Local Incentive Compatibility when the Set of
Alternatives is Infinite
Anushree Saha∗
Indian Statistical Institute, Delhi
February 15, 2016
Abstract
We consider local incentive compatibility in deterministic voting
models on a utility domain that consists of a subset of all real valued
continuous preferences over an infinite set of alternatives. We propose that local incentive compatibility is sufficient to imply incentive
compatibility if the domain is convex, and the set of alternatives is
compact. We also prove the sufficiency result for deterministic allocation rules in the with transfers setting, where utility is quasilinear in
payments.
Key words: Local incentive compatibility, infinite set of alternatives,
sufficiency, with transfers, without transfers.
JEL Classification Number: D71, D82.
†
I am indebted to my thesis advisor, Debasis Mishra, for his invaluable
comments and guidance.
∗
Email: [email protected]
1
Introduction
In the theory of social choice, incentive compatibility has been a very important topic since Gibbard-Satterthwaite theorem (Gibbard 1973, Satterthwaite 1975) established that voting schemes are manipulable, i.e., it might
be beneficial for some individuals to not reveal his true preferences. But in
several voting models, it becomes tedious to verify all the possible incentive constraints at once. Hence, coming up with weaker notions of incentive
compatibility is an interesting problem.
Carroll (2012) made a major contribution in this area by focusing on
local incentive constraints, i.e., ensuring that agents do not manipulate in
the neighbourhood of their types and then checking if local incentive compatibility is sufficient for global incentive compatibility. Carroll shows the
sufficiency of local incentive constraints for a finite domain of preferences,
both ordinal and cardinal. There are more works mentioned in the next
section, which study a similar problem but all the existing results are for
finite outcome spaces.
We extend this result to a deterministic voting model with an infinite
set of alternatives, given by a compact metric space and the type space of
an agent given by her utility profile which is a subset of all real valued continuous functions on the set of alternatives. The model that we consider
resembles that of Barbera and Peleg (1990) where they prove that all nondictatorial voting schemes for this domain with two or more alternatives are
manipulable. In this article we prove that if the utility domain is convex
then local incentive compatibility implies incentive compatibility.
Our result is relevant because in case of an infinite set of alternatives,
verifying all possible incentive constraints of a mechanism actually becomes
too demanding. If local incentive compatibility implies incentive compatibility then looking for local incentive constraints helps in simplifying the
problem. For example, suppose the coordinates of a point in the outcome
space are taken to measure the position of a candidate on different issues,
each point may be interpreted as a possible political agenda. And the agent’s
utility is given by some function which measures how much the agent sympathises with the chosen candidate’s political agenda. Here it is natural to
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assume that an agent with a socialist bent of mind will never deviate too far
to a capitalist agenda and hence it is fine to concentrate on local incentive
compatibility.
We also show that local incentive compatibility implies global incentive
compatibility in the with transfers setting, where, we have a deterministic
allocation rule with utility quasilinear in payments. For example, let the
outcome space or the set of alternatives represent different positions within
a certain geographical area. Suppose the government (planner) intends to
find a suitable position for opening a public facility in exchange for some
payment from the agent. And the agent’s utility is given by the distance
from his position to the public facility’s location. Here it is natural to
assume that the utility function will be continuous. If our result is applied
here, instead of verifying all the incentive constraints, it will be enough to
concentrate in the neighbourhood of the agent’s type.
Although we have proved the results for a single agent, it is easy to show
it for multiple agents too.
The next section provides a brief review of the literature, while Section
3 describes the model, before going on to analyse the deterministic voting
model. Section 3.3 introduces the with transfers framework. Finally, Section
4 concludes with a discussion of how our model is useful.
2
Related Literature
While incentive compatibility has been a long studied problem, there has
been a trend to find relaxed conditions that imply incentive compatibility.
Green and Laffont (1986) and Sher and Vohra (2010) are some papers that
dealt with this. Local Incentive Compatibility results have been well studied by Sato (2010) and Carroll (2012). While Sato studies deterministic
allocation rules with ordinal preferences, Caroll covers a wide range of preferences like cardinal, ordinal, single-peaked ordinal preference and successive
single-crossing ordinal preference. A recent work by Cho (2015) also studies
the conditions when local incentive compatibility, mistake monotonicity and
incentive compatibility are equivalent.
In the with transfers setting, Saks and Yu (2005), Archer and Kleinberg
2
(2008) and Carroll (2012) have made effective contributions.
3
Framework and Model
Let there be a single agent and the set of alternatives, A, a compact metric
space. The metric on A denoted by d. U is the set of all continuous real
valued functions on A. The type space of the agent is D ⊆ U, essentially
the utility profile. An agent’s type is given by her utility function u, u ∈ D.
A deterministic voting scheme is a function f : D → A.
