Randomization in Mechanism Design for Voting Environments: Some Recent Results
Randomization in Mechanism Design for
Voting Environments: Some Recent Results
Arunava Sen
Indian Statistical Institute, New Delhi
Conference Francqui, June 24, 2013
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The issue
Randomization is a natural way to construct “fair” outcomes
in the presence of conflicts of interest. Maybe particularly
important where monetary compensation is not possible.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The issue
Randomization is a natural way to construct “fair” outcomes
in the presence of conflicts of interest. Maybe particularly
important where monetary compensation is not possible.
We consider the following question: how does
randomization impact incentive-compatibility
(strategy-proofness)?
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The issue contd.
If lotteries are evaluated by the expected utility criterion,
then preferences satisfy a domain restriction - the vN-M
independence axiom.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The issue contd.
If lotteries are evaluated by the expected utility criterion,
then preferences satisfy a domain restriction - the vN-M
independence axiom.
If a player prefers lottery l to lottery l ′ , then she also prefers
λl + (1 − λ)l ′′ to λl ′ + (1 − λ)l ′′ for any λ ∈ (0, 1) and
arbitrary lottery l ′′ .
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The issue contd.
If lotteries are evaluated by the expected utility criterion,
then preferences satisfy a domain restriction - the vN-M
independence axiom.
If a player prefers lottery l to lottery l ′ , then she also prefers
λl + (1 − λ)l ′′ to λl ′ + (1 − λ)l ′′ for any λ ∈ (0, 1) and
arbitrary lottery l ′′ .
Immediate Consequence: If two social choice functions are
strategy-proof, then so is any convex combination of the
two. The set of strategy-proof social choice is convex.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The issue contd.
A natural way to characterize convex sets is in terms of
their extreme points.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The issue contd.
A natural way to characterize convex sets is in terms of
their extreme points.
A deterministic strategy-proof social choice function is
clearly an extreme-point of this set.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The issue contd.
A natural way to characterize convex sets is in terms of
their extreme points.
A deterministic strategy-proof social choice function is
clearly an extreme-point of this set.
ARE THERE ANY OTHERS?
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The issue contd.
A natural way to characterize convex sets is in terms of
their extreme points.
A deterministic strategy-proof social choice function is
clearly an extreme-point of this set.
ARE THERE ANY OTHERS?
If there are not, then randomization does not allow for any
possibilities other than picking from the set of deterministic
strategy-proof social choice functions by means of a fixed
probability distribution (the weights in the convex
combination are the probabilities.)
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The issue contd.
A natural way to characterize convex sets is in terms of
their extreme points.
A deterministic strategy-proof social choice function is
clearly an extreme-point of this set.
ARE THERE ANY OTHERS?
If there are not, then randomization does not allow for any
possibilities other than picking from the set of deterministic
strategy-proof social choice functions by means of a fixed
probability distribution (the weights in the convex
combination are the probabilities.)
If all extreme-points are deterministic, then we say that the
domain has the deterministic extreme point or DEP
property.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The issue contd.
A natural way to characterize convex sets is in terms of
their extreme points.
A deterministic strategy-proof social choice function is
clearly an extreme-point of this set.
ARE THERE ANY OTHERS?
If there are not, then randomization does not allow for any
possibilities other than picking from the set of deterministic
strategy-proof social choice functions by means of a fixed
probability distribution (the weights in the convex
combination are the probabilities.)
If all extreme-points are deterministic, then we say that the
domain has the deterministic extreme point or DEP
property.
What domains have the DEP Property?
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Model
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Model
Standard Voting model.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Model
Standard Voting model.
I = {1, 2, . . . , N}: a set of voters, |I| = N, N ≥ 2;
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Model
Standard Voting model.
I = {1, 2, . . . , N}: a set of voters, |I| = N, N ≥ 2;
A = {a1 , . . . , am }: a finite set of alternatives, |A| = m,
m ≥ 3;
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Model
Standard Voting model.
I = {1, 2, . . . , N}: a set of voters, |I| = N, N ≥ 2;
A = {a1 , . . . , am }: a finite set of alternatives, |A| = m,
m ≥ 3;
P: set of strict orderings over the elements of A;
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Model
Standard Voting model.
I = {1, 2, . . . , N}: a set of voters, |I| = N, N ≥ 2;
A = {a1 , . . . , am }: a finite set of alternatives, |A| = m,
m ≥ 3;
P: set of strict orderings over the elements of A;
D ⊂ P: an admissible domain of (strict) preference
orderings;
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Model
Standard Voting model.
