Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
On the Generic (Im)Possibility of Full Surplus Extraction in Mechanism Design Aviad Heifetz and Zvika Neeman Presented by Nick Mastronardi and Qianfeng Tang
May 6, 2009
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Background Cremer and McLean (1985, 1988) have shown that under standard assumptions of the existence of a common prior, a fixed number of types, risk neutrality, and no limited liability etc, an uninformed principal facing privately informed players can generically implement any decision rule he could implement were that private information accessible to him. Specifically, an uninformed seller, for example, is generically able to extract the full surplus of any number of privately informed bidders in an auction.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Theorem An information structure (M, π) guarantees full extraction of the surplus by a dominant strategy auction if and only if for all i ∈ N, there do not exist {ρi (si )}si ∈Mi , not all equal to zero, such that: X
ρi (si )π(s−i |si ) = 0, ∀s−i ∈ M−i .
(1)
si ∈Mi
Full-surplus-extraction is obtained whenever the belief matrix {π(s−i |si )}{i,−i∈N} is of full rank. Genericity comes with the fact that in the space of all matrices with fixed dimension, the subset of matrices with full rank is generic.
Motivation
Model Set Up
I
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Infinite Cardinality: Number of types needed to model situations with asymmetric information is not bounded.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
I
Infinite Cardinality: Number of types needed to model situations with asymmetric information is not bounded.
I
Question: How crucial the finiteness assumption is for obtaining these results?
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
I
Infinite Cardinality: Number of types needed to model situations with asymmetric information is not bounded.
I
Question: How crucial the finiteness assumption is for obtaining these results?
I
BDP: A type space (prior) has the Beliefs-determine-preferences (BDP) property if almost every possible belief of every player about others’ types pins down the player’s own preferences.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
I
Infinite Cardinality: Number of types needed to model situations with asymmetric information is not bounded.
I
Question: How crucial the finiteness assumption is for obtaining these results?
I
BDP: A type space (prior) has the Beliefs-determine-preferences (BDP) property if almost every possible belief of every player about others’ types pins down the player’s own preferences.
I
Main Result: If the family of priors P is convex and contains at least one non-BDP prior, then the collection of priors that permit FSE is ”small“ in P.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
The Wilson Doctrine
This papers makes a contribution to the growing literature on robust mechanism design following the ”Wilson Doctrine“. Specifically, by allowing convex combination of models, common knowledge of type spaces is relaxed on the side of designers. The designer know that one type space and more type spaces might be the true environment for the asymmetric information situation, but not sure about which, for this reason the convex combination of type spaces should also be considered possible.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Notations I
Denote the set of bidders N = {1, 2, . . . , n}. Player i’s valuation is denoted by vi ∈ Vi , and payoff is q · vi − m, where q is the probability of winning the object, and m is the expected payment. Analysis if confined to the private values setting, with correlated types.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Notations I
Denote the set of bidders N = {1, 2, . . . , n}. Player i’s valuation is denoted by vi ∈ Vi , and payoff is q · vi − m, where q is the probability of winning the object, and m is the expected payment. Analysis if confined to the private values setting, with correlated types.
I
We refer to vi as player i’s preference type, V = V1 × V2 × · · · × Vn is the basic space of uncertainty.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Notations I
Denote the set of bidders N = {1, 2, . . . , n}. Player i’s valuation is denoted by vi ∈ Vi , and payoff is q · vi − m, where q is the probability of winning the object, and m is the expected payment. Analysis if confined to the private values setting, with correlated types.
I
We refer to vi as player i’s preference type, V = V1 × V2 × · · · × Vn is the basic space of uncertainty.
I
Denote the set of player i’s types Θi . Define vˆi : Θi → Vi , bˆi : Θi → ∆(Θ−i ). vˆi (θi ) is player i’s preference type associated with type θi , and bˆi (θi ) is his belief type associated with type θi . We can understand θi as(ˆ vi (θi ), bˆi (θi )), no redundancy in types.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Notations I
Denote the set of bidders N = {1, 2, . . . , n}. Player i’s valuation is denoted by vi ∈ Vi , and payoff is q · vi − m, where q is the probability of winning the object, and m is the expected payment. Analysis if confined to the private values setting, with correlated types.
I
We refer to vi as player i’s preference type, V = V1 × V2 × · · · × Vn is the basic space of uncertainty.
