Starting with Gradient Flow Evolution, Quasi-Static Evolution and then going into Evolutionary

Games.

But while I was trying to prepare the talk, I realized that it was a massive amount of

work.

I could not condensate everything properly in a relatively reasonable time.

It was more a course rather than a seminar.

So I decided that I would rather present one part of what I had in mind.

It's already quite a lot of material and not all easy to digest anyway.

It will take some time to explain everything.

I will focus on especially in homogeneous evolutionary games.

So let me start then with this topic.

Let me mention that this is a joint work that I did a couple of years ago with Luigi Ambrosio,

Marco Morandotti and Giuseppe Savare.

And there are some numerical results that are about the master thesis of a student of

mine who is Nathaniel Bosch.

And I will present them at the end.

So I would say that this talk would be inspired by games.

And in particular, there are some aspects of games that attract so much attention of

game theorists, which are game equilibria.

So as physical systems tend to minimize the potential energy, they often converge to certain

steady states.

And due to the fact that somehow steady states are occurring very often, they are extremely

interesting for physicists to analyze what happens in a system when it is in equilibrium.

Similarly, game theorists have focused quite a lot on the characterization of game equilibria.

And one of the most advocated and well-known concepts of equilibrium was proposed by Nash

Brink on the work of von Neumann and Morgenstern on the notion of mixed strategy.

I will get to that in a moment.

So we will try to understand a little bit more about this in a moment.

So nevertheless, both Morgenstern and von Neumann, they felt already in their fundamental

contribution of game theory that would have been desirable to have a dynamic approach

of games rather than static game solution, considering as game solution the equilibrium

of the game.

So the idea is that can we describe the transition from maybe one equilibrium of the game to

another equilibrium of the game and describe the motion between the two states?

This is exactly what the Newton law does for physical systems.

We know that if we are in a certain equilibrium and then we are modifying the energy landscape

or maybe because boundary conditions have changed, then the Newton law will drive the

dynamics from one state equilibrium to another state equilibrium.

It's not obvious what could be a Newton law for dynamical games.

So it would be very interesting to try to see how can one model the concept of transition

towards an equilibrium in a game situation.

Well, this is not something that I invented.

I mean, it's quite a lot of theory about so-called evolutionary games.

It's all about understanding how can one make equilibria emerging in a dynamical fashion.

So one of the most advocated mechanisms for this dynamical choice of strategies is based

on a selection principle.

If you want, it's about the Darwinian evolution concept.

In order to understand this idea, one has to shift a little bit of the interpretation

of the game or a modeling of the game from the concept of mixed strategies to a population

type of dynamics.