J Optim Theory Appl (2018) 176:478–491
Population Growth and Nash Equilibria Under
Viability Constraints in the Commons
Received: 3 May 2016 / Accepted: 3 July 2017 / Published online: 11 July 2017
© Springer Science+Business Media, LLC 2017
Abstract Population growth modiﬁes the optimal equilibrium between a station-
ary population and its resource, producing instead a line of equilibria, characterized
by ﬂuctuating population size, resource quantity, harvest per head, and birth rates.
The Pontryagin procedure allows the analytical expression of the Nash equilibria for
two populations sharing a common resource and capable of growth. An alternative
procedure, which avoids solving differential equations and inherently includes state
constraints, involves building the capture-viability kernel of an auxiliary system. For
two populations, all Nash equilibria under state constraints are obtained as the intersec-
tion of the boundaries of two capture-viability kernels. The two methods, Pontryagin
and viability, yield concordant results. Viability is more ﬂexible and avoids solving
differential equations for each initial condition.
Keywords Optimal control · Nash equilibria · Viability theory · Population growth ·
Economics of the commons
Mathematics Subject Classiﬁcation 37N40 · 49M37 · 90C90
Communicated by Lionel Thibault.
Institut national d’études démographiques, 133, bld Davout, 75980 Paris cedex 20, France
École des hautes études en sciences sociales, Paris, France