I = {1, 2, . . . , N}: a set of voters, |I| = N, N ≥ 2;
A = {a1 , . . . , am }: a finite set of alternatives, |A| = m,
m ≥ 3;
P: set of strict orderings over the elements of A;
D ⊂ P: an admissible domain of (strict) preference
orderings;
Pi ∈ D: a preference ordering for voter i;
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Model
Standard Voting model.
I = {1, 2, . . . , N}: a set of voters, |I| = N, N ≥ 2;
A = {a1 , . . . , am }: a finite set of alternatives, |A| = m,
m ≥ 3;
P: set of strict orderings over the elements of A;
D ⊂ P: an admissible domain of (strict) preference
orderings;
Pi ∈ D: a preference ordering for voter i;
P = (P1 , . . . , PN ) = (Pi , P−i ) ∈ DN : a preferences profile;
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Model
Standard Voting model.
I = {1, 2, . . . , N}: a set of voters, |I| = N, N ≥ 2;
A = {a1 , . . . , am }: a finite set of alternatives, |A| = m,
m ≥ 3;
P: set of strict orderings over the elements of A;
D ⊂ P: an admissible domain of (strict) preference
orderings;
Pi ∈ D: a preference ordering for voter i;
P = (P1 , . . . , PN ) = (Pi , P−i ) ∈ DN : a preferences profile;
rk (Pi ): the kth ranked alternative in Pi , 1 ≤ k ≤ m;
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Deterministic Social Choice Functions
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Deterministic Social Choice Functions
Definition
A (Deterministic) Social Choice Function (SCF) is a mapping
f : DN → A.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Deterministic Social Choice Functions
Definition
A (Deterministic) Social Choice Function (SCF) is a mapping
f : DN → A.
Definition
A SCF f satisfies unanimity, if f (P) = a whenever r1 (Pi ) = a for
all i ∈ I.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Deterministic Social Choice Functions
Definition
A (Deterministic) Social Choice Function (SCF) is a mapping
f : DN → A.
Definition
A SCF f satisfies unanimity, if f (P) = a whenever r1 (Pi ) = a for
all i ∈ I.
Definition
A SCF f is manipulable at P via Pi′ if f (Pi′ , P−i )Pi f (Pi , P−i ). The
SCF f is strategy-proof if it is not manipulable by any voter.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Random Social Choice Functions
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Random Social Choice Functions
Let L(A) denote the set of lotteries or probability distributions
over A.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Random Social Choice Functions
Let L(A) denote the set of lotteries or probability distributions
over A.
Definition
A Random Social Choice Function (RSCF) is a mapping
ϕ : DN → L(A).
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Random Social Choice Functions
Let L(A) denote the set of lotteries or probability distributions
over A.
Definition
A Random Social Choice Function (RSCF) is a mapping
ϕ : DN → L(A).
Let ϕa (P) denote the probability assigned to a in profile P.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Random Social Choice Functions
Let L(A) denote the set of lotteries or probability distributions
over A.
Definition
A Random Social Choice Function (RSCF) is a mapping
ϕ : DN → L(A).
Let ϕa (P) denote the probability assigned to a in profile P.
Definition
A RSCF ϕ satisfies unanimity if ϕa (P) = 1 whenever r1 (Pi ) = a
for all i ∈ I.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Random Social Choice Functions
Let L(A) denote the set of lotteries or probability distributions
over A.
Definition
A Random Social Choice Function (RSCF) is a mapping
ϕ : DN → L(A).
Let ϕa (P) denote the probability assigned to a in profile P.
Definition
A RSCF ϕ satisfies unanimity if ϕa (P) = 1 whenever r1 (Pi ) = a
for all i ∈ I.
If all voters have a common top alternative, this must be chosen
with probability 1.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Strategy-proofness of RSCFs
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Strategy-proofness of RSCFs
We follow the approach of Gibbard (1977). Other
approaches are possible - see below.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Strategy-proofness of RSCFs
We follow the approach of Gibbard (1977). Other
approaches are possible - see below.
Definition
A utility function u : A → R represents Pi , if for all a, b ∈ A,
[a Pi b] ⇔ [u(a) > u(b)].
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Strategy-proofness of RSCFs
We follow the approach of Gibbard (1977). Other
approaches are possible - see below.
Definition
A utility function u : A → R represents Pi , if for all a, b ∈ A,
[a Pi b] ⇔ [u(a) > u(b)].
