By Luis C. Corchón (auth.)
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Extra info for The Theory of Implementation of Socially Optimal Decisions in Economics
Economies with Public Goods 15 W; E the relative interior of X;, 0 EX; and c/0) = 0, B(r;) is nonempty. It is also closed (as X; is closed and the c/) are continuous) and bounded, as is X. Therefore, Weierstrasss Theorem guarantees that Ml) is well-defined as it results from the maximization of a continuous function on a compact non-empty set. Furthermore, M; () is upper semi-continuous (see Kaneko (1977), p. 128 or apply Berge's maximum theorem) and, as it is single valued, it is continuous. We denote with y;(r) the components of M;() that refer to public goods.
Let N(e, M) be the set of messages which corresponds' to a Nash equilibrium for e, given a mechanism M. The concept of Nash equilibrium is adequate when there is complete information, that is, when each player knows the utility functions and strategies which can be used by other players exactly. 1 The reason for this is that if each player knows the characteristics of the others, he is able to discover the strategies they are going to play by means of the same reasoning which they used to determine their optimal strategy (this is von Neumann's and Morgenstern's so-called 'meta-argument').
It is clear that y;( ) is a continuous function. As a result of the assumption of the monotonicity ofpreferences, the r; can be assumed to be non-negative. Therefore, Y =X Y; is such that Y : S ~ fR~;n where S is the Cartesian product' of m simplices of dimension n - 1. Therefore a theorem of variational inequalities (see Herrero and Villar, 1991) assures us that there exists a y* such that Yij < yj implies r 0 = 0 and Yij 5 y j. \f i = 1, ... , n, \fj = 1, ... , m. The assumption of strict monotonicity preferences forbids the first case and therefore Yij = yr, \f j = 1, ...