Conflict Resolution Example
Two corn production companies,
Crispy
and
Sweet
, have continuous transportation flows
of their corn to two processing facilities
A
and
B
.
The fraction of corn that
Crispy
sends
to
A
is
p
, while 1
p
gets sent to
B
.
The fraction of corn that
Sweet
sends to
A
is
q
, while
1
q
gets sent to
B
.
Interviews with the management of the two companies showed the following utilities:
(i)
both companies have a utility value of 200 if both companies send all their
corn to
A
;
(ii)
both companies have a utility value of 0 if both companies send all their
corn to
B
;
(iii)
if
Crispy
sends all to
A
and, at the same time,
Sweet
all to
B
, then
Crispy
has a utility of 100 and
Sweet
of 300; and
(iv)
if
Crispy
sends all to
B
and, at the same time,
Sweet
all to
A
, then
Crispy
has a utility of 300 and
Sweet
of 100.
Assume that we see the fractions (
p
and
q
) as probabilities of shipping all corn to
A
.
For
example, if
Crispy
ships 20% to
A
, we have
p
=0.2.
We now assume that
p
=0.2 is the
probability that
Crispy
ships all to
A
.
Let's further assume that
Crispy
and
Sweet
decide
on their fractions independently of each other.
Finally, let's assume that they want to
maximize their expected utility, where
u
C
is
Crispy's
expected utility and
u
S
is
Sweet's
expected utility.
a)
Write down the formal model for
u
C
and
u
S
in terms of
p
and
q
. (e.g.,
u
C
=
apq
+
bp
(1
q
) + …, where
a
,
b
, and
c
are numbers).
b)
Draw the solution space in the
u
C

u
S
plane. Note that the solution space is defined by
all
u
C

u
S
values for varying
p
and
q
values (between 0 and 1).
c)
What are the minimum and maximum
u
C
and
u
S
values, and for which
p
and
q
values
are they obtained.
d)
Compute the fraction
q
, for which
Crispy's
expected utility,
u
C
, is independent of its
fraction
p
, what is
Crispy's
u
C
at this point?
e)
Compute the fraction
p
, for which
Sweet's
expected utility,
u
S
, is independent of its
fraction
q
, what is
Sweet's
u
S
at this point?
f)
Find the efficient (Pareto optimal) values
u
C
=
u
S
; for which fractions
p
and
q
are they
obtained?
Solutions:
a)
u
C
= 200
pq
+ 100
p
(1
q
) + 300(1
p
)
q
;
u
S
= 200
pq
+ 300
p
(1
q
) + 100(1
p
)
q
.