Information Technology Reference
In-Depth Information
In general, there will be an initial uploading agent
p
0
(in the figure
p
a
) and a set of
n
additional uploading agents,
p
1
,...,p
n
(in the figure
p
h
,
p
l
and
p
m
), all of which have
the file that has to be downloaded and they dedicate their upload capacity
b
i
i
to this
transfer. Let us call
size
(
Z
)=
T
the size of the file to download. Let us also assume
that
0
b
i
i
≤
b
ou
b
.
Then an estimation of the time necessary for the transfer is given by the ratio between
the size of the file and the coalition bandwidth(1):
T
b
i
0
+
1
b
i
i
t
S
=
(1)
On the other hand, if the coalition is not formed, the time for just an uploading agent
p
0
is given by (2):
T
b
i
0
t
0
=
(2)
Therefore the value obtained by the coalition
S
can be defined as
Δt
, the difference
between (2) and (1), as shown in (3):
1
b
i
i
=
t
0
1
b
i
i
T
b
i
0
−
T
T
b
i
0
Δt
=
+
1
b
i
i
=
0
b
i
i
0
b
i
i
(3)
b
i
0
Of course, if
p
0
/
∈
S
,
V
(
S
)=0
. To sum up, the coalitional value for every coalition is
given by (4):
V
(
S
)=
t
0
1
b
i
i
if
p
0
∈ S
0
b
i
i
(4)
0
if
p
0
/
∈
S
Now we address the following problem, to define a stable payoff division of
V
(
S
)
between agents, i. e., given a partition of the set of all agents into different coalitions,
to assign an amount
x
(
i
)
to every agent
i
. The problem is to distribute the utility in
a fair and stable way, so that the agents in the coalition are not motivated to abandon
it. Game theory provides different concepts (core, kernel, Shapley value, etc.) for the
stability of coalitions [4]. The core is the simplest to define. A payment configuration
belongs to the core if there is no other coalition that can improve on the payoffs of all
of its members.
Formally, let
N
be the set of all agents, and let us denote
x
(
S
)=
i∈S
x
(
i
)
.The
payoff division lies inside the core iff the following holds: i) for all sets of agents
S
⊆
N
,
V
(
S
)
x
(
S
)
(group rationality); and ii)
V
(
N
)=
x
(
N
)
(global rationality). The
existence of the core is not guaranteed in the general case, in a given situation the core
may be empty. However, we will show for our case a payoff division that lies inside the
core. We will expand on the ideas presented in [15].
The proposed payoff division scheme is calculated by means of the marginal profit
concept. Thus, the payment to each agent is given by the marginal profit according to the
resources that describe that agent, and multiplied by the value of the resources. Since
the concept of marginal profit is really that of partial derivative, the payment vector
x
will be computed as follows: for the original uploading agent,
p
0
, there are 2 parameters
≤
