipman and Ben-Porath (2011). Good to be allocated to agent with the highest value. Transfers not permitted. n risk neutral agents Value each agent assigns to the good is called their type Types are independent draws from T = { 1 , . . . , m } f t > 0 is probability that buyer is of type t For a cost K , planner can verify an agents report of his type. Rakesh V. Vohra () LP&Mech August 2012 9 / 47
Direct Mechanism Planner announces three functions whose argument is the profile of types reported. Allocation rule: specifies what ‘fraction’ of the object goes to each agent as a function of profile of reported types. Payment rule: specifies payment of each agent as a function of profile of reported types. Inspection rule: specifies probability that an agent will be inspected as a function of profile of reported types. Rakesh V. Vohra () LP&Mech August 2012 10 / 47
Allocation Rules For simplicity assume 2 agents. a is an allocation rule a i ( t , s ) is probability good is allocated to agent i when agent 1 reports t and agent 2 reports s . Feasibility: a 1 ( t , s ) + a 2 ( t , s ) ≤ 1 ∀ t , s a i ( t , s ) ≥ 0 ∀ i , ∀ t , s Rakesh V. Vohra () LP&Mech August 2012 11 / 47
Interim Allocations A i � t = a i ( t , s ) f s s ∈ T A i t is the interim allocation probability to agent i when she reports t . An interim allocation probability is implementable if there exists an allocation rule that corresponds to it. Characterize the implementable A ’s. Rakesh V. Vohra () LP&Mech August 2012 12 / 47
Border-Maskin-Matthews-Riley Suppose allocation rule is anonymous, i.e., does not depend on names. t = A j A i t = A t A t is implementable iff. f t ) n ∀ S ⊆ T . � � n f t A t ≤ 1 − ( t ∈ S i �∈ S i �∈ S f t ) n is monotone and submodular. g ( S ) = 1 − ( � Rakesh V. Vohra () LP&Mech August 2012 13 / 47
Interim Allocations Assume T = { t , t ′ } and S = { s , s ′ } . Here are all the inequalities: a 1 ( t , s ) + a 2 ( t , s ) ≤ 1 a 1 ( t ′ , s ) + a 2 ( t ′ , s ) ≤ 1 a 1 ( t ′ , s ′ ) + a 2 ( t ′ , s ′ ) ≤ 1 a 1 ( t , s ′ ) + a 2 ( t , s ′ ) ≤ 1 a 1 ( t , s ) f s + a 1 ( t , s ′ ) f s ′ = A 1 t a 1 ( t ′ , s ) f s + a 1 ( t ′ , s ′ ) f s ′ = A 1 t ′ a 2 ( t , s ) f t + a 2 ( t ′ , s ) f t ′ = A 2 s a 2 ( t , s ′ ) f t + a 2 ( t ′ , s ′ ) f t ′ = A 2 s ′ Rakesh V. Vohra () LP&Mech August 2012 14 / 47
Interim Allocations f t f s a 1 ( t , s ) + f t f s a 2 ( t , s ) ≤ f t f s f t ′ f s a 1 ( t ′ , s ) + f t ′ f s a 2 ( t ′ , s ) ≤ f t ′ f s f t ′ f s ′ a 1 ( t ′ , s ′ ) + f t ′ f s ′ a 2 ( t ′ , s ′ ) ≤ f t ′ f s ′ f t f s ′ a 1 ( t , s ′ ) + f t f s ′ a 2 ( t , s ′ ) ≤ f t f s ′ f t f s a 1 ( t , s ) + f t f s ′ a 1 ( t , s ′ ) = f t A 1 t f t ′ f s a 1 ( t ′ , s ) + f t ′ f s ′ a 1 ( t ′ , s ′ )) = f t ′ A 1 t ′ f t f s a 2 ( t , s ) + f t ′ f s a 2 ( t ′ , s ) = f s A 2 s f t f s ′ a 2 ( t , s ′ ) + f t ′ f s ′ a 2 ( t ′ , s ′ ) = f s ′ A 2 s ′ x i ( u , v ) = f u f v a i ( u , v ). Rakesh V. Vohra () LP&Mech August 2012 15 / 47
