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:

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:

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:

- \(x_1=0\): Everyone drives on the right.
- \(x_1=1\): Everyone drives on the left.
- \(x_1=1/2\): Half the population drives on the left and half on the right.

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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