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{
(
5
,
4
)
,
(
5
,
4
)
,
(
3
,
2
)
,
(
3
,
2
)
,
(
1
,
0
)
,
(
1
,
0
)
,
(
0
,
0
)
},
{
(
5
,
4
)
,
(
5
,
4
)
,
(
3
,
2
)
,
(
3
,
2
)
,
(
1
,
0
)
,
(
1
,
0
)
,
(
0
,−
1
)
},
{
(
5
,
4
)
,
(
5
,
4
)
,
(
3
,
2
)
,
(
3
,
2
)
,
(
1
,
0
)
,
(
1
,
0
)
,
(
0
,−
2
)
}
We now have that
1
7
−
7
×
1
7
=
−
8
R
(
0;
{
(
5
,
4
)
,
(
5
,
4
)
,
(
3
,
2
)
,
(
3
,
2
)
,
(
1
,
0
)
,
(
1
,
0
)
,
(
0
,
0
)
}
)=
−
49
×
2
3
−
49
3
×
1
7
−
7
3
×
1
7
=
−
10
3
R
(
0;
{
(
5
,
4
)
,
(
5
,
4
)
,
(
3
,
2
)
,
(
3
,
2
)
,
(
1
,
0
)
,
(
1
,
0
)
,
(
0
,−
1
)
}
)=
−
6
5
−
49
5
×
1
7
−
7
5
×
1
7
=
−
14
5
.
R
(
0;
{
(
5
,
4
)
,
(
5
,
4
)
,
(
3
,
2
)
,
(
3
,
2
)
,
(
1
,
0
)
,
(
1
,
0
)
,
(
0
,−
2
)
}
)=
−
The expected utility of a type
[
0
,
3
]
male under
{
(
5
,
4
)
,
(
5
,
4
)
,
(
3
,
2
)
,
(
3
,
2
)
,
(
1
,
0
)
,
(
1
,
0
)
,
(
0
,−
2
)
}
is greater than the utility obtained from pairing with females of type
[
0
,
0
]
and
[
0
,
6
]
.
Hence, at a symmetric equilibrium males of type
[
0
,
3
]
should not accept such fe-
males.
It remains to consider the set of females that a male of attractiveness 0 should
solicit a date with according to the conditions based on the concept of a trembling
hand perfect equilibrium. At equilibrium, no female of attractiveness greater than 2
would ever pair with a male of attractiveness 0 in the dating subgame. Hence, males
of attractiveness 0 should never solicit dates with females of attractiveness above
2. Females of type
male in the dating
subgame. Arguing as in the case of males of attractiveness 4 soliciting dates with
females of attractiveness 5 and 6, males of attractiveness 0 should solicit dates with
females of attractiveness 1 and 2.
Tab le
17.1
gives a synopsis of the equilibrium strategy profile. Each individual
should accept a prospective partner in the dating subgame if the utility from such a
matching is at least as great as the utility from search. For ease of presentation, the
set of such partners is not presented.
[
2
,
3
]
and
[
1
,
3
]
would pair with a type
[
0
,
3
]
17.9 Generalizing the Model
Suppose that character is placed along a line instead of around a circle, i.e. the
difference between two characters is calculated according to the standard abso-
lute difference. Considering the game presented in Sect.
17.8.2
(with unspecified
search and dating costs), there is still a large degree of symmetry with respect to
sex and character (e.g. the character levels
j
and 6
−
j
can be interchanged without
essentially changing the game).
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