3.1
Definitions
Definition 1 An voting scheme is manipulable if there exists u, u0 ∈ D such
that,
u f (u0 ) > u f (u)
Definition 2 A voting scheme is incentive compatible if it is not manipulable for any u ∈ D. In other words, f is incentive compatible if for all
(u, u0 ) ∈ D × D if
u f (u) ≥ u f (u0 )
∀u0 ∈ D
Definition 3 A set S of incentive constraints satisfy local incentive compatibility if every u ∈ D has an open neighbourhood Nu ∈ D such that
(u, u0 ) ∈ S and (u0 , u) ∈ S
3.2
∀u0 ∈ Nu
Result
The following proposition gives sufficient conditions for local incentive compatibility to imply incentive compatibility in the above stated model.
Proposition 1 Suppose the set of alternatives, A, is a compact metric
space and the agent’s type is given by her utility u ∈ D, where D ⊆ U : set
of real valued continuous functions on A. If the utility domain D is convex
and local incentive compatibility holds, then the voting scheme f is incentive
compatible, i.e., for all (u, u0 ) ∈ D × D,
u f (u) ≥ u f (u0 )
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∀u0 ∈ D
Proof. Let S be the set of LIC’s and f be the mechanism satisfying S.
For types u and v let u ↔ v if (u, v) and (v, u) are both in S. Assume that
D is convex. Consider any 2 arbitrary functions, u, v ∈ U . By convexity We
have,
For λ ∈ [0, 1], if uλ = (1 − λ)u + λv, then uλ ∈ D.
Now consider two elements, x, y ∈ A such that u(x) ∈ R and v(y) ∈ R.
We know that u, v being continuous real valued functions defined on
a compact metric space attain their bound (by Extreme Value Theorem).
Hence we can write,
argmax|u(x) − v(y)|
x,y∈A
This gives us 2 points, say x̄, ȳ ∈ A, such that the functions u and v are
farthest at these points. So, we can restrict attention to ū and v̄, i.e., if
we can use local incentive constraints to prove that ū and v̄ are incentive
compatible then we can do so for any arbitrary points on u and v.
Let ū satisfy local incentive constraints in the neighbourhood Nū ().
Define uλ = (1 − λ)ū + λv̄
, λ ∈ [0, 1]
We need to show that there exists a suitable λ for which uλ ∈ Nū (), i.e.,
||uλ − ū|| < ||uλ − ū|| =||(1 − λ)ū + λv̄ − ū||∞
=|λ| ||v̄ − ū||∞
≤|λ| ||v̄||∞ + ||ū||∞
Choosing λ =
(||v̄||∞ +||ū||∞ ) ,
we get uλ ∈ Nū (). The denominator exists
because u and v are bounded functions.
Now define a set,
L = {λ|∃
0 = λ0 < λ1 < ... < λr ≤ 1 with uλ0 ↔ uλ1 ↔ ... ↔ uλr }
If λ ∈ L, λ < λ0 ≤ 1 and uλ ↔ uλ0 then λ0 ∈ L. Let λ̄ = sup L. λ̄ 6= 0
because our choice of λ above gives atleast one λ > 0. Hence, λ̄ > 0. Now
for λ sufficiently close to λ̄, we have uλ ↔ uλ̄ since we can choose λ ∈ L
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arbitrarily close we again get λ̄ ∈ L. Now if λ̄ < 1, then uλ̄ ↔ uλ for λ
slightly larger than λ̄ ⇒ λ ∈ L contradicting λ̄ = sup L.
Hence, λ̄ = 1 and 1 ∈ L.
From the local incentive constraints we have,
uλk f (uλk ) ≥ uλk f (uλk+1 )
uλk+1 f (uλk+1 ) ≥ uλk+1 f (uλk )
Writing these in terms of ū and v̄. Let f (uλk ) = p and f (uλk+1 ) = q.
(1 − λk )ū(p) + λk v̄(p) ≥ (1 − λk )ū(q) + λk v̄(q)
(1 − λk+1 )ū(q) + λk+1 v̄(q) ≥ (1 − λk+1 )ū(p) + λk+1 v̄(p)
Multiplying by λk+1 and λk respectively and adding gives,
(λk+1 − λk )ū(x) ≥ (λk+1 − λk )ū(y)
Since (λk+1 − λk ) ≥ 0, we get
ū f (uλk ) − ū f (uλk+1 ) ≥ 0
Adding for k = 0, 1, ....r, we are left with,
ū f (ū) ≥ ū f (v̄)
This proves that incentive constraint for any arbitrary (u, v) ∈ D × D holds
and hence f is incentive compatible.
In what follows, we will first study the case of deterministic allocation
rules with transfers.
4
Allocation Rules with Transfers
We consider the setting in which the agent has quasilinear utility in payment.
There is a single agent and the set of alternatives, A, a compact metric
space. The metric on A is denoted by d. U is the set of continuous real
valued functions on A. An agent’s type is given by her utility function
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u, u ∈ D, D ⊆ U .
A deterministic allocation rule is a function f : D → A. Let the payment
be p : D → R. Hence the realized utility is given by,
u f (u) + p(u)
4.1
Definitions
Definition 4 (f, p) is said to be incentive compatible if for all u ∈ D,
u f (u) + p(u) ≥ u f (u0 ) + p(u0 )
∀u0 ∈ D
Definition 5 A set S of incentive constraints satisfy local incentive compatibility if every u ∈ D has an open neighbourhood Nu ∈ D such that
(u, u0 ) ∈ S and (u0 , u), ∈ S
4.2
∀u0 ∈ Nu .
