Game Development Reference

In-Depth Information

Table 1.1
The pivotal mechanism for the sealed-bid auction

player

type

tax

u
i

A

18

0

0

B

21

0

0

C

24

−

21

3

offered (so the first, or the highest price), is not incentive compatible. Indeed,

reconsider the above example. If player C submits 22 instead of his true type

24, he then wins the object but needs to pay 22 instead of 24. More formally,

in the direct mechanism corresponding to the first-price auction we have

u
C
((
f, t
)(18
,
21
,
22)
,
24)=24

−

22=2
>
0=
u
C
((
f, t
)(18
,
21
,
24)
,
24)
,

which contradicts incentive compatibility for the joint type (18
,
21
,
24).

Re: Example 1.27
To compute the taxes in the public project problem in

the case of the pivotal mechanism we use the following observation.

Note 1.30

In the public project problem we have for the pivotal mechanism

⎧

⎨

if
j
=
i
θ
j
≥

n
c
and
j
=1
θ
j
≥ c

n
−
1

0

j
=
i
θ
j
−

if
j
=
i
θ
j
<
n
−
1

c
and
j
=1
θ
j
≥

n
−
1

n

c

c

n

t
i
(
θ
)=

if
j
=
i
θ
j
≤

c
and
j
=1
θ
j
<c

⎩

n
−
1

n

0

n
c −
j
=
i
θ
j

if
j
=
i
θ
j
>
n
−
n
c
and
j
=1
θ
j
<c.

n
−
1

Exercise 1.11

Provide the proof.

To illustrate the pivotal mechanism suppose that
c
= 30 and that there

are three players, A, B, and C whose true types are respectively 6, 7, and

25. When these types are announced the project takes place and Table 1.2

summarises the taxes that players need to pay and their final utilities. The

taxes were computed using Note 1.30.

Suppose now that the true types of players are respectively 4, 3 and 22

and, as before,
c
= 30. When these types are also the announced types, the

project does not take place. Still, some players need to pay a tax, as Table 1.3

illustrates. One can show that this deficiency is shared by all feasible incentive

compatible direct mechanisms for the public project, see [Mas-Collel et al.,

1995, page 861-862].

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