Interim Allocations x 1 ( t , s ) + x 2 ( t , s ) ≤ f t f s x 1 ( t ′ , s ) + x 2 ( t ′ , s ) ≤ f t ′ f s x 1 ( t ′ , s ′ ) + x 2 ( t ′ , s ′ ) ≤ f t ′ f s ′ x 1 ( t , s ′ ) + x 2 ( t , s ′ ) ≤ f t f s ′ x 1 ( t , s ) + x 1 ( t , s ′ ) = f t A 1 t x 1 ( t ′ , s ) + x 1 ( t ′ , s ′ ) = f t ′ A 1 t ′ x 2 ( t , s ) + x 2 ( t ′ , s ) = f s A 2 s x 2 ( t , s ′ ) + x 2 ( t ′ , s ′ ) = f s ′ A 2 s ′ Rakesh V. Vohra () LP&Mech August 2012 16 / 47
Allocation with Inspection A t is interim allocation probability to type t ∈ T . 1 − c ( t ) is the probability of checking a report of type t conditional on the good being allocated to a type t . Total value less the cost of inspection is m m � � f t t A t − K f t A t (1 − c ( t )) t =1 t =1 Rakesh V. Vohra () LP&Mech August 2012 17 / 47
Allocation with Inspection m m � � max f t t A t − K f t A t (1 − c ( t )) t =1 t =1 s.t. t A t ≥ t A s c ( s ) ∀ t , s ∈ T (1) 0 ≤ c ( t ) ≤ 1 ∀ t ∈ T (2) f t ) n = g ( S ) ∀ S ⊆ T � � f t A t t ≤ 1 − ( (3) n t ∈ S t �∈ S Rakesh V. Vohra () LP&Mech August 2012 18 / 47
Allocation with Inspection: IC Bayesian incentive compatibility constraint captured here by: t A t ≥ t A s c ( s ) ∀ t , s ∈ T ⇒ A t ≥ A s c ( s ) ∀ t , s ∈ T This is dual to problem of finding a feasible flow in a generalized network. Associate node with each member of T and for each ordered pair ( t , s ) insert a directed edge from t to s with flow multiplier c ( s ). System is feasible iff the associated network has no flow generating cycles. For any subset R of types we must have Π t ∈ R c ( t ) ≤ 1 . Rakesh V. Vohra () LP&Mech August 2012 19 / 47
Allocation with Inspection A t ≥ A s c ( s ) ⇒ A s ≤ A t c ( s ) Never good to inspect t = 1. So, c (1) = 1. Therefore, A t ≥ A 1 for all t ∈ T . A s ≤ A 1 c ( s ) ∀ s ∈ T . Rakesh V. Vohra () LP&Mech August 2012 20 / 47
Allocation with Inspection: Relaxation � max f t A t [ t − K + Kc ( t )] t ∈ T s.t. c ( t ) ≤ A 1 ∀ t ∈ T A t A t ≥ A 1 ∀ t ∈ T 0 ≤ c ( t ) ≤ 1 ∀ t ∈ T � f t A t ≤ g ( S ) ∀ S ⊆ T n t ∈ S c ( t ) = min { A 1 A t , 1 } = A 1 A t . Rakesh V. Vohra () LP&Mech August 2012 21 / 47
Allocation with Inspection: Relaxation m � max f t A t [ t − K ] + K A 1 t =1 s.t. A 1 ≤ A t ∀ t ∈ T � f t A t ≤ g ( S ) ∀ S ⊆ T n t ∈ S 1 A t = x t + A 1 for all t ≥ 2 2 H ( S ) = g ( S ) − n A 1 � i ∈ S f i . 3 H is submodular. g ( S ) 4 For A 1 ≤ min S t ∈ S f t , H is monotone. n P Rakesh V. Vohra () LP&Mech August 2012 22 / 47
Allocation with Inspection: Relaxation m m � � ( tf t ) A 1 + max f t x t [ t − K ] t =1 t =2 � s.t. n f t x t ≤ H ( S ) ∀ S ⊆ T \ { 1 } t ∈ S One more change of variables: z t = f t x t for all t ≥ 2. Rakesh V. Vohra () LP&Mech August 2012 23 / 47
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