Let U(Pi ) denote the set of utility functions that represent Pi .
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Strategy-proofness of RSCFs
We follow the approach of Gibbard (1977). Other
approaches are possible - see below.
Definition
A utility function u : A → R represents Pi , if for all a, b ∈ A,
[a Pi b] ⇔ [u(a) > u(b)].
Let U(Pi ) denote the set of utility functions that represent Pi .
Definition
A RSCF ϕ : DN → L(A) is strategy-proof, if for all i ∈ I;
Pi , Pi′ ∈ D, P−i ∈ DN−1 and u ∈ U(Pi ), we have
X
a∈A
u(a) ϕa (Pi , P−i ) ≥
X
a∈A
u(a) ϕa (Pi′ , P−i )
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Strategy-proofness for RSCFs contd.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Strategy-proofness for RSCFs contd.
According to Gibbard’s definition, truth-telling must yield higher
expected utility than misrepresentation for every possible utility
representation of the true preference.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Strategy-proofness for RSCFs contd.
According to Gibbard’s definition, truth-telling must yield higher
expected utility than misrepresentation for every possible utility
representation of the true preference.
This is equivalent to the following: truth-telling stochastically
dominates the lottery from misrepresentation according to the
true preference.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Strategy-proofness for RSCFs contd.
According to Gibbard’s definition, truth-telling must yield higher
expected utility than misrepresentation for every possible utility
representation of the true preference.
This is equivalent to the following: truth-telling stochastically
dominates the lottery from misrepresentation according to the
true preference.
Weaker requirement: Misrepresentation should not give higher
expected utility for all possible utility representation.
Definition
A RSCF ϕ : DN → L(A) is strategy-proof, if for all i ∈ I;
Pi , Pi′ ∈ D and P−i ∈ DN−1 , we have
t
X
k =1
ϕrk (Pi ) (Pi , P−i ) ≥
t
X
k =1
ϕrk (Pi ) (Pi′ , P−i ),
t = 1, . . . , m
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Observations
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Observations
Fix a domain D.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Observations
Fix a domain D.
Let F denote the set of all deterministic strategy-proof
SCFs satisfying unanimity defined on D.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Observations
Fix a domain D.
Let F denote the set of all deterministic strategy-proof
SCFs satisfying unanimity defined on D.
Let Φ denote the set of strategy-proof RSCFs satisfying
unanimity defined on D.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Observations
Fix a domain D.
Let F denote the set of all deterministic strategy-proof
SCFs satisfying unanimity defined on D.
Let Φ denote the set of strategy-proof RSCFs satisfying
unanimity defined on D.
Let conv[F] denote the set of all convex combinations of
the elements of F.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Observations
Fix a domain D.
Let F denote the set of all deterministic strategy-proof
SCFs satisfying unanimity defined on D.
Let Φ denote the set of strategy-proof RSCFs satisfying
unanimity defined on D.
Let conv[F] denote the set of all convex combinations of
the elements of F.
It is easy to verify that conv[F] ⊂ Φ.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Observations
Fix a domain D.
Let F denote the set of all deterministic strategy-proof
SCFs satisfying unanimity defined on D.
Let Φ denote the set of strategy-proof RSCFs satisfying
unanimity defined on D.
Let conv[F] denote the set of all convex combinations of
the elements of F.
It is easy to verify that conv[F] ⊂ Φ.
We say that D has the deterministic extreme-point or DEP
property if
Φ = conv[F].
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Results
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Results
Domains that have the DEP Property
P with |A| ≥ 3. (Gibbard (1977)).
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Results
Domains that have the DEP Property
P with |A| ≥ 3. (Gibbard (1977)).
The binary domain, i.e. |A| = 2. (Picot-Sen (2012)).
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Results
Domains that have the DEP Property
P with |A| ≥ 3. (Gibbard (1977)).
The binary domain, i.e. |A| = 2. (Picot-Sen (2012)).
The single-peaked domain, (Peters-Roy-Storcken-Sen
(2011), Pycia-Unver (2012)).
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Results
Domains that have the DEP Property
P with |A| ≥ 3. (Gibbard (1977)).
The binary domain, i.e. |A| = 2. (Picot-Sen (2012)).
The single-peaked domain, (Peters-Roy-Storcken-Sen
(2011), Pycia-Unver (2012)).
The product domain with lexicographically separable
preferences (Chatterji-Roy-Sen (2012)).
However, not all domains have the DEP Property.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Complete Domain with |A| ≥ 3
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Complete Domain with |A| ≥ 3
Definition
A SCF f is dictatorial if there exists a voter i such that at all
profiles P, f (P) = r1 (Pi ).
