Game Development Reference
In-Depth Information
of 'how to arrange our economic interactions so that, when everyone behaves
in a self-interested manner, the result is something we all like'. So these
interactions are supposed to yield desired social decisions when each agent is
interested in maximising only his own utility.
In mechanism design one is interested in the ways of inducing the players
to submit true information. This subject is closely related to game theory,
though it focuses on other issues. In the next section we shall clarify this
connection. To discuss mechanism design in more detail we need to introduce
some basic concepts.
Assume a set
of players with n> 1, a non-empty set of deci-
sions D , and for each player i
{
1 ,...,n
}
a non-empty set of types Θ i , and
an initial utility function v i : D
×
Θ i
R
.
In this context a type is some private information known only to the player,
for example, in the case of an auction, the player's valuation of the items for
sale.
When discussing types and sets of types we use then the same abbreviations
as in Section 1.2. In particular, we define Θ := Θ 1 ×···×
Θ n and for θ
Θ
we have ( θ i −i )= θ .
A decision rule is a function f
D . We call the tuple
( D, Θ 1 ,..., Θ n ,v 1 ,...,v n ,f )
a decision problem .
Decision problems are considered in the presence of a central authority
who takes decisions on the basis of the information provided by the play-
ers. Given a decision problem the desired decision is obtained through the
following sequence of events, where f is a given, publicly known, decision
rule:
Θ i ,
each player i announces to the central authority a type θ i Θ i ; this yields
a joint type θ := ( θ 1 ,...,θ n ),
each player i receives (becomes aware of) his type θ i
the central authority then takes the decision d := f ( θ ) and communicates
it to each player,
the resulting initial utility for player i is then v i ( d, θ i ).
The di culty in taking decisions through the above described sequence
of events is that players are assumed to be rational, that is they want to
maximise their utility. As a result they may submit false information to