RESEARCH
BEST THEORETICAL WORK
The MPhys project has been awarded the Franz Mandl Prize for "best theoretical work carried out in the final year of an undergraduate degree". Obviously, Dr Tobias Galla should also get credit for the achievements of the project. I was presented with the award by Prof Fred Loebinger on my graduation day. (photo courtesy of Karina Rudzinska)
Chaos in learning dynamics (click for larger version)
MPHYS IN BRIEF
We are looking at modeling risk in evolutionary game theory (EGT) where normally only the maximum average payoff in considered. No attention is paid to the spread of the elements in the payoff matrix and there is no theory with would take this into the account. The utility function is usually the choice of risk modellers in economics but it seems a bit ill suited for EGT. Initial investigations show that the notion of utility are not producing any new outcomes and results only in shifts of the Nash equilibria which already existed in the standard "risk-ignorant" EGT, leaving the long term state of the system qualitatively unchanged. Our approach, however, produces new fixed points and the dynamics is certainly different to that of normal replicator equations. The cooperation is maintained in games such as Prisoner's Dilemma and in the dynamics of learning, the existence of chaos has been confirmed. I also worked on the van Kampen system size expansion for our games with the aim of calculating the spectra of the stochasticly induces oscillations. These have been confirmed in numerical simulations at the last stage of the project.
MPHYS PROPOSAL
This is the original proposal for my MPhys project with Dr Tobias Galla. Just to give an idea of what the research is about, although some things were added as the project progressed over the last few months.
"Evolutionary game theory describes populations of agents who repeatedly play a given game. A payoff is then paid, and individuals reproduce in proportion to their success and pass on their strategies to their offspring. Over time the composition of the population changes, with more successful strategies becoming more abundant, and less successful ones dying out. These processes are described by so-called replicator equations. These are coupled ordinary differential equations, and can be analysed using the tools of nonlinear dynamics. One current area of interest is concerned with stochastic effects in game theory. Here one does not consider differential equations, but studies the dynamics as a process of interacting particles (the players), and methods from statistical physics are immensely successful in characterising such systems (e.g. Master equations, Langevin dynamics etc).
"In this project you will consider an extension of standard replicator dynamics which includes a notion of risk. A given strategy may pay a higher payoff on average, but it may also be associated with a higher risk. Greedy players will ignore the risk and only take into account the average expected payoff. Risk averse agents will be more careful.
"In detail the work programme could develop as follows:
1. Textbook reading: basics of game theory
2. Analysis of replicator equations with tools from nonlinear dynamics
3. Computer simulation of the interacting-agent system
4. Notions of risk in game dynamical learning
5. Potentially: stochastic dynamics and statistical mechanics analysis
6. Studies of systems with coupled evolutionary dynamics and learning
"This is an ambitious, but feasible project, which, depending on progress and results, may form the basis of a scientific publication. If steps 1-3 are carried out successfully in the first semester, an extension towards a full-year project is possible upon mutual agreement.
"In order to be able to carry out the project applicants MUST have successfully completed and enjoyed the third year course on nonlinear physics (PHYS 30471), and they must attend the course "Advanced statistical physics" in their 4th year (parallel to the project). An interest in mathematical aspects of physics, as well as extended programming skills in at least one higher programming language (Fortran, C, C++) are essential.
MPHYS REPORT
The entire body of the report from first semester will not be available before the publication of the results but the introduction and reference/bibliograpy section can be download via the links below.
Risk vs Greed in EGT (front matter)
Risk vs Greed in EGT (back matter)
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