̅, where the corresponding equilibrium
Theorem 4.4. Under Assumption 1, consider a point 𝐩
fulfils the Multi-Class SC Condition (MCSCC) and the Cycle Independence Condition (CIC).
̅ is generic, and the derivatives of 𝐟, 𝐓 and TVT w.r.t. positive tolls are given by (19), (20)
Then, 𝐩
and (21), respectively.
Proof. By Prop. 2.3, the total link flows 𝐟(𝐩) and the link travel times 𝐓(𝐩) are continuous as
functions of p. This implies that in the SP problem for any origin q and class k the node prices and
hence also the reduced costs are continuous as functions of p.
̅, the reduced cost is still positive for 𝐩 = 𝐩
̅ + 𝑑𝐩
Thus, for any link with positive reduced cost at 𝐩
̅. Hence no new links appear in the corresponding reduced SP graph. Therefore the (q,k)
close to 𝐩
flow change, corresponding to such a toll change, will be a circulation in the reduced SP graph.
And as such it is a sum of cycle flows. Thus the total flow change can be written in the form 𝑑𝐟 =
𝐆𝑑𝛄 as above.
Based on this relationship, let’s compute a tentative solution which we will later check for
optimality and feasibility.
̅ all equilibrium cycles 𝐠 𝑖 at 𝐩
̅ remain
So assume tentatively that in a neighbourhood of 𝐩
equilibrium cycles. This implies that condition (11) is valid for all of them. Reasoning as in the
derivation of Thm. 4.2, we see that condition (17) must be fulfilled for the resulting cycle flow
̃𝑑𝐩. So the computed 𝑑𝛄 is of the same order as 𝑑𝐩.
change 𝑑𝛄, i.e. 𝑑𝛄 ∼ 𝐆
̅, where all SPs carry positive
Since the MCSCC is fulfilled, there is a route flow equilibrium at 𝐩
flows. This implies that for all origins q and classes k, the (q,k) reduced SP graph has positive flow
in all links. This, in turn, implies that all, equilibrium cycles and hence also all basis cycles carry
̅ + 𝑑𝐩, in 𝒫+ , the cycle flow changes 𝑑γ𝒊 will
flow. Thus for a sufficiently small change 𝑑𝐩 with 𝐩
fit into the current flows, and can feasibly be allocated to the (q,k)-flows. Thus the computed flow
change is feasible. Since it was computed as the optimum over a potentially larger set it is then in
fact optimal.
Hence, the computed solution is feasible and optimal in (7), and hence it is the equilibrium.
Since the computed solution fulfils, conditions (11) and (12), the reduced SP graph in fact remains
̅ is generic, and the other conclusions follow by Thms. 4.2 and 4.3. ∎
unchanged, and a fortiori 𝐩