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Complete Domain with |A| ≥ 3
Definition
A SCF f is dictatorial if there exists a voter i such that at all
profiles P, f (P) = r1 (Pi ).
Definition
A RSCF ϕ : DN →PL(A) is random dictatorship, if there exist
N
{εk }N
k =1 ≥ 0 with Pk =1 εk = 1 such that for all a ∈ A and
P ∈ DN , ϕa (P) = {k :r1 (Pk )=a} εk .
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Complete Domain with |A| ≥ 3
Definition
A SCF f is dictatorial if there exists a voter i such that at all
profiles P, f (P) = r1 (Pi ).
Definition
A RSCF ϕ : DN →PL(A) is random dictatorship, if there exist
N
{εk }N
k =1 ≥ 0 with Pk =1 εk = 1 such that for all a ∈ A and
P ∈ DN , ϕa (P) = {k :r1 (Pk )=a} εk .
Theorem
Gibbard-Sattherthwaite (1973) Let f : PN → A be a
strategy-proof SCF satisfying unanimity. Then f must be
dictatorial.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Complete Domain with |A| ≥ 3 contd.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Complete Domain with |A| ≥ 3 contd.
Theorem
Gibbard (1977) Let ϕ : PN → L(A) be a strategy-proof RSCF
satisfying unanimity. Then f must be a random dictatorship.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Complete Domain with |A| ≥ 3 contd.
Theorem
Gibbard (1977) Let ϕ : PN → L(A) be a strategy-proof RSCF
satisfying unanimity. Then f must be a random dictatorship.
Clearly P has the DEP property.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Complete Domain with |A| ≥ 3 contd.
Theorem
Gibbard (1977) Let ϕ : PN → L(A) be a strategy-proof RSCF
satisfying unanimity. Then f must be a random dictatorship.
Clearly P has the DEP property. Other (simpler) proofs of the
DEP Property of P: Duggan (1996), Sen (2012).
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Binary Domain
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Binary Domain
Picot-Sen (2012)
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Binary Domain
Picot-Sen (2012)
The set of deterministic strategy-proof SCFs is the set of
committees.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Binary Domain
Picot-Sen (2012)
The set of deterministic strategy-proof SCFs is the set of
committees.
The question can be given a matrix algebra formulation.
Farkas’ Lemma can be applied to show that the domain
has the DEP Property.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Binary Domain
Picot-Sen (2012)
The set of deterministic strategy-proof SCFs is the set of
committees.
The question can be given a matrix algebra formulation.
Farkas’ Lemma can be applied to show that the domain
has the DEP Property.
Valid even without unanimity.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Single-Peaked Domains
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Single-Peaked Domains
The set of strategy-proof SCFS on the single-peaked
domain is well-understood. Moulin (1980) etc.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Single-Peaked Domains
The set of strategy-proof SCFS on the single-peaked
domain is well-understood. Moulin (1980) etc.
Peters-Roy-Sen-Storcken (2011) show that this domain
has the DEP property using Farkas Lemma and the
max-flow min-cut theorem from network flow analysis.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Single-Peaked Domains
The set of strategy-proof SCFS on the single-peaked
domain is well-understood. Moulin (1980) etc.
Peters-Roy-Sen-Storcken (2011) show that this domain
has the DEP property using Farkas Lemma and the
max-flow min-cut theorem from network flow analysis.
Pycia-Unver (2012) prove the same result using direct
methods. Related paper: (Morimoto 2012).
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Single-Peaked Domains
The set of strategy-proof SCFS on the single-peaked
domain is well-understood. Moulin (1980) etc.
Peters-Roy-Sen-Storcken (2011) show that this domain
has the DEP property using Farkas Lemma and the
max-flow min-cut theorem from network flow analysis.
Pycia-Unver (2012) prove the same result using direct
methods. Related paper: (Morimoto 2012).
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
Chatterji-Roy-Sen (2012)
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
Chatterji-Roy-Sen (2012)
A = A1 × A2 × ... × Am with |Aj | ≥ 3 for all j.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
Chatterji-Roy-Sen (2012)
A = A1 × A2 × ... × Am with |Aj | ≥ 3 for all j.
Voter preferences on A assumed to be separable in some
sense. An implication of separability is that every ordering
on A induces preferences on every component.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
Chatterji-Roy-Sen (2012)
A = A1 × A2 × ... × Am with |Aj | ≥ 3 for all j.
