On the implementability of the Lindahl correspondence by means of an anonymous mechanism Sébastien Rouillon GREThA, Bordeaux 4 Journées 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 X i, an initial endowment w i (of the private good) and a preference R i, defined over X i ; The set of admissible economies is denoted E, with generic elements e = (R i, w i ) 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,  i x i + y =  i w i.

Lindahl Allocations A Lindahl equilibrium is a vector of personal prices (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, y), for all (x i, y)  X i such that x i + p i * y ≤ w i ;  i x i * + y* = w i. The allocation ((x i *) 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 = X i M i is called the message space, h is called the outcome function and associates messages m  M to allocations h(m)  IR n+1. More precisely, we will use the notations: h(m) = ((w i – T i (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 M i ; player i’s preference R i * over M follows from his endowment w i, from his preference R i over X i, and from the outcome function h: m R i * m’  (w i – T i (m), Y(m)) R i (w i – T i (m’), Y(m’)).

… Nash Equilibrium A Nash equilibrium of this game is a strategy profile m* such that: For all i and all m i  M i, m* R i * (m*/m i ), where (m*/m i ) = (m 1 *, …, m i, …, m n *). 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) = ((T i (m)) i, Y(m)), for all m  M. Definition 1. An economic mechanism (M, h) is said to be anonymous if: (i) M 1 = … = M n, (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 T i (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

Existence of a continuous and weakly balanced mechanism? D C S UWB D = Discontinuous C = Continuous S = Smooth U = unbalanced W = 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 M i = IR 2, for all i, with generic elements denoted m i = (p i, y i ); The outcome function is defined by (with  > 0): T i (m) = (1 –  j  i p j )  i y i + (1 – 1/n) |  i p i – 1| (|  i y i | +  ), for all i, Y(m) =  i y i.

… Continuous implementation Proposition 1. Let E* be the set of economies such that: for all i, R i 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? T i (m) = (1 –  j  i p j )  i y i + (1 – 1/n) |  i p i – 1| (|  i y i | +  ),  i, Y(m) =  i y i. T i (m) = (1 –  j  i p j )  i y i,  i, Y(m) =  i y i.

Existence of a Smooth and Weakly Balanced Mechanism? D C S UWB Set of Anonymous Mechanisms to Nash- implement the Lindahl Correspondence ? Proposition 1 yes ?

Impossibility of a smooth implementation … Property 1’. The mechanism in Definition 4 satisfies: M 1 = … = M n = IR 2, and, for all m, T i (m) = T(m i,  j  i m j ), for all i, Y(m) = G(m i +  j  i m j ), where T and G are functions from M to IR.

… Impossibility of a smooth implementation Proposition 2. Let E C be the set of economies such that, for all a = (a i ) i  (0,1) n, R i can be represented by a Cobb-Douglas utility function: U i (x i, y) = x i a i y 1–a i, for all i, and the endowments are fixed at a given (w i ) i. Assume that E C  E. Then, there exists no smooth mechanism (M, h), which satisfies the definitions 1’, 2 and 3.

Conclusion D C S UWB Set of Anonymous Mechanisms to Nash- implement the Lindahl Correspondence yes ?  no Proposition 2   no ?

Thank you for your attention

How does it work? T i (m) = (1 –  j  i p j )  i y i + (1 – 1/n) |  i p i – 1| (|  i y i | +  ),  i, Y(m) =  i y i. T i (m) = (1 –  j  i p j )  i y i,  i, Y(m) =  i y i.