Solution
Who Pays for the Bridge
Show the problem again
A town can build a bridge costing 90, paid for by an outside funder (the town's budget is not involved). Three citizens have private valuations of 50, 30 and 25 for the bridge. They each report a value and the town uses the Clarke pivot (VCG) mechanism: build the bridge if the reported total value exceeds its cost, so that total reported value minus cost is maximised; and charge each citizen the amount by which their presence lowers the best attainable (reported value minus cost) welfare of the *other two* citizens, relative to what the others get with the bridge-or-not decision actually made. Assume everyone reports truthfully. What is the sum of the three payments?
Worked solution
The reports total 105 > 90, so the bridge is built. Others' welfare at the actual decision, for each citizen i, is (the others' values) minus 90. Citizen with value 50: the others hold 55 in value, so with the bridge they get 55 - 90 = -35. Without the citizen, the best decision for the others is not to build (the surplus would be -35 < 0), giving 0. Payment 0 - (-35) = 35. Citizen with 30: the others hold 75, so with the bridge they get -15; without the citizen they would not build, giving 0. Payment 15. Citizen with 25: the others hold 80, giving -10 with the bridge and 0 without. Payment 10. The total is 35 + 15 + 10 = **60**. Truthful reporting is a dominant strategy, but the mechanism collects only 60 of the 90 cost, a classic illustration that VCG does not balance the budget.
Source: Clarke pivot (VCG) mechanism for public goods (Vickrey 1961, Clarke 1971, Groves 1973); the valuations are invented for AxiomIQ. Statement written for AxiomIQ.