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anonymity model that allows each user to have its specific anonymity set. For
the SGUM-based PCG, we show that there exists a SNE. Then we develop an
algorithm that greedily determines users' strategies, based on the social group
utility derived from only the users whose strategies have been determined.
We show that this algorithm can efficiently find a Pareto-optimal SNE with
social welfare higher than that corresponding to the socially-oblivious PCG.
Numerical results demonstrate that social welfare can be significantlyimproved
by exploiting users' social ties.
6.2
Future Work
In Chap. 2, we develop the SGUM framework which leverage user's “positive” social
ties to stimulate their cooperative behaviors. In general, depending on the nature of
social relationship, the social tie between two users can be “negative” (e.g., between
opponents or enemies) such that one user intends to damage the other's welfare. It is
thus natural to extend the SGUM framework to capture negative social ties. Similar
to positive social ties, negative social ties can also be diverse such that a user intends
to damage others at different levels.
To capture both positive and “negative” social ties, we can extend the SGUM
framework by defining the social group utility
f
i
as
N
f
i
(
x
i
,
x
−
i
)
s
ij
u
i
(
x
i
,
x
−
i
)
j
=
1
where
s
ij
(0, 1], it represents the extent to which user
i
cares
about user
j
's utility, and it reaches the maximum when
s
ij
∈
(
−∞
, 1]: when
s
ij
∈
=
1 (i.e., user
i
cares
about user
j
's utility as much as its own utility); when
s
ij
, 0), it pinpoints to
how much user
i
intends to damage user
j
's utility, and reaches the extreme as
s
ij
goes to
∈
(
−∞
(i.e., user
i
would sacrifice its utility to damage user
j
's utility).
For the extended SGUM game, if the total social tie level to each user is 0, i.e.,
−∞
s
ji
=
0,
∀
i
∈
N
,
j
∈
N
then the SGUM game degenerates to a zero-sum game (ZSG), where each user views
the total gain of other users as its loss. Formally, a game is a ZSG if all users' payoff
functions are summed up to 0. For example, an SGUM game of two users with
f
1
u
1
, is a zero-sum game.
Note that we obtain an equivalent game when a user's payoff function is multiplied
by a number or added by a function independent of that user's strategy. For example,
an SGUM game of two users with
f
1
=
u
1
−
u
2
and
f
2
=
u
2
−
u
1
,or
f
1
=
u
1
and
f
2
=−
=
=
u
2
−
ₒ∞
u
1
and
f
2
s
21
u
1
where
s
21
=
=−
is equivalent to that with
f
1
u
1
, and thus is a zero-sum game.
Therefore, the extended SGUM framework encompasses not only NCG and NUM
but also ZSG as special cases (as illustrated in Fig.
6.1
). Furthermore, it spans the
continuum from ZSG to NCG to NUM (as illustrated in Fig.
6.2
).
u
1
and
f
2
