Mechanism design and the provision of public goods

1. Mechanism design and the provision of public goods Ilaria Petrarca

2. Players of the mechanism • Principal : tax autority (government) • Agents : taxpayers

3. The principal • The principal would like to condition her actions on some information that is privately known by the agents (individual’s willingness to pay for a public project)

4. Principal’s utility to maximize • Self-interest principal • Benevolent principal where

5. The agents • Types Θi are single dimensional; • Preferences are quasi-linear (there are no income effects on the demand for PG); • Bernoulli utility function =

6. Two approaches • Dominant strategy; • Mechanism for Bayesian environment.

7. 1. Dominant strategy • Each agent’s optimal announcement is independent of the announcements of the others; • A DS mechanism is a function y(Θ) such that • DIC:

8. Clarke-Groves mechanism 1 • Binary decision k=(0,1); • C: cost of the project; • Θi: individual willingness to pay; • The project is undertaken if : And each individual will pay a sum: But Θiis a private information

9. Clarke-Groves mechanism 2 are now transfers over and above the payments => loss of utility for the agents. Agent i’s net benefit from project level k Agent i’s gross benefit from project level k Equal contribution required to provide k

10. Social choice function • f:Θ1x… ΘxI→X such that, for each possible agent’s type Θ1, assigns a collective choice f(Θ1… ΘI) X

11. CGM definition • Direct revelation mechanism in which the social function satisfies both: (s.f. is ex post efficient); (s.f. is truthfully implementable in DS).

12. Comments on C-G mechanism • Truth telling is a dominant strategy; • Each individual ‘internalizes’ the total social surplus; but • The mechanism does not satisfy budget balance • Green-Laffont thm.(1979): no social choice function that is (ex post) welfare maximizing (taking into account money burning as a loss) is implementable in dominant strategies.

13. 2. Mechanisms for Bayesian environment • Extension of Clarke-Groves mechanism in Bayesian context; • Weaker requirement than dominant strategy implementation (an agent’s best response strategy may depend on others’ strategies); • D’Aspremont, Gérard-Varet mechanism (1979)

14. Agents • Bernoulli utility function: • Agent’s types are statistically independent; • Utilitarian setting: the output maximizes the sum of the agent’s utilities.

15. Assumptions • for all k (the social choice function is ex post efficient) • budget balance condition BB for all Θ

16. Expected externality mechanism • There is a social function in which Expected benefits of agent j when i announces his type and j tell the truth It is an arbitrary function of Θ-i

17. IC in the model • The IC constraint have to hold only in expectation:

18. Budget balance constraint • We can choose hi(ּ) satisfiyng also BB:

19. D’Aspremont, Gérard-Varet (1979) • Assume quasilinear preferences and statistically independent types. A utilitarian social choice function f: Θ -> (x(Θ), t(Θ)) can be implemented in Bayes-Nash equilibrium if ti(Θi)= (Θi) + hi(Θ-i) for arbitrary function h.

20. Conclusions • D’Aspremont, Gérard-Varet (1979) show that ‘using the Bayesian approach to incomplete information, one may find mechanism to solve efficiently a collective decision problem, which ensure simoultaneusly IC and budget equilibrium’; • Limit: participation constraints are not satisfied.

21. Participation constraints • Agents can’t be forced to participate in a mechanism: it must be in their own best interest; • A mechanism is individually rational if an agent’s (expected) utility from participating is (weakly) better than what it could get by not participating.

22. Example • Asumptions: • Two agents : 1,2; • Decision set: K={0,1} • Types:Θi= {Θa,Θb} and Θa>2Θb>0 • Cost of the project: c(Θa,Θb) • If , then 1 participates; • If both have valuation Θa, the feasiblility condition is not satisfied. Hence, it is impossible to implement a s.c. function with an ex post efficient project when the agents can withdraw from the mechanism at any time.

23. Conclusions • CG and DAGV mechanisms give an explanation for observed failures to provide PG: efficiency and BB are not obtained with voluntary participation.

24. A wise man said… • P. Samuelson (1954) conjectured that no resource allocation mechanism can ensure a fully efficient level of public goods because “it is in the selfish interest of each person to give false signals, to pretend to have less interest in a given collective activity that he really has”=> free riding

25. References • Fudenberg-Tirole, ch.7.4; • MGW, ch. 23C, 23D; • D’Aspremont, Gérard-Varet: Incentives and incomplete information, 1979 JPE; • Mechanism design theory, Compiled by the Prize Committee of the Royal Swedish Academy of Sciences, 2007.