Voter preferences on A assumed to be separable in some
sense. An implication of separability is that every ordering
on A induces preferences on every component.
Large literature on such models staring with
Barberà-Sonnenschein-Zhou (1991) (where |Aj | = 2). For
example Le-Breton-Sen (1999).
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
Chatterji-Roy-Sen (2012)
A = A1 × A2 × ... × Am with |Aj | ≥ 3 for all j.
Voter preferences on A assumed to be separable in some
sense. An implication of separability is that every ordering
on A induces preferences on every component.
Large literature on such models staring with
Barberà-Sonnenschein-Zhou (1991) (where |Aj | = 2). For
example Le-Breton-Sen (1999).
We assume lexicographic separability.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
Chatterji-Roy-Sen (2012)
A = A1 × A2 × ... × Am with |Aj | ≥ 3 for all j.
Voter preferences on A assumed to be separable in some
sense. An implication of separability is that every ordering
on A induces preferences on every component.
Large literature on such models staring with
Barberà-Sonnenschein-Zhou (1991) (where |Aj | = 2). For
example Le-Breton-Sen (1999).
We assume lexicographic separability.
Applying Le-Breton-Sen (1999), it follows that every
strategy-proof SCF satisfying unanimity on this domain is a
component dictatorship.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
Chatterji-Roy-Sen (2012)
A = A1 × A2 × ... × Am with |Aj | ≥ 3 for all j.
Voter preferences on A assumed to be separable in some
sense. An implication of separability is that every ordering
on A induces preferences on every component.
Large literature on such models staring with
Barberà-Sonnenschein-Zhou (1991) (where |Aj | = 2). For
example Le-Breton-Sen (1999).
We assume lexicographic separability.
Applying Le-Breton-Sen (1999), it follows that every
strategy-proof SCF satisfying unanimity on this domain is a
component dictatorship.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
contd.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
contd.
A voter sequence i is an sequence of voters (possibly
involving repetition) of length m.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
contd.
A voter sequence i is an sequence of voters (possibly
involving repetition) of length m.
Fix a profile P. The component dictatorship outcome w.r.t a
voter sequence i is defined as follows: the value of
component k, k = 1, . . . , m is the maximal outcome of
component k of the kth voter in the sequence i.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
contd.
A voter sequence i is an sequence of voters (possibly
involving repetition) of length m.
Fix a profile P. The component dictatorship outcome w.r.t a
voter sequence i is defined as follows: the value of
component k, k = 1, . . . , m is the maximal outcome of
component k of the kth voter in the sequence i.
A generalized random dictatorship GRS is a probability
distribution over voter sequences.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
contd.
A voter sequence i is an sequence of voters (possibly
involving repetition) of length m.
Fix a profile P. The component dictatorship outcome w.r.t a
voter sequence i is defined as follows: the value of
component k, k = 1, . . . , m is the maximal outcome of
component k of the kth voter in the sequence i.
A generalized random dictatorship GRS is a probability
distribution over voter sequences.
CRS show that a strategy-proof RSCF over this domain is
a GRS.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Product Domains and Lexicographic Preferences
contd.
A voter sequence i is an sequence of voters (possibly
involving repetition) of length m.
Fix a profile P. The component dictatorship outcome w.r.t a
voter sequence i is defined as follows: the value of
component k, k = 1, . . . , m is the maximal outcome of
component k of the kth voter in the sequence i.
A generalized random dictatorship GRS is a probability
distribution over voter sequences.
CRS show that a strategy-proof RSCF over this domain is
a GRS.
Corollary: the product domain with lexicographically
separable preferences has the DEP property.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Generalized Random Dictatorships: Observations
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Generalized Random Dictatorships: Observations
A random dictatorship is a special case of a GRP where
the only voter sequences that get strictly positive
probability are the sequences (i, i, . . . , i), i = 1, . . . , n, i.e.
the diagonal elements.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Generalized Random Dictatorships: Observations
A random dictatorship is a special case of a GRP where
the only voter sequences that get strictly positive
probability are the sequences (i, i, . . . , i), i = 1, . . . , n, i.e.
the diagonal elements.
A GRP is not a tops-only RSCF. However it is top-product
set only. Suppose there are two voters and two
components. Suppose the voter tops are (a1 , a2 ) for voter
1 and (b1 , b2 ) for voter 2. Then there can be positive
probability only on the sets (a1 , a2 ), (a1 , b2 ), (b1 , a2 ) and
(b1 , b2 ).
