Secure Implementation Experiments: Do Strategy-proof Mechanisms ...

where r i is agent i's most preferred level of the public good. We can represent these singlepeaked
preferences by the r i instead of the v i . The optimal output level of the public good
satisfying (4.1) is given by y( r1, r2 ) = (r 1+r 2 )/2. In this case any SCF f meeting (4.1) and (4.2)
satisfies the rectangular property and is therefore securely implementable (Saijo et al., 2003).
Consider an example that will be used in our experimental design later, in which h i = 0.
Then,
u ( ~ r , ~ r ) = v ( y( ~ r , ~ r )) + t ( ~ r , ~ r ) = − (( ~ r + ~ r )/ 2− r ) − (( ~ r + ~ r )/ 2 −~ r )
1 1 2 1 1 2 1 1 2
{ }
=− ( ~ r − r ) + ( ~ r −r
) /
1 1 2 2 1 2 2
1 2 1 2 1 2 2 2
where r 1 is player 1’s true peak and ( ~ r1, ~ r2 ) is a vector of reported peaks. Clearly agent 1’s
payoff is maximized at r 1 . Since the payoff function is quadratic, no other maximizers exist.
Furthermore, the payoff is maximized at r 1 regardless of ~ r 2 . Figure 3 shows agent 1’s payoff
when r 1 = 12. If ~ r2 = 4 , the maximizer is a, and if ~ r 2 = 12 , it is b. Both are maximized at r 1 = 12.
Therefore, the best response curve is a line parallel to the ~ r 2 axis. This indicates that truthtelling
is the dominant strategy. In fact, it is strictly dominant. However, this is true only as long
as the public goods level is continuously variable. In our experiment, we will discretize the
public goods level and the payoff functions, and truth-telling will not be strictly dominant even
though preferences are single-peaked. 8 However, with single-peaked preferences
implementation will still be secure, because there will be a unique dominant strategy
equilibrium which is also a unique Nash equilibrium (Treatment S). When preferences are not
single peaked, there will be multiple Nash equilibria and implementation is not secure
(Treatment P).
8 In general, with a discrete public good, single-peaked preferences will not assure the existence of a strictly
dominant strategy. However, secure implementation will be assured.
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