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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