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Generalized Random Dictatorships: Observations
A random dictatorship is a special case of a GRP where
the only voter sequences that get strictly positive
probability are the sequences (i, i, . . . , i), i = 1, . . . , n, i.e.
the diagonal elements.
A GRP is not a tops-only RSCF. However it is top-product
set only. Suppose there are two voters and two
components. Suppose the voter tops are (a1 , a2 ) for voter
1 and (b1 , b2 ) for voter 2. Then there can be positive
probability only on the sets (a1 , a2 ), (a1 , b2 ), (b1 , a2 ) and
(b1 , b2 ).
No component independence as in the deterministic case,
i.e. it is not a product of probability distributions across
components. Counterexample: random dictatorship.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Generalized Random Dictatorships: Observations
A random dictatorship is a special case of a GRP where
the only voter sequences that get strictly positive
probability are the sequences (i, i, . . . , i), i = 1, . . . , n, i.e.
the diagonal elements.
A GRP is not a tops-only RSCF. However it is top-product
set only. Suppose there are two voters and two
components. Suppose the voter tops are (a1 , a2 ) for voter
1 and (b1 , b2 ) for voter 2. Then there can be positive
probability only on the sets (a1 , a2 ), (a1 , b2 ), (b1 , a2 ) and
(b1 , b2 ).
No component independence as in the deterministic case,
i.e. it is not a product of probability distributions across
components. Counterexample: random dictatorship.
Separable preferences? Characterization open.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Dictatorial and Random Dictatorship Domains,
Chatterji-Sen-Zeng (2012) (CSZ)
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Dictatorial and Random Dictatorship Domains,
Chatterji-Sen-Zeng (2012) (CSZ)
The previous discussion may suggest that “all” domains have
the DEP Property.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Dictatorial and Random Dictatorship Domains,
Chatterji-Sen-Zeng (2012) (CSZ)
The previous discussion may suggest that “all” domains have
the DEP Property. NOT TRUE!!
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Dictatorial and Random Dictatorship Domains,
Chatterji-Sen-Zeng (2012) (CSZ)
The previous discussion may suggest that “all” domains have
the DEP Property. NOT TRUE!!
Definition
A domain D is dictatorial (resp. a random dictatorship domain)
if every strategy-proof SCF (resp. RSCF) satisfying unanimity
defined on it, is dictatorial (resp. a random dictatorship).
We have seen that P has the DEP property. However all
dictatorial domains are not random dictatorship domains,
i.e. do not have the DEP Property.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Dictatorial and Random Dictatorship Domains,
Chatterji-Sen-Zeng (2012) (CSZ)
The previous discussion may suggest that “all” domains have
the DEP Property. NOT TRUE!!
Definition
A domain D is dictatorial (resp. a random dictatorship domain)
if every strategy-proof SCF (resp. RSCF) satisfying unanimity
defined on it, is dictatorial (resp. a random dictatorship).
We have seen that P has the DEP property. However all
dictatorial domains are not random dictatorship domains,
i.e. do not have the DEP Property.
CSZ provide sufficient conditions for a dictatorial domain to
be a random dictatorship domain.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Linked Domains; Aswal-Chatterji-Sen (2003)
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Linked Domains; Aswal-Chatterji-Sen (2003)
General Approach to dictatorial domains.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Linked Domains; Aswal-Chatterji-Sen (2003)
General Approach to dictatorial domains.
Definition
A pair of alternatives: a, b ∈ A, is connected, denoted a ∼ b, if
there exist Pi , Pi′ ∈ D such that
(i) r1 (Pi ) = a and r2 (Pi ) = b
(ii) r1 (Pi′ ) = b and r2 (Pi′ ) = a.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Linked Domains; Aswal-Chatterji-Sen (2003)
General Approach to dictatorial domains.
Definition
A pair of alternatives: a, b ∈ A, is connected, denoted a ∼ b, if
there exist Pi , Pi′ ∈ D such that
(i) r1 (Pi ) = a and r2 (Pi ) = b
(ii) r1 (Pi′ ) = b and r2 (Pi′ ) = a.
Definition
A subset B ⊂ A and an alternative a ∈ A − B, are linked,
denoted a ∼ B, if there exist b, c ∈ B such that a ∼ b and a ∼ c.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Linked Domains (contd.)
Definition
A domain D is linked, if there exists an one to one function
σ : {1, . . . , m} → {1, . . . , m} such that
(i) aσ(1) ∼ aσ(2) ,
(ii) aσ(j) ∼ {aσ(1) , aσ(2) , . . . , aσ(j−1) }, j = 3, . . . , m.
