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value created by acting alone. Condition (5.22) requires that the share of value
allocated to c
1
should be at least the amount of its contributed effort.
In other words, the share of value allocated c
1
, denoted by v(c
1
), should
be:
e≤v(c
1
)≤V (G
2
)−V (G
1
)−e
(5.23)
Case 3 G
a
={p, c
1
, c
2
}
The set of players now includes p and two potential children, i.e., P =
{p, c
1
, c
2
}. If the parent accepts both peers, they form a larger coalition, G
3
,
and create a value of V (G
3
). This is to be distributed among the three players:
V (G
3
) = v(p) + v(c
1
) + v(c
2
)
(5.24)
It should be ensured that G
3
is a stable coalition where the parent and the
two children have no incentive to leave. This requires the following conditions
to be satisfied:
v(p)−2e≥V (G
1
)
(5.25)
v(c
1
)−e≥0
(5.26)
v(c
2
)−e≥0
(5.27)
v(p) + v(c
1
)≥V ({p, c
1
})
(5.28)
v(p) + v(c
2
)≥V ({p, c
2
})
(5.29)
Condition (5.25) ensures that the parent would not drop the two chil-
dren. Conditions (5.26) and (5.27) lead to non-negative utilities for c
1
and c
2
,
respectively. In other words, these two conditions are the incentive compati-
bility constraint in (5.18). The last two conditions, on the other hand, cause
dropping one of the two children an undesirable move. The conditions can be
simplified as follows:
v(c
1
)≤V (G
3
)−V ({p, c
2
})
(5.30)
v(c
2
)≤V (G
3
)−V ({p, c
1
})
(5.31)
v(c
1
) + v(c
2
)≤V (G
3
)−V (G
1
)−2e
(5.32)
v(c
1
), v(c
2
)≥e
(5.33)
Case n G
a
={p, c
1
,, c
n−1
}
This is the general scenario where the parent is encountered with (n−1)
potential children. If they form a single coalition of size n, this creates a value
of V (G
n
), which is to be distributed among the members, i.e.,
V (G
n
) = v(p) +
v(c
i
)
(5.34)
c
i
∈G
n
For G
n
to be stable, peers should have no incentive to leave the coalition
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