## Guest talks: Prof. Dr. Sofia Castro (04.+ 05.09.2017)

On 04. and 05. September, 2017, Professor Sofia Castro, University of Porto, is visiting the research training group.
We cordially invite you to her two talks titled:

*Stability of cycles and dynamics near heteroclinic networks*

Monday 04th September 2017, 14:00, room MZH 2340, MZH building, Bibliothekstr. 1.

*The dynamics of learning in game theory*

Tuesday 05th September 2017,14:00, room MZH 4140, MZH building, Bibliothekstr. 1.

** Abstracts **

* Stability of cycles and dynamics near heteroclinic networks *

In ordinary differential equations, a saddle-sink connection is generically not robust. However, when symmetry is present giving rise to flow-invariant spaces, a saddle-sink connection may be robust. A sequence of connections between consecutive equilibria is called a heteroclinic cycle. A heteroclinic network is a connected union of heteroclinic cycles.
When a cycle is part of a heteroclinic network it cannot be asymptotically stable. It can nevertheless exhibit some stability that may make the cycle visible in experiments and simulations. I shall describe several intermediate notions of stability and ways to determine them. Then I shall show how the stability of the connections in the network can be used to describe nearby dynamics.

This talk will be more pedagogical than a standard scientific seminar. Towards the end of it I shall report on joint work with Alexander Lohse (U. of Hamburg).

* The dynamics of learning in game theory *

Game theory studies the interactions among agents that try to maximize their payoffs by choosing from a set of actions. When this interaction occurs over time, the agents can change their choice of action depending on what they believe is the best action at any given time. This process constitutes a learning mechanism and can be described by various dynamical systems. Different dynamical systems arise as a result of different forms of belief update.
I shall look at two classic learning mechanisms: replicator dynamics (RD) and best-response dynamics (BRD). It is known that the Nash equilibria, describing the outcome of the game, are the same for these two learning mechanisms. However, even when the Nash equilibrium is locally stable its basins of attraction for RD and BRD can be very different. In such cases, the learning outcome can be distinct depending on the learning mechanism and no prediction about the game's outcome can be made. We provide sufficient conditions that guarantee a considerable intersection of the basins of attraction, including full coincidence. These conditions depend on the indifference of the agents with respect to the available actions.
In the context of RD, I shall present an example where learning leads to periodic behaviour.
Some of the work is joint with Liliana Garrido-da-Silva (U. of Porto).