Okay, so welcome everyone. Today we have Professor Marco Marantotti from the, yeah, it's quite

difficult to say, Polytechnico di Turinno in Italy. And today he will be speaking about

the specially inhomogeneous evolutionary games place at the U of T Club. Thank you very much.

So first of all, let me warmly thank the organizers, Marius and Enrico, for the invitation to join

this seminar. It's a pleasure for me to contribute to this series of online talks, which even

though they are not optimal given the online situation, I think they are really, really

fundamental to keep collaboration and science going during these troubled times. I hope

you can see my screen properly and fully. So I also want to start by acknowledging my

collaborators and co-authors on this research project. They are Stefano Almi from Vienna,

Luigi Ambrosio from Pisa, Massimo Fornasier from Munich, Giuseppe Savare from Milan, and

Francesco Solombrino from Naples. And I would like to take this opportunity to start by

describing what specially inhomogeneous evolutionary games are, which will occupy the first part

of my talk so that we can get started with the applications. And so what I'm going to

talk about today is about replicator dynamics and the way we phrased it and the model we

provided for it in suitable probability spaces with Ambrosio Fornasier and Savare. And then

there's an application of this dynamics to social dynamics with label switching. And

then as a last topic, I would like to mention a recent work with Stefano and Francesco on

an alternate Lagrangian scheme which can serve for the numerical approximation of these dynamics.

So the idea is that we want to study a modeling for agents that can interact. So the word

games comes from game theory, of course, and this can be seen as a group of interacting

agents that can communicate with one another. And they are characterized by the spatial

position and also by a strategy. So everybody is probably familiar, at least in a very wide

sense with the work of Nash on games. And then strategies are what characterizes the

behavior of these agents. And in our model, agents can both move in space so they can

change their position and there can change the strategy. So the idea is that they do

this in an adaptive way, meaning there's an explicit time dependence on this, which was

absent in the notion, for instance, of Nash equilibrium. And they do this in order to

maximize a payoff. So as a mathematician, especially coming from calculus of variations,

I'm used to minimizing costs, but maximizing the payoff is pretty much the same thing up

to a minus sign. So the principle is the same. So why are we interested in this? Because

we would like to study a dynamical approach to the convergence to game equilibrium. As

I mentioned before, Nash equilibrium is a particular state of a game, of a mathematical

game which is characterized by the fact that if anybody changes their strategy, then there

is a loss in the payoff. So then somebody is going to lose something, meaning that the

strategies that are being played need to remain constant in order for everybody to get the

best that they can get. But then, of course, the system is not always in a steady state.

The system happens to be in a different state, and then it wants to converge to the equilibrium.

So this contribution that we gave takes care of this part and describes the evolution of

the game or say of the distribution of agents with their strategies or labels, because in

the second part, the strategies would be called labels, in the process of converging to the

equilibrium. And then Nash equilibrium can be just one of them. There can be different

notions of equilibrium according to what we define to be an equilibrium. And the way the

strategies change is determined by the so-called replicator equation, which is why at the beginning

here I wrote replicator dynamics, which is a concept that's been introduced, I think,

in the 70s in mathematical biology. And we like to think this as giving a Darwinian flavor

to our game, meaning that a strategy which is not very successful gets progressively,

say its occurrence is progressively diminished, whereas the strategies that are successful

will be enhanced by the dynamics. So if you realize that playing a certain strategy gives

you a higher payoff, then you want to pay that one more and more. So the model that

we propose is a main field model. So we introduce it and we study this main field model for