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
,θ
−i
)=
θ
.

A
decision rule
is a function
f
:Θ

→

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

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