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Nash Implementation
of Lindahl Equilibria
Sébastien Rouillon
Journées LAGV, 2007
Economic Model
Let e be an economy with:
 one private good x and one public good y;
 n consumers, indexed i, each characterized by
an endowment wi of private good, a
consumption set Xi and a preference Ri;
 a production set Y = {(x, y); x + y ≤ Si wi}.
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,
x + y = Si wi.
Economic Mechanisms
The institutions of the economy are specified as
an economic mechanism D = (M, r), which:
 endows each consumer with a message space
Mi, with generic elements mi;
 associates to any joint message m in M = Xi Mi,
an economic allocation r(m).
The function r(m) is called an outcome function.
Economic Equilibrium…
A game can be associated to D = (M, r), 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 preference Ri over Xi and the outcome
function r:
m Ri* m’  r(m) Ri r(m’).
…Economic Equilibrium
A Nash equilibrium of this game is a strategy
profile m* such that:
for all i and all mi, m* Ri* (m*/mi),
where (m*/mi) = (m1*, …, mi, …, mn*).
Lindahl Mechanism
Lindahl (1919) defines institutions, where the
consumers are:
 first given personal tax rates pi, chosen such
that Si pi = 1 to ensure budget balancing,
 and then are asked to tell which amount yi of
public goods they demand, knowing that they
will pay pi per unit.
Lindahl Equilibrium
Lindahl (1919) forecasts that, endowed with
such institutions, the behaviors of the
consumers will drive the economy to a Lindahl
equilibrium, defined as a list of personal tax
rates (pi*)i, with Si pi* = 1, and an allocation
((xi*)i, y*), such that, for all i:
(xi*, y*) Ri ((xi)i, y),
for all (xi, y)  Xi such that xi + pi* y ≤ wi.
The Free-Riding Problem
Samuelson (1966) rejects this forecast, by noting
that Lindahl’s definition relies on the belief
that each consumer truly thinks that he alone
determines the supply of the public good and,
thus, demands the amount that maximizes his
preference, subject to his budget constraint.
This is unrealistic, for rational consumers should
notice that they have an incentive to free-ride.
Mechanisms yielding Lindahl
Equilibria…
After Samuelson’s critic, Hurwicz (1979),
Walker (1981) and Kim (1993) found
economic mechanisms such that:
 The outcome functions r have the form:
r(m) = ((wi – pi(m) y(m) – ri(m))i, y(m));
 If m* is a Nash equilibrium of the associated
game, then the personal taxes (pi(m*))i and the
allocation r(m*) is a Lindahl equilibrium.
…Mechanisms yielding Lindahl Eq.
(H)urwicz (1979)
A player i’s message is an element mi = (pi, yi)
in IR2
 For n ≥ 3, r is defined by:
pi(m) = 1/n + pi+1 – pi+2,
y(m) = (1/n) Si yi,
ri(m) = pi+1 (yi – yi+1)2 – pi+2 (yi+1 – yi+2)2.

…Mechanisms yielding Lindahl Eq.
(W)alker (1981)
A player i’s message is an element mi in IR
 For n ≥ 3, r is defined by:
pi(m) = 1/n – mi+1 + mi+2,
y(m) = Si mi,
ri(m) = 0.

…Mechanisms yielding Lindahl Eq.
(K)im (1993)
A player i’s message is an element mi = (pi, yi)
in IR2
 For n ≥ 2, r is defined by:
pi(m) = 1/n – Sji yj + (1/n) Sji pj,
y(m) = (1/n) Si yi,
ri(m) = (1/2) (pi – Si yj )2.

The Interpretation Problem
These mechanisms share the following defect:
 Let m* be a Nash equilibrium, and
 Let (pi*)i and ((xi*)i, y*) be the associated
Lindahl equilibrium.
The equilibrium strategy m* cannot be expressed
from (pi*)i and ((xi*)i, y*) by means of familiar
economic notions.
Definition of a new mechanism
Our mechanism D = (M, r) is defined as follows:
 A player i’s message is an element mi = (pi, y)
in IR2
 For n ≥ 2, r is defined by:
r(m) = ((wi – pi(m) y(m) – fi(m)2/2)i, y(m)),
where: pi(m) = 1 – Sji pj,
y(m) = Si yi/n,
fi(m) = 1 – Si pi + Si yji/(n – 1) – yj.
Using our mechanism…
Avoiding the fines
The term (1/2) fi(m)2 plays the role of a fine.
A player i’s goal is to escape it.
Therefore, at a Nash equilibrium m*:
fi(m*) = 0, for all i.
This implies that, at a Nash equilibrium m*:
pi* = pi(m*) and yi* = y(m*), for all i,
Si pi* = 1.
Using our mechanism…
Choosing the level of public good
Whatever m–i, a player i can find mi such that:
 he always pays pi(m) = 1 – Sji pj per unit;
 he freely sets the supply y(m) = (1/n) Si yi;
 he is not fined, i.e. (1/2) fi(m)2 = 0.
Therefore, a Nash equilibrium m* will be such
that, for all i, y(m*) maximizes Ri, given
pi(m*), and fi(m*) = 0.
Main results
Theorem 1. For an economy e, if the joint
strategy m* = (pi*, yi*)i is a Nash equilibrium
(of the game associated to D), then:
 pi* = pi(m*) and yi* = y(m*), for all i;
 the individualized prices (pi(m*))i and the
allocation r(m*) form a Lindahl equilibrium.
Main results
Theorem 2. For an economy e, if the
individualized prices (pi*)i and the allocation
((xi*)i, y*) form a Lindahl equilibrium, then:
 The joint strategy m* = (pi*, y*)i is a Nash
equilibrium (of the game associated to D);
 The mechanism D implements the Lindahl
equilibrium, i.e. r(m*) = ((xi*)i, y*).
Critic of the Nash equilibrium
The use of the Nash equilibrium concept to solve
the game relies on the assumption that the
players know both the rules of the game
derived from D and the economy e.
This is most of the time unrealistic. In this case,
Hurwicz (1972) argues that a Nash equilibrium
could nevertheless result as an equilibrium
point of a tâtonnement process.
Gradient process…
Let e be economy such that, for all i, the
preferences Ri can be represented as a
differentiable utility function Ui(xi, y).
If our mechanism D is used, for all m, the utility
of i will be equal to:
ui(m) = Ui(wi – pi(m) y(m) – fi(m)2/2, y(m)).
…Gradient process
A gradient process describes the behavior of
players who, in continuous time, adjust their
strategy mi = (pi, yi), in the direction that
maximizes the instantaneous increase of their
utility ui(m), taking others’ strategies as given.
It is formalized as a dynamic system:
dmi/dt = dui(m)/dmi, mi(0) = mi0, for all i. (S)
Main results
Theorem 3. Let e be such that, for all i:
Ui(xi, y) = xi – vi(y), vi’(y) > 0 > vi’’(y).
If one exists, a Lindahl equilibrium of e, defined
by (pi*)i and ((xi*)i, y*), is unique.
Then, m* = (pi*, y*)i is the unique stationary
point of (S) and is globally stable.