Game Development Reference

In-Depth Information

Table 1.2
The pivotal mechanism for the public project problem

player

type

tax

u
i

A

6

0

−

4

B

7

0

−

3

C

25

−

7

8

Table 1.3
The pivotal mechanism for the public project problem

player

type

tax

u
i

A

4

−

5

−

5

B

3

−

6

−

6

C

22

0

0

1.8 Pre-Bayesian games

Mechanism design, as introduced in the previous section, can be explained

in game-theoretic terms using pre-Bayesian games, introduced by Ashlagi

et al. [2006] (see also Hyafil and Boutilier [2004] and Aghassi and Bertsimas

[2006]). In strategic games, after each player selected his strategy, each player

knows the payoff of
every other player
. This is not the case in pre-Bayesian

games in which each player has a private type on which he can condition

his strategy. This distinguishing feature of pre-Bayesian games explains why

they form a class of games with
incomplete information
. Formally, they are

defined as follows.

Assume a set
{
1
,...,n}
of players, where
n>
1. A
pre-Bayesian game

for
n
players consists of

•

a non-empty set
A
i
of
actions
,

•

a non-empty set Θ
i
of
types
,

•
a
payoff function
p
i
:
A
1
× ...× A
n
×
Θ
i
→
R
,

for each player
i
.

Let
A
:=
A
1
×

A
n
. In a pre-Bayesian game Nature (an external agent)

moves first and provides each player
i
with a type
θ
i
∈

...

×

Θ
i
. Each player

knows only his type. Subsequently the players simultaneously select their

actions. The payoff function of each player now depends on his type, so after

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