A study of Altruistic Behaviour from a Control Theory Perspective

The paper shows that the altruistic behaviour can be interpreted as the solution of an optimal control problem. To this purpose, two discrete-time dynamic models for the interaction between individuals' fitnesses are described. For the first model, which is linear with constant coefficients, it is possible to prove that the optimal control input is a classic bang-bang function with one switching (at maximum) that can be explicitly located in time. To show that a solution exists also for the second model, which is a non-linear extension of the first one, we adapt to the discrete-time case a result available in the literature and concerning the implicit expression of the solution of optimal control problems for continuous-time positive systems. Numerical examples are also added to highlight the results. Finally, we show that the models can also fit other real problems, among which we consider, in particular, the optimal investment of a capital.