# Nash Implementation of Lindahl Equilibria Sébastien Rouillon Journées LAGV, 2007.

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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 w i of private good, a consumption set X i and a preference R i ; a production set Y = {(x, y); x + y ≤  i w i }.

Economic Allocations An allocation is a vector ((x i ) i, y) in IR n+1, where: x i = i’s private consumption, y = public consumption. It is possible if: For all i, ((x i ) i, y)  X i, x + y =  i w i.

Economic Mechanisms The institutions of the economy are specified as an economic mechanism D = (M,  ), which: endows each consumer with a message space M i, with generic elements m i ; associates to any joint message m in M = X i M i, an economic allocation  (m). The function  (m) is called an outcome function.

Economic Equilibrium … A game can be associated to D = (M,  ), where: the set of players is {1, …, n}; player i’s strategic space is M i ; player i’s preference R i * over M follows from his preference R i over X i and the outcome function  : m R i * m’   (m) R i  (m’).

… Economic Equilibrium A Nash equilibrium of this game is a strategy profile m* such that: for all i and all m i, m* R i * (m*/m i ), where (m*/m i ) = (m 1 *, …, m i, …, m n *).

Lindahl Mechanism Lindahl (1919) defines institutions, where the consumers are: first given personal tax rates p i, chosen such that  i p i = 1 to ensure budget balancing, and then are asked to tell which amount y i of public goods they demand, knowing that they will pay p i 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 (p i *) i, with  i p i * = 1, and an allocation ((x i *) i, y*), such that, for all i: (x i *, y*) R i ((x i ) i, y), for all (x i, y)  X i such that x i + p i * y ≤ w i.

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  have the form:  (m) = ((w i – p i (m) y(m) – r i (m)) i, y(m)); If m* is a Nash equilibrium of the associated game, then the personal taxes (p i (m*)) i and the allocation  (m*) is a Lindahl equilibrium.

… Mechanisms yielding Lindahl Eq. (H)urwicz (1979) A player i’s message is an element m i = (p i, y i ) in IR 2 For n ≥ 3,  is defined by: p i (m) = 1/n + p i+1 – p i+2, y(m) = (1/n)  i y i, r i (m) = p i+1 (y i – y i+1 ) 2 – p i+2 (y i+1 – y i+2 ) 2.

… Mechanisms yielding Lindahl Eq. (W)alker (1981) A player i’s message is an element m i in IR For n ≥ 3,  is defined by: p i (m) = 1/n – m i+1 + m i+2, y(m) =  i m i, r i (m) = 0.

… Mechanisms yielding Lindahl Eq. (K)im (1993) A player i’s message is an element m i = (p i, y i ) in IR 2 For n ≥ 2,  is defined by: p i (m) = 1/n –  j  i y j + (1/n)  j  i p j, y(m) = (1/n)  i y i, r i (m) = (1/2) (p i –  i y j ) 2.

The Interpretation Problem These mechanisms share the following defect: Let m* be a Nash equilibrium, and  Let (p i *) i and ((x i *) i, y*) be the associated Lindahl equilibrium. The equilibrium strategy m* cannot be expressed from (p i *) i and ((x i *) i, y*) by means of familiar economic notions.

Definition of a new mechanism Our mechanism D = (M,  ) is defined as follows: A player i’s message is an element m i = (p i, y) in IR 2 For n ≥ 2,  is defined by:  (m) = ((w i – p i (m) y(m) – f i (m) 2 /2) i, y(m)), where:p i (m) = 1 –  j  i p j,y(m) =  i y i /n, f i (m) = 1 –  i p i +  i y j  i /(n – 1) – y j.

Using our mechanism … Avoiding the fines The term (1/2) f i (m) 2 plays the role of a fine. A player i’s goal is to escape it. Therefore, at a Nash equilibrium m*: f i (m*) = 0, for all i. This implies that, at a Nash equilibrium m*: p i * = p i (m*) and y i * = y(m*), for all i,  i p i * = 1.

Using our mechanism … Choosing the level of public good Whatever m –i, a player i can find m i such that: he always pays p i (m) = 1 –  j  i p j per unit; he freely sets the supply y(m) = (1/n)  i y i ; he is not fined, i.e. (1/2) f i (m) 2 = 0. Therefore, a Nash equilibrium m* will be such that, for all i, y(m*) maximizes R i, given p i (m*), and f i (m*) = 0.

Main results Theorem 1. For an economy e, if the joint strategy m* = (p i *, y i *) i is a Nash equilibrium (of the game associated to D), then: p i * = p i (m*) and y i * = y(m*), for all i; the individualized prices (p i (m*)) i and the allocation  (m*) form a Lindahl equilibrium.

Main results Theorem 2. For an economy e, if the individualized prices (p i *) i and the allocation ((x i *) i, y*) form a Lindahl equilibrium, then: The joint strategy m* = (p i *, y*) i is a Nash equilibrium (of the game associated to D); The mechanism D implements the Lindahl equilibrium, i.e.  (m*) = ((x i *) 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 R i can be represented as a differentiable utility function U i (x i, y). If our mechanism D is used, for all m, the utility of i will be equal to: u i (m) = U i (w i – p i (m) y(m) – f i (m) 2 /2, y(m)).

… Gradient process A gradient process describes the behavior of players who, in continuous time, adjust their strategy m i = (p i, y i ), in the direction that maximizes the instantaneous increase of their utility u i (m), taking others’ strategies as given. It is formalized as a dynamic system: dm i /dt = du i (m)/dm i, m i (0) = m i0, for all i.(S)

Main results Theorem 3. Let e be such that, for all i: U i (x i, y) = x i – v i (y), v i ’(y) > 0 > v i ’’(y). If one exists, a Lindahl equilibrium of e, defined by (p i *) i and ((x i *) i, y*), is unique. Then, m* = (p i *, y*) i is the unique stationary point of (S) and is globally stable.

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