Abstract
In this article, we propose a mean field game approach for modeling the flows of excursionists within a network of tourist attractions. We prove the existence of an equilibrium within the network using a balance ordinary differential equation together with optimality conditions in terms of the value function. We also propose a bi-level formulation of the problem where we aim at achieving a sustainable-oriented control strategy in the upper level and at maximizing excursionists’ satisfaction in the lower level. Our proposed model may provide an effective management tool for local authorities who deal with the challenging problem of finding an optimal control policy to the often conflicting objectives of ensuring the maximum excursionists’ satisfaction while pursuing the highest sustainability benefits.
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Appendix
Appendix
The function \({{{\mathcal {V}}}}_r^{\xi ,e_{11}}\) as defined in (21) is of the form
only if
from which we get
Considering the first factor of the left hand side of (53), we get that
with \(h'>0\) independent of r, on \(\rho \) and of the control. If instead, we analyze the second factor of the left hand side of (53) we have
with \(h''\) independent of r, on \(\rho \) and on the control. If in (54) \(h''>0\) then \({{{\mathcal {V}}}}_r^{e_{11},\xi }\) has the shape of (52) only if \(T-\max \{ h', h''\}\le t\le T-\min \{h', h'' \}\). If, instead, \(h''\le 0\) then \(\mathcal{V}_r^{e_{11},\xi }\) is not defined as in (52) but rather by (see (21)):
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Andria, J., Maggistro, R. & Pesenti, R. Sustainable Management of Tourist Flow Networks: A Mean Field Model. J Optim Theory Appl 196, 730–761 (2023). https://doi.org/10.1007/s10957-023-02158-8
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DOI: https://doi.org/10.1007/s10957-023-02158-8