Theorem
(ACS, 2003) A linked domain is a dictatorial domain.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Example
Let A = {a1 , a2 , a3 , a4 , a5 , a6 , a7 }. Domain DL :
P1
a1
a3
·
..
.
P2
a1
a4
a2
..
.
P3
a1
a5
a2
..
.
P4
a2
a6
·
..
.
P5
a2
a7
·
..
.
P6
a3
a1
·
..
.
P7
a3
a4
a2
..
.
P8
a4
a1
a2
..
.
P9
a4
a3
a2
..
.
P10
a4
a5
a2
..
.
P11
a4
a7
a2
..
.
P12
a5
a1
a2
..
.
P13
a5
a4
a2
..
.
a2
·
·
·
·
a2
·
·
·
·
·
·
·
P20
a7
a4
a2
..
.
P21
a7
a5
a2
..
.
P22
a7
a6
a2
..
.
P14
a5
a6
a2
..
.
P15
a5
a7
a2
..
.
P16
a6
a2
·
..
.
P17
a6
a5
a2
..
.
P18
a6
a7
a2
..
.
P19
a7
a2
·
..
.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Example contd.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Example contd.
The connectivity graph of DL is shown below:
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Example contd.
The connectivity graph of DL is shown below:
a_3
a_4
a_1
a_2
a_7
a_5
a_6
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Example contd.
The connectivity graph of DL is shown below:
a_3
a_4
a_1
a_2
a_7
a_5
a_6
Observe that DL is linked. It follows from ACS that it is
dictatorial.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Example contd.
The connectivity graph of DL is shown below:
a_3
a_4
a_1
a_2
a_7
a_5
a_6
Observe that DL is linked. It follows from ACS that it is
dictatorial.
However, it is not a random dictatorship domain. In fact,
“well-behaved” RSCFs (anonymous) SCFs can be
constructed on this domain.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Sufficient Conditions for Random Dictatorship
Domains
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Sufficient Conditions for Random Dictatorship
Domains
CSZ provide two sufficient conditions for a domain to be a
random dictatorship domain.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Sufficient Conditions for Random Dictatorship
Domains
CSZ provide two sufficient conditions for a domain to be a
random dictatorship domain.
The first strengthens the connectivity requirement.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Sufficient Conditions for Random Dictatorship
Domains
CSZ provide two sufficient conditions for a domain to be a
random dictatorship domain.
The first strengthens the connectivity requirement.
The second strengthens the requirement for linked-ness
and a relatively weaker strengthening of connectivity.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Sufficient Conditions for Random Dictatorship
Domains
CSZ provide two sufficient conditions for a domain to be a
random dictatorship domain.
The first strengthens the connectivity requirement.
The second strengthens the requirement for linked-ness
and a relatively weaker strengthening of connectivity.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Hub Condition: Condition H
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Hub Condition: Condition H
Definition
A Domain satisfies Condition H, if there exists a ∈ A such that
for all b ∈ A − {a}, b ∼ a. (Alternative a is referred to as a hub.)
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Hub Condition: Condition H
Definition
A Domain satisfies Condition H, if there exists a ∈ A such that
for all b ∈ A − {a}, b ∼ a. (Alternative a is referred to as a hub.)
a_1
a_3
a_3
a_2
a_3
a_5
a_1
a_1
a_5
a_3
a_2
a_4
a_1
a_2
a_4
a_2
a_4
Randomization in Mechanism Design for Voting Environments: Some Recent Results
The Hub Condition: Condition H
Definition
A Domain satisfies Condition H, if there exists a ∈ A such that
for all b ∈ A − {a}, b ∼ a. (Alternative a is referred to as a hub.)
a_1
a_3
a_3
a_2
a_3
a_5
a_1
a_1
a_5
a_3
a_2
a_4
a_1
a_2
a_4
a_2
a_4
Theorem
A linked domain satisfying Condition H is a random dictatorship
domain.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Another Sufficient Condition: Strong Connectedness
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Another Sufficient Condition: Strong Connectedness
Definition (Chatterji, Sanver and Sen (2010))
A pair of alternatives a, b ∈ A, is strongly connected, denoted
by a ≈ b, if there exist Pi , Pi′ ∈ D such that
(i) r1 (Pi ) = a and r2 (Pi ) = b,
(ii) r1 (Pi′ ) = b and r2 (Pi′ ) = a,
(iii) rk (Pi ) = rk (Pi′ ), k = 3, . . . , m.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Another Sufficient Condition: Strong Connectedness
Definition (Chatterji, Sanver and Sen (2010))
A pair of alternatives a, b ∈ A, is strongly connected, denoted
by a ≈ b, if there exist Pi , Pi′ ∈ D such that
(i) r1 (Pi ) = a and r2 (Pi ) = b,
(ii) r1 (Pi′ ) = b and r2 (Pi′ ) = a,
(iii) rk (Pi ) = rk (Pi′ ), k = 3, . . . , m.