I
Denote the set of player i’s types Θi . Define vˆi : Θi → Vi , bˆi : Θi → ∆(Θ−i ). vˆi (θi ) is player i’s preference type associated with type θi , and bˆi (θi ) is his belief type associated with type θi . We can understand θi as(ˆ vi (θi ), bˆi (θi )), no redundancy in types. Q A prior pi on the private-values type space Θ = i∈N Θi is called a prior for bidder i if pi (·|θi ) = bˆi (θi ) for pi − a.e.θi ∈ Θi . A probability measure p ∈ ∆Θ is a common
I
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Definition Convexity of priors: If pi0 ∈ ∆(Θ0 ) and pi00 ∈ ∆(Θ00 ) are two priors for bidder i, then so is αpi0 + (1 − α)pi00 ∈ ∆(Θ0 ∪ Θ00 ) for every α ∈ [0, 1].
Definition A prior p ∈ ∆(Θ) satisfies the beliefs-determine-preferences property for bidder i ∈ N if there exists a measurable subset Θpi ⊂ Θi such that the marginal p|Θi assigns probability 1 to Θpi , and no pair of distinct types θi 6= θi0 in Θpi hold the same beliefs, ˆ i ) 6= b(θ ˆ 0 ) for every two different types θi , θ0 ∈ Θp . i.e., b(θ i i i
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Theorem Let Θ0 , Θ00 be two type spaces and Θ = Θ0 ∪ Θ00 . A nondegenerate convex combination p = αp 0 + (1 − α)p 00 ∈ ∆(Θ) of two priors p 0 ∈ ∆(Θ0 ) and p 00 ∈ ∆(Θ00 ) is BDP if and only if both p 0 and p 00 are BDP. In particular, a nondegenerate convex combination of a BDP prior and a non-BDP prior (or of two non-BDP priors) is a non-BDP prior.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
p0 θ10 = (v1 , b10 ) θ˜10 = (˜ v1 , b˜10 )
θ20 = (v2 , b20 ) a0 c0
θ˜20 = (˜ v2 , b˜20 ) b0 d0
p 00 θ100 = (v1 , b100 ) θ˜100 = (˜ v1 , b˜100 )
θ200 = (v2 , b200 ) a00 c 00
θ˜200 = (˜ v2 , b˜200 ) b 00 d 00
Smallness of the Set of NBDP Priors
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
In the tables above, entries in each matrix are positive and sum to one, and either v1 6= v˜1 or v2 6= v˜2 . Then priors p 0 and p 00 are BDP 0 0 00 00 if and only if ca0 6= db 0 and ca00 6= db 00 A nondegenerate convex combination p = αp 0 + (1 − α)p 00 is represented by the matrix below, it is BDP if and only if both p 0 and p 00 are BDP. p θ2 = (v2 , b20 ) θ˜2 = (˜ v2 , b˜20 ) θ20 = (v2 , b200 ) θ˜20 = (˜ v2 , b˜ 0 0 0 θ1 = (v1 , b1 ) αa αb 0 0 0 0 ˜ ˜ θ1 = (˜ v 1 , b1 ) αc αd 0 θ10 = (v1 , b100 ) 0 0 (1 − α)a00 (1 − α)b 0 00 00 ˜ ˜ θ1 = (˜ v 1 , b1 ) 0 0 (1 − α)c (1 − α)d
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
IC and IR I
A direct revelation auction mechanism is a profile < qi , mi >i∈N . Where qi : Θ → [0, 1], mi : Θ → [0, 1]. Bidder i is asked to report their types and then to participate in a lottery in which he or she pays an amount mi (θ), and wins the object with prob. qi (θ).
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
IC and IR I
A direct revelation auction mechanism is a profile < qi , mi >i∈N . Where qi : Θ → [0, 1], mi : Θ → [0, 1]. Bidder i is asked to report their types and then to participate in a lottery in which he or she pays an amount mi (θ), and wins the object with prob. qi (θ).
I
A mechanism is incentive-compatible if every bidder maximizes its expected payoff by truthfully reporting type, i.e., for every θi , θi0 ∈ Θi , Z (qi (θi , θ˜−i )ˆ vi (θi ) − mi (θi , θ˜−i ))d bˆi (θi )(θ˜−i ) Θ−i Z ≥ (qi (θi0 , θ˜−i )ˆ vi (θi ) − mi (θi0 , θ˜−i ))d bˆi (θi )(θ˜−i ) Θ−i
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
A mechanism is individually rational if every type θi ∈ Θi prefers to participate, Z (qi (θi , θ˜−i )ˆ vi (θi ) − mi (θi , θ˜−i ))d bˆi (θi )(θ˜−i ) ≥ 0. (2) Θ−i
Definition A prior p permits the full surplus extraction from a set K ⊂ N of bidders if there exists an incentive-compatible and individually rational mechanism < qi , mi >i∈N that generates an expected payment to the seller that is equal to the full surplus generated by the bidders in K , i.e., Z XZ mi (θ)dp(θ) = max{ˆ vi (θi )} dp(θ). (3) i∈K
Θ
Θ i∈K
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
A prior that permits the full surplus extraction from the K bidders is called a full-surplus extraction prior for K . Then case of K containing a single player i is given by, Z Z mi (θ)dp(θ) = vˆi (θi ) dp(θ). (4) Θ
Θ
We show that BDP is necessary for full surplus extraction. Specifically, if prior p permits the extraction of bidder i’s full surplus, then p is a BDP prior for player i.
