In class today Michalis Panayides presented research from his PhD. Michalis’ work uses queuing theory to build a Normal Form Game between two hospitals. This is used to identify a good set of incentives/targets to help reduce ambulances being blocked outside of Accident and Emergency departments.
Once the game is built, based on a specific set of incentives, an evolutionary algorithm based on Replicator Dynamics is used to see what would happen. This helps to understand the effects of the incentives.
You can see a recording of this here: cardiff.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=cb2afc49-e02a-4753-8f9e-afb900a56bca
Here is a publication if you would like to read more: A game theoretic model of the behavioural gaming that takes place at the EMS - ED interface
After this we discussed potential reasons for the emergence of the social convention of riding on a particular site of the road (on the left in the UK for example).
We used the following game to model this using Replicator Dynamics:
\[A = \begin{pmatrix} 1 & -1 \\ -1 & 1\\ \end{pmatrix}\]This game is meant to model the interaction of individuals in a given population who interact (by driving past each other). If both these individuals drive according to the same convention then they get a utility of 1 but if not they get a utility of -1.
We then model a given population using a vector \(x=(x_1, x_2)\) where \(x_1\) corresponds to the proportion of individuals driving according to the first convention (say: the left) and \(x_2=1-x_1\) is the proportion driving according to the second convention.
We can then compute the average utility of an individual who drives using the first convention (we can refer to this as the first type and to utility as fitness). They will interact with another individual of the first type \(x_1\) of the time getting a fitness of \(1\) and an individual of the second type \(x_2\) of the time getting a fitness of \(-1\). The average utility is then:
\[f_1 = x_1 - x_2\]The average utility of the individuals of the second type are:
\[f_2 = - x_1 + x_2\]The average utility over the entire population is then given by:
\[\phi=x_1f_1+x_2f_2\]In the notes on Replicator Dynamics you can find linear algebraic expressions of these quantities \(f\) and \(\phi\) that extend naturally to populations with more than just 2 types.
The actual Replicator Dynamics equation is then given by:
\[\frac{dx_i}{dt} = x_i(f_i-\phi)\text{ for all}i\]In the case of our game this corresponds to:
\[\begin{align} \frac{dx_1}{dt} =& x_1 ((x_1 - x_2) - (x_1 - x_2) ^2)\\ \frac{dx_2}{dt} =& x_2 ((x_2 - x_1) - (x_1 - x_2) ^2) \end{align}\]Substituting \(x_2=1-x_1\) we have:
\[\begin{align} \frac{dx_1}{dt} =& x_1 (2x_1 - 1)2(1-x_1))\\ \frac{dx_2}{dt} =& -x_1 (2x_1 - 1)2(1-x_1)) \end{align}\]And we see (setting the derivatives to be equal to 0) that there are 3 stable populations:
The fact that this is stable mathematically (ie the derivaties are zero) corresponds to the game theoretic fact that in these populations every type of individual has the same fitness: so no one behaviour has an evolutionary advantage.
With the final minutes of the class I showed how we could use Nashpy to solve these equations numerically. You can find the notebook with that here.
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