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

4. Secure Implementation in Public Good Economies
The previous section presented two examples drawn from many strategy-proof
mechanisms that may have “bad” Nash equilibria. They implement the social choice function
(SCF) in dominant strategies, but not in Nash equilibria. Saijo et al. (2003) introduce a new
concept of implementation, called secure implementation, which does not share this
shortcoming.
We introduce notation and definitions here to describe the concept of secure
implementation in the context of public good economies with two agents and quasi-linear
preferences. Denote the set of feasible allocations by
A= {( y, t , t ) y∈Y, t , t ∈R}
1 2 1 2 ,
where Y ⊆Ris a production possibility set, y
∈ Y is an output level of a public good, and t i is a
transfer of a private good to agent i. For simplicity, we assume that there is no cost involved in
producing y. Each agent i’s utility function, u i : A → R, is selfish and quasi-linear:
u ( y, t1, t2 ) = u ( y, t ) = v ( y)
+ t , i = 12 , .
i i i i i
The class of valuation functions, v i : Y → R, admissible for agent i is denoted by V i . Following
Holmström (1979), suppose V i is smoothly connected. Let v = ( v1, v2 ) ∈ V ≡ V 1× V 2 be a
valuation profile.
A social choice function (SCF) is a function f : V → A that associates with every list of
valuation functions, v ∈ V, a unique feasible allocation f (v) in A. The allocation f (v) is said to be
f-optimal for v.
A mechanism (or game form) is a function g: S
12
× S → A that assigns a unique element of
1 2
A to every ( s , s ) ∈ S × S , where S i is the strategy space of agent i. For a strategy profile
1 2 1 2
1 2 1 2
s= ( s , s ) ∈ S × S , the outcome of g for the profile s is denoted by gs () = ( y (), s t ()) s , where
g
y () s is the level of the public good and t () s = ( t1 (), s t2 ()) s is the transfer vector.
g g g
The strategy profile s= ( s , s ) ∈ S × S is a Nash equilibrium of g at v ∈ V if
1 2 1 2
g g g g
1 1 2 1 1 2 1 1 2 1 1 2
v ( y ( s , s )) + t ( s , s ) ≥ v ( y ( s′ , s )) + t ( s′ , s ) for all s1′ ∈S1 , and
g
g