Information Technology Reference
In-Depth Information
-
U
i
:
i
∈
N
S
i
−→ R
is agent i's personal utility function mapping every strategy
profile in
i
∈
N
S
i
to a real number (
i.e
., personal utility of the strategy profile for
player i).
Let 2
Agt
∗
=
2
N
be the set of all non-empty sets of players (
alias
coalitions). For
notational convenience we write
\{∅}
2
Agt
∗
,wedefinethe
set of strategies for the coalition
J
to be
S
J
=
i
∈
J
S
i
.Elementsof
S
J
are denoted by
s
J
,
−
i
instead of
N
\{
i
}
.Forevery
J
∈
s
J
,...
For notational convenience, we write
S
instead of
S
N
and we denote elements
s
,...
of
S
by
s
Every strategy
s
J
of coalition
J
can be seen as a tuple (
s
i
)
i
∈
J
where
player
i
chooses the individual strategy
s
i
∈
,
S
i
.
The following definition extends the definition of normal form game with a
moral
component. Namely we assume that players in a game also evaluates outcomes with
respect to their ideality degree,
i.e
., how much a given outcome conforms to the player's
moral values.
Definition 2 (Normal form game with moral values).
A normal form game with
moral values is a tuple
Γ
+
=
(
N
,{
S
i
}
i
∈
N
,{
U
i
}
i
∈
N
,{
I
i
}
i
∈
N
)
where:
-
(
N
,{
S
i
}
i
∈
N
,{
U
i
}
i
∈
N
)
is a normal form game;
-
I
i
:
i
∈
N
S
i
−→ R
is agent i's ideality function mapping every strategy profile in
i
∈
N
S
i
to a real number (
i.e
., the ideality of the strategy profile for player i).
The preceding notion of ideality corresponds to Harsanyi's notion of ethical preference.
We define guilt as the emotion which arises from the comparison between the ideality
of the current situation and the ideality of a counterfactual situation that could have been
achieved had the player chosen a different action. In particular, intensity of guilt feeling
is defined as the difference between the ideality of the current state and the ideality of
the best alternative state that could have been achieved had the player chosen a different
action.
Γ
+
Definition 3 (Guilt).
Given
a
normal
form
game
with
moral
values
=
(
N
I
i
}
i
∈
N
)
the guilt player i will experience after the strategy profile
s is played, denoted by Guilt
(
i
,{
S
i
}
i
∈
N
,{
U
i
}
i
∈
N
,{
,
s
)
, is defined as follows:
I
i
a
i
,
,
=
−
Guilt
(
i
s
)
I
i
(
s
)
max
a
i
∈
S
i
s
−
i
)
(1)
We assume that guilt affects the utility function of a certain player depending on the
player's degree of guilt aversion. More precisely, the higher the influence of guilt on the
utility of a given decision option, the more guilt averse the player. The extent to which
a player's utility is affected by his guilt feeling is called
degree of guilt aversion
.The
following definition describes how a player's utility function is transformed depending
on the player's guilt and on the player's degree of guilt aversion.
Definition 4 (Guilt-dependent utility).
Given a normal form game with moral values
Γ
+
=
I
i
}
i
∈
N
)
the guilt-dependent utility of the strategy profile s for
agent i is defined as follows:
(
N
,{
S
i
}
i
∈
N
,{
U
i
}
i
∈
N
,{
U
i
(
s
)
=
U
i
(
s
)
+δ
i
(
Guilt
(
i
,
s
))
(2)
where
δ
i
is a nondecreasing function
δ
i
:
R −→ R
such that
δ
i
(0)
=
0
.
