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].