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
,θ
i
)
.

v
i
(
f
(
θ
)
,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
:Θ

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
,θ
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
,θ
−i
)
,θ
i
)=
v
i
(
f
(
θ
i
,θ
−i
)
,θ
i
)+
t
i
(
θ
i
,θ
−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
):=

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