Theorem A prior p that is a FSE prior for bidder i is a BDP prior for bidder i.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Proof Suppose that p is a FSE prior for bidder i. Let < qi , mi >i∈N be a mechanism satisfies IC and IR that extracts the full surplus of bidder i. This implies that bidder i must win with prob. 1 under the mechanism and IR must be binding with prob. 1. Suppose that p is not a BDP prior for bidder i. It follows that there exist two disjoint measurable subsets of bidder i’s types, Ai , A0i ⊂ Θi , that each have a positive p-probability, p|Θi (Ai ) > 0, p|Θi (A0i ) > 0,
(5)
and the same range of beliefs bˆi (Ai ) = bˆi (A0i ) ⊂ ∆(Θ−i ), but different valuations, i.e., if θi ∈ Ai and bˆi (θi0 ) = bˆi (θi ), then
θi0
∈
(6) A0i
are such that (7)
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Proof Ctn’d In particular, for every type θi ∈ Ai there exists a type θi0 ∈ A0i such that bˆi (θi0 ) = bˆi (θi ), but vˆi (θi0 ) < vˆi (θi ). So Z
(qi (θi , θ˜−i )ˆ vi (θi ) − mi (θi , θ˜−i ))d bˆi (θi )(θ˜−i )
Θ−i
Z ≥
(qi (θi0 , θ˜−i )ˆ vi (θi ) − mi (θi0 , θ˜−i ))d bˆi (θi )(θ˜−i )
Θ−i
Z =
(qi (θi0 , θ˜−i )ˆ vi (θi ) − mi (θi0 , θ˜−i ))d bˆi (θi0 )(θ˜−i )
Θ−i
Z > Θ−i
Contradicts.
(qi (θi0 , θ˜−i )ˆ vi (θi ) − mi (θi0 , θ˜−i ))d bˆi (θi0 )(θ˜−i )
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Purpose: show that within a convex family P of priors that contains at least one non-BDP (NBDP) prior for bidder i, the subset F of FSE priors for bidder i is small in two different senses:
I
Geometric: The set of FSE priors is contained in a proper face of the convex body of priors P.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Purpose: show that within a convex family P of priors that contains at least one non-BDP (NBDP) prior for bidder i, the subset F of FSE priors for bidder i is small in two different senses:
I
Geometric: The set of FSE priors is contained in a proper face of the convex body of priors P.
I
Measure-theoretic: The set F of FSE priors is finitely shy in P.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Geometric Smallness I
Definition A convex subset F of a convex set C is called a face if whenever f ∈ F is a convex combination of x, y ∈ C , then x, y ∈ F .
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Geometric Smallness I
Definition A convex subset F of a convex set C is called a face if whenever f ∈ F is a convex combination of x, y ∈ C , then x, y ∈ F .
I
Definition A face F of C is called a proper face if F is a proper subset of C . If C is a convex subset of a vector space X , then the codimension of F in C is the dimension of the minimal subspace Y of X such that C is contained in the subspace spanned by F and Y .
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Geometric Smallness I
Definition A convex subset F of a convex set C is called a face if whenever f ∈ F is a convex combination of x, y ∈ C , then x, y ∈ F .
I
Definition A face F of C is called a proper face if F is a proper subset of C . If C is a convex subset of a vector space X , then the codimension of F in C is the dimension of the minimal subspace Y of X such that C is contained in the subspace spanned by F and Y .
I
Theorem Let P be a convex family of priors that includes at least one NBDP prior for bidder i. Then the subset B of BDP priors for bidder i is a proper face of P.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Families of Priors There are two important families of priors: I
Pu : the family of priors on the universal type space. The universal type space is the type space into which any other type space can be mapped in a belief-preserving way. (More to be added.)
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Families of Priors There are two important families of priors: I
Pu : the family of priors on the universal type space. The universal type space is the type space into which any other type space can be mapped in a belief-preserving way. (More to be added.)
