Here we present several paradoxes regarding environmental emissions in supernetworks, in which certain links correspond to telecommunication links and are characterized by a zero emission factor. By a zero emission factor, we mean that, irregardless of the flow on such links the total emissions generated by the use of such links is zero.

We identify three distinct paradoxical phenomena that can occur in such
networks, which demonstrate that so-called *improvements* to the network
may result in increases in total emissions generated. These paradoxes were
discovered by Nagurney and Dong and published in a paper in the journal
Transportation Research D in 2001.

**Paradox 1: The Addition of a Zero Emission Link Results
in an Increase in Total Emissions with No Change in Demand**

We first consider the network below which has the Braess (1968) network topology and user link cost structure.

There are 4 links: a, b, c, d; 4 nodes: 1, 2, 3, 4, and a single origin/destination
pair of nodes (1,4) in the first network. There are, thus, 2 paths available
to users between this O/D pair: p_{1}=(a,c) and p_{2}=(b,d).
The user link cost functions are: c_{a}(f_{a})=10f_{a},
c_{b}(f_{b})=f_{b}+50, c_{c}(f_{c})=f_{c}+50,
c_{d}(f_{d})=10f_{d}, where f_{a }denotes
the flow on link a, f_{b} denotes the flow on link b, and so on.

The demand for the origin/destination pair is given by d_{w}=6.
Assume that the emission factors on the links are given by: h_{a}=.2,
h_{b}=.1, h_{c}=.1, and h_{d}=.2. This means that
the total emissions generated on link a, given a flow of f_{a}
on it would be h_{a}f_{a}, and so on, for the other links.

The user-optimized flow pattern, under which the network is in equilibrium
with no user or traveller having any incentive to alter his travel path,
is given by: x_{p1}*=3, x_{p2}*=3, with equilibrium link
flows: f_{a}*=3, f_{b}*=3, f_{c}*=3, f_{d}*=3,
and with associated equilibrium path costs: C_{p1}=83 and C_{p2}=83.
The total emissions generated by the equilibrium link flow pattern is:
E=.2(3)+.1(3)+.1(3)+.2(3)=1.8.

Now, as illustrated in the figure above, consider the addition of a
new link e, joining node 2 to node 3 to the original network. Assume that
link e has an emission factor h_{e}=0 and a user cost function
associated with traveling on link e given by c_{e}(f_{e})=f_{e}+10.
The addition of this link creates a new path p_{3}=(a,e,d) that
is available to the users of the network. Assume also that the demand d_{w}
remains unchanged at 6 units of flow.

The equilibrium path flow pattern on the new network is: x_{p1}**=2,
x_{p2}**=2, x_{p3}**=2, with equilibrium link flows: f_{a}**=4,
f_{b}**=2, f_{c}**=2, f_{e}**=2, f_{d}**=4,
and with associated equilibrium path costs: C_{p1}=C_{p2}=C_{p3}=92.

The total emissions generated in the new network are equal to 2, a value
greater than the total generated in the original network. Hence, the addition
of a new link with zero emissions makes not only everyone worse off in
terms of cost but also in terms of the emissions generated.

**Paradox 2 - A Decrease in Demand on a Supernetwork
with a Zero Emission Link May Result in an Increase in Emissions**

We now illustrate a second paradox regarding a supernetwork with a zero emission link in which a decrease in the demand for an origin/destination pair, joined by a single path consisting of a single link with a zero emission factor, results in an increase in total emissions.

Consider the supernetwork depicted in the Figure below. The user link
cost functions are: c_{a}(f_{a})=f_{a}+1, c_{b}(f_{b})=f_{b}+4,
and c_{c}(f_{c})=f_{c}+1. There are two origin/destination
pairs: w_{1}=(1,2) and w_{2}=(1,3). The path connecting
O/D pair w_{1}, p_{1}, consists of the single link a. The
paths connecting O/D pair w_{2} are: p_{2}=(a,c) and p_{3}=b.
The demands in the original problem are: d_{w1}=1 and d_{w2}=2.
The emission factors on the links are: h_{a}=0., h_{b}=.01,
and h_{c}=.5. Hence, link *a* may correspond to a telecommunication
link.

The network equilibrium path flow pattern is: x_{p1}*=1, x_{p2}*=1,
x_{p3}*=1, with induced link flow pattern: f_{a}*=2, f_{b}*=f_{c}*=1.
The path user costs are: For O/D pair w_{1}: C_{p1}=3,
and for origin/destination (O/D) pair w_{2}: C_{p2}=C_{p3}=5.
The total emissions generated: E=.02+.01+.5=.51.

We now consider a decrease in demand associated with O/D pair w_{1}
with the new demand d_{w1}=.5 and all other data remain the same.
The new network equilibrium path flow pattern is: x_{p1}**=.5,
x_{p2}**=1.1666..., x_{p3}**=.833..., with induced equilibrium
link flow pattern: f_{a}**=1.666..., f_{b}**=.833..., f_{c}**=1.166...
The new path user costs are: For O/D pair w_{1}: C_{p1}=2.666...,
and for O/D pair w2: C_{p2}=C_{p3}=4.833.... The total
emissions now generated: E=0.000...+.00833...+.5830=.59133....

Hence, the total emissions have increased from .51 to .59133... even though the demand has decreased. Moreover, the costs for paths between the two origin/destination pairs have decreased.

**Paradox 3 - Adding A New Path which Consists Solely
of a Zero Emission Link and which Shares No Links With Any Other Path May
Result in an Increase in Emissions**

We now illustrate a third paradox regarding a supernetwork in which the addition of a new path consisting solely of a zero emission links results in an increase in total emissions.

Consider the first network depicted in the Figure below. The user link
cost functions are: c_{a}(f_{a})=f_{a}+10, c_{b}(f_{b})=3f_{a}+3f_{b}.
There is a single origin/destination pair: w_{1}=(1,2). The paths
connecting the O/D pair are: p_{1}=a and p_{2}=b. The demands
in this network is: d_{w1}=5. The emission factors on the links
are: h_{a}=0.1, h_{b}=.5.

The network equilibrium path flow pattern is: x_{p1}*=5, x_{p2}*=0,
with induced link flow pattern: f_{a}*=5, f_{b}*=0. The
path user costs are: C_{p1}=15, and C_{p2}=15. The total
emissions generated E*=.5+0=.5.

The new network equilibrium link flow pattern is: f_{a}**=2,
f_{b}**=2, and f_{c}**=1. The user path travel costs are:
C_{p1}=C_{p2}=C_{p3}=12. The total emissions now
generated are given by: E=0.2 + 1.0+0.0=1.2. Hence, the equilibrium path
costs have decreased. However, the total emissions have increased from
.5 to 1.2 even though a new disjoint path (that is, a path that has no
links in common with any other path) with zero emissions was added.