Problem Statement SWO Axioms for SWO Main Result
Roberts’ Theorem with Neutrality:
A Social Welfare Ordering Approach
Debasis Mishra
Indian Statistical Institute
joint work with
Arunava Sen, ISI
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Outline
Objectives of this research
Characterize (dominant strategy) implementable social choice
functions in quasi-linear environments when agents have
multidimensional types.
Use social welfare ordering approach, as in aggregation
problems with utility information (“richer” versions of the
Arrovian aggregation problem).
Investigate if this approach works for any restricted domain.
Our results
A “simple” proof of Roberts’ theorem with neutrality.
Roberts’ theorem with neutrality holds for a particular
restricted domain.
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Outline
Objectives of this research
Characterize (dominant strategy) implementable social choice
functions in quasi-linear environments when agents have
multidimensional types.
Use social welfare ordering approach, as in aggregation
problems with utility information (“richer” versions of the
Arrovian aggregation problem).
Investigate if this approach works for any restricted domain.
Our results
A “simple” proof of Roberts’ theorem with neutrality.
Roberts’ theorem with neutrality holds for a particular
restricted domain.
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Our Approach
Step 1: Every implementable social choice function satisfies a
condition called “positive association of differences” (PAD) Roberts, 1979.
Step 2: If the domain is unrestricted or the set of all
non-negative types then:
Step 2a: If a social choice function satisfies PAD and is
neutral, then it induces an ordering on the domain.
Step 2b: This ordering satisfies three axioms: weak Pareto,
invariance, and continuity.
Step 2c: Every ordering satisfying these axioms can be
characterized as weighted welfare maximizers. We can use
classical results in aggregation theory such as Blackwell and
Grishick (1954) and Milnor (1954).
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
The Model
Finite set of m alternatives: A = {a, b, c, . . .}. Assume m ≥ 3.
Set of agents N = {1, 2, . . . , n}.
Type of agent i: ti = (tia , tib , . . . , ) - a vector in Rm .
Type profile of agents: t (n × m matrix) - n vectors in Rm .
Possible types of agent i is T and possible type profiles of
agents is T.
t a is the column vector corresponding to alternative a.
Possible column vector of any type profile matrix is CT .
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Dominant Strategy Implementation
A social choice function (SCF) is a mapping f : T → A.
A payment function is a mapping p : T → Rn , where pi (t)
denotes the payment of agent i when the type profile is t ∈ T.
Definition
A social choice function f is implementable if there exists a
payment function p such that for all i ∈ N, for all t−i ∈ T−i we
have
f (t)
ti
f (si ,t−i )
− pi (t) ≥ ti
− pi (si , t−i )
∀ si , ti ∈ T .
What social choice functions are implementable?
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
A Property of Implementable SCFs
Definition
A social choice function f satisfies positive association of
differences (PAD) if for every s, t ∈ T such that f (t) = a with
s a − t a s b − t b for all b 6= a, we have f (s) = a.
Lemma (Roberts, 1979)
Every implementable social choice function satisfies PAD.
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Roberts’ Theorem
Definition
A social choice function f satisfies non-imposition if for every
a ∈ A, there exists t ∈ T such that f (t) = a.
Theorem (Roberts, 1979)
Suppose CT = Rn . If an implementable social choice function
satisfies non-imposition, then there exists weights λ ∈ Rn+ \ {0}
and a deterministic real-valued function κ : A → R such that for all
t ∈ T,
X a
λi ti + κ(a)
f (t) ∈ arg max
a∈A
Debasis Mishra
i∈N
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
A Choice Set
Assumption: CT = Rn (unrestricted domain) or Rn+ .
Definition
The choice set of an SCF f at every type profile t is
C f (t) = {a ∈ A : ∀ ε 0, f (t a + ε, t −a ) = a}.
Lemma
If f is implementable, then for all type profiles t, f (t) ∈ C f (t).
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
A Choice Set
Assumption: CT = Rn (unrestricted domain) or Rn+ .
Definition
The choice set of an SCF f at every type profile t is
C f (t) = {a ∈ A : ∀ ε 0, f (t a + ε, t −a ) = a}.
Lemma
If f is implementable, then for all type profiles t, f (t) ∈ C f (t).
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Social Welfare Ordering
Definition
A social welfare ordering (SWO) R f induced by a social choice
function f is a binary relation on CT defined as follows. The
symmetric component of R f is denoted by I f and the
antisymmetric component of R f is denoted by P f . Pick x, y ∈ CT .
We say xP f y if and only if there exists a profile t with t a = x
and t b = y for some a, b ∈ A such that a ∈ C f (t) but
b∈
/ C f (t).
We say xI f y if and only if there exists a profile t with t a = x
and t b = y for some a, b ∈ A such that a, b ∈ C f (t).
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Why R f is an Ordering
Lemma
Let f be an implementable social choice function. Consider two
type profiles t = (t a , t b , t −ab ) and s = (s a = t a , s b = t b , s −ab ).
a) Suppose a, b ∈ C f (t) and a ∈ C f (s). Then b ∈ C f (s).
b) Suppose a ∈ C f (t) but b ∈
/ C f (t). Then b ∈
/ C f (s).
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Why R f is an Ordering - Neutrality
Definition
A social choice function f is neutral if for every type profile t ∈ T
and for all permutations % on A such that t 6= s, where s is the
type profile due to permutation %, we have %(f (t)) = f (s).
Proposition
Suppose f is an implementable and neutral social choice function.
The relation R f induced by f on CT is an ordering.
