Suppose the government proposes the following mechanism

Subject:PhilosophyPrice: Bought3

Share With

Suppose the government proposes the following mechanism. Each neighborhood will state an amount that it values the school. The neighborhoods can say any number between 0 and one million, and they need not tell the truth. If the sum of the two statements is greater than one million, the school will be build with neighborhood 1 paying $600,000 and neighborhood 2 paying $400,000. If the two statements sum to less than or is equal to one million, the school will not be built. In addition, the government will set up a incentive payments to each neighborhood t1 and t2 as a function of the stated valuations (s1 and s2). Let us consider how the payments can align the incentives of a utilitarian government and the neighborhoods. The government will build the school if s1 + s2 −1, 000, 000 > 0 which is the same as s1 > 1, 000, 000 −s2. Neighborhood 1 would like the school to be built if v1 −600000 + t1 > 0 which is the same as v1 > 600000 −t1.

Comparing these two inequalities, one might conjecture that neighborhood 1 cannot benefit from lying if the two right-hand sides are the same—i.e,. if t1 = s2 −400000 so that 1, 000, 000 −s2 = 600000 −t1. If this is the case, the argument goes, then if player 1 tells the truth, the government will build exactly when player 1 wants the government to build.
 

Show that this conjecture is correct. That is, show that if t1 = s2−400000, then for any v1 and any statement of neighborhood 2, s2, neighborhood 1 cannot benefit from lying to the government about its valuation.