Refining Nash Implementation

In the two preceding chapters we have studied the Nash equilibrium approach to the problem of implementation. Various authors have put forward certain undesirable consequences of the property of monotonicity which, as you will remember, is a necessary condition for implementation in Nash equilibria. Firstly, monotonicity prohibits any type of consideration based on the cardinality of utility functions. Secondly, in some cases, distributional considerations may collide with monotonicity. The following example (taken from Moore and Repullo, 1988) will illustrate this point. We assume that there is a public good (which can take two values, 0 or 1), and a private good. The utility functions are quasi linear of the form u i = a i y + x i and the cost of 1 (resp. 0) is 1 (resp. 0). An allocation is a list (y, t 1 ,…, t n ) where y ∈ {0, 1} and t i is the tax paid by i. An economy u is a list (a i ,…, a n ) (the parameter a i is called the marginal propensity to pay). Consider an economy u for which the allocation (1, t1,…, t n ) is optimal. We now consider an economy u′ such that all the marginal propensities to pay, apart from that of the first individual, increase. Then, monotonicity implies that (1, t1,…, t n ) is also optimal for u′ no matter how much the marginal propensities to pay of all the other consumers have increased.