Two alternatives are strongly connected if they can be ranked
first and second in an admissible ordering. Moreover their
positions can be switched without changing the positions of
other alternatives.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Another Sufficient Condition: Strong Connectedness
Definition (Chatterji, Sanver and Sen (2010))
A pair of alternatives a, b ∈ A, is strongly connected, denoted
by a ≈ b, if there exist Pi , Pi′ ∈ D such that
(i) r1 (Pi ) = a and r2 (Pi ) = b,
(ii) r1 (Pi′ ) = b and r2 (Pi′ ) = a,
(iii) rk (Pi ) = rk (Pi′ ), k = 3, . . . , m.
Two alternatives are strongly connected if they can be ranked
first and second in an admissible ordering. Moreover their
positions can be switched without changing the positions of
other alternatives.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Strong Connectedness contd.
Definition
A strongly linked domain is defined in exactly the same way as
a linked domain except that the notion of connectedness is
replaced by strong connectedness.
There exist strongly linked domains which are not random
dictatorship domains. i.e., Domain DL .
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Condition TS
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Condition TS
Definition
A domain satisfies Condition TS if for all a, b ∈ A, either a ≈ b,
or there exists c ∈ A such that a ≈ c and b ≈ c.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Condition TS
Definition
A domain satisfies Condition TS if for all a, b ∈ A, either a ≈ b,
or there exists c ∈ A such that a ≈ c and b ≈ c.
a_7
a_3
a_3
a_1
a_1
a_2
a_5
a_2
a_4
a_5
a_4
a_6
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Condition TS
Definition
A domain satisfies Condition TS if for all a, b ∈ A, either a ≈ b,
or there exists c ∈ A such that a ≈ c and b ≈ c.
a_7
a_3
a_3
a_1
a_1
a_2
a_5
a_5
a_2
a_4
a_6
a_4
Theorem
A strongly linked domain satisfying Condition TS is a random
dictatorship domain.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Additional Results
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Additional Results
The Circular Domain is a random dictatorship domain. It
has been shown to be a minimal dictatorship domain (Sato
(2010), Chatterji-Sen (2011)). It is therefore also a minimal
random dictatorship domain (size 2m).
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Additional Results
The Circular Domain is a random dictatorship domain. It
has been shown to be a minimal dictatorship domain (Sato
(2010), Chatterji-Sen (2011)). It is therefore also a minimal
random dictatorship domain (size 2m).
A critical tool for these results is a ramification theorem that
says that a random dictatorship domain for 2 voters is a
random dictatorship domain for an arbitrary number of
voters. Ramification results for dictatorial domains proved
by Kalai-Muller (1977) and Kim-Roush (1980).
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Additional Results
The Circular Domain is a random dictatorship domain. It
has been shown to be a minimal dictatorship domain (Sato
(2010), Chatterji-Sen (2011)). It is therefore also a minimal
random dictatorship domain (size 2m).
A critical tool for these results is a ramification theorem that
says that a random dictatorship domain for 2 voters is a
random dictatorship domain for an arbitrary number of
voters. Ramification results for dictatorial domains proved
by Kalai-Muller (1977) and Kim-Roush (1980).
CSZ prove a ramification result that requires an additional
restriction on domains.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Conclusion
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Conclusion
The DEP Property is important in understanding the role of
randomization on incentive-compatibility.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Conclusion
The DEP Property is important in understanding the role of
randomization on incentive-compatibility.
Whether or not a domain exhibits the DEP property is a
subtle matter. Some sort of richness is clearly important
but not fully understood.
Randomization in Mechanism Design for Voting Environments: Some Recent Results
Conclusion
The DEP Property is important in understanding the role of
randomization on incentive-compatibility.
Whether or not a domain exhibits the DEP property is a
subtle matter. Some sort of richness is clearly important
but not fully understood.
Next Question: if a domain does not exhibit the DEP
Property what are the extreme-point (some of these will
involve randomization).