On the implementability of the Lindahl correspondence by means of an anonymous mechanism

Sébastien RouillonGREThA, Bordeaux 4Journées LAGV, 2008

Topic and results

This paper deals with the Nash implementation of the Lindahl allocations (assuming perfect information and no auctioneer).

Precisely, we search for anonymous mechanisms.We derive two results. We prove that: Such mechanisms exist which are also continuous

and weakly balanced. No such mechanism exists which are also smooth and

weakly balanced.

Economic Model

Let e be an economy with: one private good x and one public good y, which is

produced from the former under constant return to scales (one unit of y costs one unit of x);

n consumers, indexed i, each characterized by a consumption set Xi, an initial endowment wi (of the private good) and a preference Ri, defined over Xi;

The set of admissible economies is denoted E, with generic elements e = (Ri, wi)i.

Economic Allocations

An allocation is a vector ((xi)i, y) in IRn+1, where:

xi = i’s private consumption,

y = public consumption.

It is possible if:

For all i, ((xi)i, y) Xi,

i xi + y = i wi.

Lindahl Allocations

A Lindahl equilibrium is a vector of personal prices (pi*)i, with i pi* = 1, and an allocation ((xi*)i, y*), such that:

for all i, (xi*, y*) Ri (xi, y), for all (xi, y) Xi such that xi + pi* y ≤ wi;

i xi * + y* = wi.The allocation ((xi*)i, y*) is then called a Lindahl

allocation. For all e E, the set of Lindahl allocations is denoted

L(e). L is said to be the Lindahl correspondence.

Economic mechanism

An economic mechanism is a pair (M, h), where: M = Xi Mi is called the message space, h is called the outcome function and associates

messages m M to allocations h(m) IRn+1 .

More precisely, we will use the notations:

h(m) = ((wi – Ti(m))i, Y(m)).

Nash Equilibrium…

The pair (M, h) defines a game form, where: the set of players is {1, …, n}; player i’s strategic space is Mi; player i’s preference Ri* over M follows from his

endowment wi, from his preference Ri over Xi, and from the outcome function h:

m Ri* m’ (wi – Ti(m), Y(m)) Ri (wi – Ti(m’), Y(m’)).

…Nash Equilibrium

A Nash equilibrium of this game is a strategy profile m* such that:

For all i and all mi Mi, m* Ri* (m*/mi),

where (m*/mi) = (m1*, …, mi, …, mn*).

For all e E, the set of Nash equilibriums is denoted v(e). The set of the corresponding allocations h(v(e)) is denoted N(e).

N is called the Nash correspondence.

Anonymous mechanisms

Let g be the net trade function associated to h, defined by: g(m) = ((Ti(m))i, Y(m)), for all m M.

Definition 1. An economic mechanism (M, h) is said to be anonymous if:

(i) M1 = … = Mn,

(ii) g(m(1), …, m(n)) = ((T(i)(m))i, Y(m)), m M,

where denotes any permutation of the set {1, …, n}.

Weakly balanced mechanisms

Definition 2. An economic mechanism (M, h) is said to be weakly balanced if:

i Ti(m) ≥ Y(m), for all m M.

(It is said balanced if the inequality ≥ is replaced by the equality =.)

Lindahl implementation

Definition 3. An economic mechanism (M, h) is said to (fully) implement the Lindahl correspondence if:

L(e) = N(e), for all e E.

Existence of anonymous mechanisms?

For large economies (n > 2), Hurwicz (1979) and Walker (1981) constructed balanced and smooth mechanisms to implement the Lindahl correspondence.

However, they use cycles…

and thus are not anonymous.

1

23

Existence of a continuous and weakly balanced mechanism?

D

C

S

U W B

D = DiscontinuousC = ContinuousS = Smooth

U = unbalancedW = Weakly Bal.B = Balanced

Legend

Kim (1993)

Maskin (1999)

Set of Anonymous Mechanisms to Nash-implement the Lindahl

Correspondence

?

Continuous implementation…

Definition 4. For n ≥ 2, let (M, h) be such that: The player’s strategic spaces are Mi = IR2, for all i,

with generic elements denoted mi = (pi, yi); The outcome function is defined by (with > 0):

Ti(m) = (1 – ji pj) i yi

+ (1 – 1/n) |i pi – 1| (|i yi| + ), for all i,

Y(m) = i yi.

…Continuous implementation

Proposition 1. Let E* be the set of economies such that: for all i, Ri is complete, transitive and (strictly) increasing in the private good. Assume that E E*.The mechanism (M, h) in definition 4 is anonymous, continuous (not smooth), weakly balanced and (fully) implements the Lindahl correspondance.

How does it work?

Ti(m) = (1 – ji pj) i yi

+ (1 – 1/n) |i pi – 1| (|i yi| + ), i,

Y(m) = i yi.

Ti(m) = (1 – ji pj) i yi, i,

Y(m) = i yi.

Existence of a Smooth and Weakly Balanced Mechanism?

D

C

S

U W B

Set of Anonymous Mechanisms to Nash-implement the Lindahl

Correspondence

? Proposition 1yes

?

Impossibility of a smooth implementation…

Property 1’. The mechanism in Definition 4 satisfies:

M1 = … = Mn = IR2,and, for all m,

Ti(m) = T(mi, ji mj), for all i,

Y(m) = G(mi + ji mj),where T and G are functions from M to IR.

…Impossibility of a smooth implementation

Proposition 2. Let EC be the set of economies such that, for all a = (ai)i (0,1)n, Ri can be represented by a Cobb-Douglas utility function: Ui(xi, y) = xi

ai y1–ai, for all i, and the endowments are fixed at a given (wi)i.

Assume that EC E. Then, there exists no smooth mechanism (M, h), which satisfies the definitions 1’, 2 and 3.

Conclusion

D

C

S

U W B

Set of Anonymous Mechanisms to Nash-implement the Lindahl

Correspondence

yes

?no Proposition 2

no

?

Thank you for your attention

How does it work?

Ti(m) = (1 – ji pj) i yi

+ (1 – 1/n) |i pi – 1| (|i yi| + ), i,

Y(m) = i yi.

Ti(m) = (1 – ji pj) i yi, i,

Y(m) = i yi.