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Why R f is an Ordering - Neutrality
Definition
A social choice function f is neutral if for every type profile t ∈ T
and for all permutations % on A such that t 6= s, where s is the
type profile due to permutation %, we have %(f (t)) = f (s).
Proposition
Suppose f is an implementable and neutral social choice function.
The relation R f induced by f on CT is an ordering.
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Three Axioms for an Ordering
Definition
An ordering R on CT satisfies weak Pareto (WP) if for all
x, y ∈ CT with x y we have xPy .
Definition
An ordering R on CT satisfies invariance (INV) if for all
x, y ∈ CT and all z ∈ Rn such that (x + z), (y + z) ∈ CT we have
xPy implies (x + z)P(y + z) and xIy implies (x + z)I (y + z).
Definition
An ordering R on CT satisfies continuity (C) if for all x ∈ CT , the
sets U x = {y ∈ CT : yRx} and Lx = {y ∈ CT : xRy } are closed in
Rn .
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
An SWO Satisfies these Axioms
Proposition
Suppose f is an implementable and neutral social choice function.
Then the social welfare ordering R f induced by f on CT satisfies
weak Pareto, invariance, and continuity.
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Characterizing the Ordering
Suppose an ordering on CT satisfies WP, INV, and C. Can we
say anything about the ordering? See d’Aspremont and
Gevers (2002).
Proposition (Blackwell and Girshick (1954))
Suppose an ordering R on Rn satisfies weak Pareto, invariance,
and continuity. Then there exists weights λ ∈ Rn+ \ {0} and for all
x, y ∈ Rn
X
X
xRy ⇔
λi xi ≥
λi yi .
i∈N
i∈N
We show that this result holds even if R is an ordering on Rn+ .
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Characterizing the Ordering
Suppose an ordering on CT satisfies WP, INV, and C. Can we
say anything about the ordering? See d’Aspremont and
Gevers (2002).
Proposition (Blackwell and Girshick (1954))
Suppose an ordering R on Rn satisfies weak Pareto, invariance,
and continuity. Then there exists weights λ ∈ Rn+ \ {0} and for all
x, y ∈ Rn
X
X
xRy ⇔
λi xi ≥
λi yi .
i∈N
i∈N
We show that this result holds even if R is an ordering on Rn+ .
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Characterizing the Ordering
Suppose an ordering on CT satisfies WP, INV, and C. Can we
say anything about the ordering? See d’Aspremont and
Gevers (2002).
Proposition (Blackwell and Girshick (1954))
Suppose an ordering R on Rn satisfies weak Pareto, invariance,
and continuity. Then there exists weights λ ∈ Rn+ \ {0} and for all
x, y ∈ Rn
X
X
xRy ⇔
λi xi ≥
λi yi .
i∈N
i∈N
We show that this result holds even if R is an ordering on Rn+ .
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Roberts’ Theorem with Neutrality
Theorem
Suppose f is an implementable and neutral social choice function.
Then there exists weights λ ∈ Rn+ \ {0} such that for all t ∈ T,
f (t) ∈ arg max
a∈A
Debasis Mishra
X
λi tia .
i∈N
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Final Comments and Future Directions
If we impose anonymity (permuting rows), then λs become
equal (i.e., f is efficient) in Roberts’ theorem. An easy proof
using our approach and an elegant result of Milnor (1954)
exists for this case.
Can we drop neutrality and prove the general version of
Roberts theorem using our approach?
Note: Our results are slightly more general than Roberts’
theorem with neutrality since our result holds for Rn+ also.
Can we extend this approach to private goods settings (where
every SCF satisfies PAD)?
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Final Comments and Future Directions
If we impose anonymity (permuting rows), then λs become
equal (i.e., f is efficient) in Roberts’ theorem. An easy proof
using our approach and an elegant result of Milnor (1954)
exists for this case.
Can we drop neutrality and prove the general version of
Roberts theorem using our approach?
Note: Our results are slightly more general than Roberts’
theorem with neutrality since our result holds for Rn+ also.
Can we extend this approach to private goods settings (where
every SCF satisfies PAD)?
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Anonymity
Definition
A social choice function f is anonymous if for every t ∈ T and
every permutation σ on the row vectors (agents) of t we have
f (σ(t)) = f (t).
Definition
An ordering R on CT satisfies anonymity if for every x, y ∈ CT
and every permutation σ on agents we have xIy if x = σ(y ).
Lemma
Suppose f is implementable and anonymous. Then, R f satisfies
anonymity.
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
A Characterization of Efficiency
Theorem
Suppose f is implementable, neutral, and anonymous. Then f is
the efficient social choice function.
Debasis Mishra
Roberts’ Theorem with Neutrality
Problem Statement SWO Axioms for SWO Main Result
Sketch of Proof
The proof is an adaptation of an elegant proof of Milnor (1954).
Suppose we have an ordering R which satisfies WP, INV, and
Anonymity. Consider n = 3 and take x = (5, 3, 4) and
y = (6, 1, 5). We show that xIy .
Consider x 0 = (3, 4, 5) and y 0 = (1, 5, 6). By Anonymity, xIx 0
and yIy 0 . So x 0 and y 0 ranked same way as x and y .
Consider x 00 = (2, 0, 0) and y 00 = (0, 1, 1). By INV, x 00 and y 00
ranked same way as x and y .
Repeating this, we will finally get (0, 0, 0) and (0, 0, 0) to
conclude xIy .
If x = (5, 3, 7) and y = (6, 1, 5), we construct x = (5, 3, 7)
and y 0 = (7, 2, 6). By WP y 0 Py , and like before we show xIy 0
to conclude xPy .
Debasis Mishra
Roberts’ Theorem with Neutrality