I've got sort of a cute example/illustration to start things off. And the government is nowhere to be found in it!
Suppose there is afirm looking to provide a particular public good. It can be supplied in different levels (say in hour long increments). Two people in town, Alice and Bob. Their actual values--also known as marginal benefits (and willingness to pay for the good) along with the firm's marginal cost per hour are listed below
Hours........Alice......Bob....Marginal Cost
1...........$18.......$10..........$2
2...........$16........$9..........$5
3...........$14........$8..........$8
4...........$12........$6..........$11
5...........$10........$4..........$14
6...........$8..........$3..........$17
So here, the optimal amount of the public good would be 5 hours, with Alice paying $10 and Bob paying $4. The marginal cost of providing that fifth hour equals the marginal benefit, which is the decision making metric which maximizes social welfare (it maximizes consumer surplus plus producer surplus).
Now suppose Alice is a lying bitch
. She under reports her willingness to pay for this public good by $7. Now, the firm can only supply four hours of protection, and collects $5 from Alice and $6 from Bob for their good.
Is this in Alice's best interest? well, she loses the 5th hour of the public good, which means she loses something she values at $10. But instead of paying $50 (5 hours x $10/hr) she pays only $5/hour x 4 hours, or $20. So she is better off by $30, so she has incentive to lie, and to cause underproduction.
Note: the demand curve in this case makes it so the firm may want to under-produce if it is a monopoly (another market failure), but the willingnesses to pay could be adjusted to make the demand curve more inelastic so that the firm--even if it is a monopoly--will want to produce as much of the good as possible (possible meaning that the money/hour it can collect is greater than the Marginal cost)