I
Pf : the family of priors on finite type spaces. Note that this set of priors contain all the priors with finite support, which differs from that of Cremer and McLean, who work with fixed type spaces.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Families of Priors There are two important families of priors:
I
I
Pu : the family of priors on the universal type space. The universal type space is the type space into which any other type space can be mapped in a belief-preserving way. (More to be added.)
I
Pf : the family of priors on finite type spaces. Note that this set of priors contain all the priors with finite support, which differs from that of Cremer and McLean, who work with fixed type spaces.
Theorem For Pu and Pf , the subset of full-surplus-extraction (FSE) priors for bidder i is contained in a proper face of infinite codimension in Pu and Pf , respectively.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Proof It remains to show that the codimension of the set of BDP priors B in Pu and Pf is infinite. This follows from the fact that there are infinitely many (in fact, a continuum of) finite-support NBDP priors that are not convex combinations of other priors. pr ,s v +i vi0
vj rs (1 − r )s
vj0 r (1 − s) (1 − r )(1 − s)
With different valuations, under prior pr ,s each bidder has the same belief about the other’s types, so pr ,s is not a BDP. Moreover, it’s not a convex combination of other common priors in P, because if (r , s) 6= (r 0 , s 0 ), then the priors pr ,s and pr 0 ,s 0 on the universal space T have disjoint supports.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Shyness: Measure-theoretic Smallness First notice that the dimensions of both Pu and Pf are infinite. However, unlike in a finite-dimensional Euclidean space Rk where Lebesgue measure can be defined, in infinite-dimension spaces there does not exist any translation invariant measure. Eg., the unit ball in any infinite dimensional space is not compact; Any open r ball contain infinitely many disjoint 4r balls. A translation of E ⊂ Rk is a set {x + y : x ∈ E }, y ∈ Rk .
Definition Let X be a completely metrizable topological vector space. A Borel set E ⊂ X is shy if there exists a regular Borel probability measure µ on X with compact support such that µ(E + x) = 0 for every x ∈ X . A subset F ⊂ X is shy if it is contained in a shy Borel set. A subset Y ⊂ X is prevalent if X \ Y is shy.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Definition Let C be a completely metrizable convex subset of the topological vector space X . A universally measurable subset E ⊂ C is finitely shy in C ∈ X if there exists a finite-dimensional subspace H ⊂ X such that λH (C + p) > 0 for some p ∈ X and λH (E + x) = 0 for every x ∈ X . A arbitrary subset F ⊂ X is finitely shy in C if it is contained in a finitely shy universally measurable set. Intuitively, a set is finitely shy if all of its translations are of measure zero in some finite-dimensional subset of X , which itself is not of measure zero.
Theorem Let P be a completely metrizable convex set of priors with a topology that satisfies the continuity and Borel requirements. Suppose that P contains at least one NBDP prior for bidder i. If the subset B ⊂ P of BDP priors for bidder i is universally measurable, then both B and the set of FSE priors for bidder i, F, are finitely shy in P.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Discussions and Future Research
I
Heifetz and Neeman Vs. Cremer and McLean. Not comparable. Cremer and McLean case the set of priors is not convex across models, actually the model is fixed.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Discussions and Future Research
I
Heifetz and Neeman Vs. Cremer and McLean. Not comparable. Cremer and McLean case the set of priors is not convex across models, actually the model is fixed.
I
Robust Implementation. What happens if the designer does not know the exact beliefs that players hold, but a neighborhood of the actual beliefs? Redundant types?
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Discussions and Future Research
I
Heifetz and Neeman Vs. Cremer and McLean. Not comparable. Cremer and McLean case the set of priors is not convex across models, actually the model is fixed.
I
Robust Implementation. What happens if the designer does not know the exact beliefs that players hold, but a neighborhood of the actual beliefs? Redundant types?
I
Sufficient conditions. While Cremer and McLean (1988) proved both necessity and sufficiency for FSE, in this paper only necessity is mentioned.
Motivation
Model Set Up
BDP Priors
Full Surplus Extraction
Smallness of the Set of NBDP Priors
Discussions and Future Research
I
Heifetz and Neeman Vs. Cremer and McLean. Not comparable. Cremer and McLean case the set of priors is not convex across models, actually the model is fixed.
I
Robust Implementation. What happens if the designer does not know the exact beliefs that players hold, but a neighborhood of the actual beliefs? Redundant types?
I
Sufficient conditions. While Cremer and McLean (1988) proved both necessity and sufficiency for FSE, in this paper only necessity is mentioned.
I
Type spaces without a common prior...
