Game Development Reference
In-Depth Information
f ( θ ):= 1 f i =1 θ i
c
0
otherwise.
c
n
If the project takes place ( d = 1),
is the cost share of the project for
each player.
Let us return now to the decision rules. We call a decision rule f e cient
if for all θ
Θ and d
D
n
n
v i ( d i ) .
v i ( f ( θ ) i )
i =1
i =1
Intuitively, this means that for all θ
Θ, f ( θ ) is a decision that maximises
the initial social welfare from a decision d , defined by i =1 v i ( d, θ i ). It
is easy to check that the decision rules used in Examples 1.26 and 1.27 are
e cient.
Let us return now to the subject of manipulations. As an example, con-
sider the case of the public project problem. A player whose type (that is,
appreciation of the gain from the project) exceeds the cost share
c
n should
manipulate the outcome and announce the type c . This will guarantee that
the project will take place, irrespective of the types announced by the other
players. Analogously, a player whose type is lower than
c
n should submit the
type 0 to minimise the chance that the project will take place.
To prevent such manipulations we use taxes , which are transfer payments
between the players and central authority. This leads to a modification of the
initial decision problem ( D, Θ 1 ,..., Θ n ,v 1 ,...,v n ,f ) to the following one:
n ,
the set of decisions is D × R
n , where t
n and
the decision rule is a function ( f, t ):Θ
D
× R
R
( f, t )( θ ):=( f ( θ ) ,t ( θ )) ,
n
the final utility function of player i is the function u i : D
× R
×
Θ i R
defined by
u i ( d, t 1 ,...,t n i ):= v i ( d, θ i )+ t i .
n , Θ 1 ,..., Θ n ,u 1 ,...,u n , ( f, t )) a direct mechanism
and refer to t as the tax function .
So when the received (true) type of player i is θ i and his announced type
is θ i , his final utility is
u i (( f, t )( θ i −i ) i )= v i ( f ( θ i −i ) i )+ t i ( θ i −i ) ,
We call then ( D × R
where θ −i are the types announced by the other players.
In each direct mechanism, given the vector θ of announced types, t ( θ ):=