Result
Here we show the sufficient conditions for local incentive compatibility to
imply incentive compatibility in the with transfers case.
Proposition 2 Suppose the set of alternatives, A, is a compact metric
space and the agent’s type is given by her utility u ∈ D, where D ⊆ U : set
of real valued continuous functions on A. Let f : D → A be a deterministic allocation rule and p : D → R is the payment. Assume agents have
quasilinear utility in payment. If D is convex and local incentive compatibility holds, then the mechanism (f, p) is incentive compatible, i.e., for all
(u, u0 ) ∈ D × D,
u f (u) + p(u) ≥ u f (u0 ) + p(u0 )
∀u0 ∈ D
Proof. Let S be the set of LIC’s and (f, p) be the mechanism satisfying
S. For types u and v let u ↔ v if (u, v) and (v, u) are both in S.
Consider any 2 arbitrary functions, u, v ∈ D. Consider their convex
combination,
For λ ∈ [0, 1], if uλ = (1 − λ)u + λv
Then uλ ∈ D (by convexity of D).
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Proceeding in the same way as we did in Proposition 1, we consider two
elements, x, y ∈ A such that u(x) ∈ R and v(y) ∈ R.
We know that u, v being continuous real valued functions defined on a compact metric space attain their bound (by Extreme Value Theorem). Hence
we can write,
argmax|u(x) − v(y)|
x,y∈A
This gives us 2 points, say x̄, ȳ ∈ A, such that the functions u and v are
farthest at these points. So, we can restrict attention to ū and v̄, i.e., if
we can use local incentive constraints to prove that ū and v̄ are incentive
compatible then we can do so for any arbitrary points on u and v.
Let ū satisfy local incentive constraints in the neighbourhood Nū ().
Define uλ = (1 − λ)ū + λv̄
, λ ∈ [0, 1]
We need to show that there exists a suitable λ for which uλ ∈ Nū (), i.e.,
||uλ − ū|| < ||uλ − ū|| =||(1 − λ)ū + λv̄ − ū||∞
=|λ| ||v̄ − ū||∞
≤|λ| ||v̄||∞ + ||ū||∞
Choosing λ =
(||v̄||∞ +||ū||∞ ) ,
we get uλ ∈ Nū (). The denominator exists
because u and v are bounded functions.
Now define a set,
L = {λ|∃
0 = λ0 < λ1 < ... < λr ≤ 1 with uλ0 ↔ uλ1 ↔ ... ↔ uλr }
If λ ∈ L, λ < λ0 ≤ 1 and uλ ↔ uλ0 then λ0 ∈ L. Let λ̄ = sup L. λ̄ 6= 0
because our choice of λ above gives atleast one λ > 0. Hence, λ̄ > 0. Now
for λ sufficiently close to λ̄, we have uλ ↔ uλ̄ since we can choose λ ∈ L
arbitrarily close we again get λ̄ ∈ L. Now if λ̄ < 1, then uλ̄ ↔ uλ for λ
slightly larger than λ̄ ⇒ λ ∈ L contradicting λ̄ = sup L.
Hence, λ̄ = 1 and 1 ∈ L.
From the local incentive constraints Ihave,
uλk f (uλk ) + p(uλk ) ≥ uλk f (uλk+1 ) + p(uλk+1 )
7
uλk+1 f (uλk+1 ) + p(uλk+1 ) ≥ uλk+1 f (uλk ) + p(uλk )
Writing these in terms of ū and v̄. Let f (uλk ) = a and f (uλk+1 ) = b.
(1 − λk )ū(a) + λk v̄(a) + p(uλk ) ≥ (1 − λk )ū(b) + λk v̄(b) + p(uλk+1 )
(1 − λk+1 )ū(b) + λk+1 v̄(b) + p(uλk+1 ) ≥ (1 − λk+1 )ū(a) + λk+1 v̄(a) + p(uλk )
Multiplying by λk+1 and λk respectively and adding gives,
(λk+1 − λk ) ū(a) + p(uλk ) ≥ (λk+1 − λk ) ū(b) + p(uλk+1 )
Since (λk+1 − λk ) ≥ 0, Iget
ū f (uλk ) + p(uλk ) − ū f (uλk+1 ) − p(uλk+1 ) ≥ 0
Adding for k = 0, 1, ....r, Iare left with,
ū f (ū) + p(ū) ≥ ū f (v̄) + p(v̄)
This shows that incentive constraint for (u, v) ∈ D × D is satisfied and since
we chose (u, v) arbitrarily, wesay that (f, p) is incentive compatible.
5
Conclusion
We show that in the setting where the set of alternatives form a compact
metric space and the utilities are the convex set of real valued continuous
functions on that set, local incentive compatibility is sufficient to imply
incentive compatibility both in the with and without transfers case. Caroll
(2012) in his paper allows for randomisation, i.e., lotteries over the outcome
space but that is not equivalent to having an infinite outcome space.
Our result is applicable in the multiple agents model as well.
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6
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