The tragedy of the commons (TOTC, introduced by Hardin, 1968) states that the individual incentives will result in overusing common pool resources which in turn may have detrimental future consequences that affect everyone involved negatively. However, in many real-life situations this does not happen and researchers such as the Nobel laureate Elinor Ostrom suggested mutual restraint by individuals can be the preventing factor. In mean field games (MFGs), since individuals are insignificant and fully non-cooperative, the TOTC is inevitable. This shows that MFG models should incorporate a mixture of self- ishness and altruism to capture real-life situations that include common pool resources. Motivated by this, we will discuss different equilibrium notions to capture the mixture of cooperative and non-cooperative behavior in the population. First, we will introduce mixed individual MFGs and mixed population MFGs where we also include the common pool resources. The former captures altruistic tendencies at the individual level and the latter models a population that is a mixture of fully cooperative and non-cooperative individuals. For both cases, we will briefly discuss definitions and characterization of equi- librium with the forward backward stochastic differential equations. Later, we will discuss a real-life inspired example of fishers where the fish stock is the common pool resource. We will analyze the existence and uniqueness results, and discuss the experimental results.[-]

We discuss the natural Lagrangian and Eulerian formulations of multi-agent deterministic optimal control problems, analyzing their relations with a novel Kantorovich formulation. We exhibit some equivalence results among the various representations and compare the respective value functions, by combining techniques and ideas from optimal transport, control theory, Young measures and evolution equations in Banach spaces. We further exploit the connections among Lagrangian and Eulerian descriptions to derive consistency results as the number of particles/agents tends to infinity. (In collaboration with Giulia Cavagnari, Stefano Lisini and Carlo Orrieri)[-]

We investigate the mean-field limit of large networks of interacting biological neurons. The neurons are represented by the so-called integrate and fire models that follow the membrane potential of each neuron and captures individual spikes. However we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. We are able to extend the concept of extended graphons, introduced in Jabin-Poyato-Soler, by introducing a novel notion of discrete observables in the system. This is a joint work with D. Zhou.[-]

We investigate the mean-field limit of large networks of interacting biological neurons. The neurons are represented by the so-called integrate and fire models that follow the membrane potential of each neuron and captures individual spikes. However we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. We are able to extend the concept of extended graphons, introduced in Jabin-Poyato-Soler, by introducing a novel notion of discrete observables in the system. This is a joint work with D. Zhou.[-]