In class we spoke about the replicator dynamics equation: a differential equation that is a building block of evolutionary game theory.
You can see a recording of this here: https://cardiff.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=5cc70927-0793-4dd5-9690-b11a00c682d0.
After this we discussed potential reasons for the emergence of the social convention of walking 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 walking past each other). If both these individuals walk 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 walking according to the first convention (say: the left) and \(x_2=1-x_1\) is the proportion walking according to the second convention.
We can then compute the average utility of an individual who walks 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:
\[\begin{align} \frac{dx_1}{dt} =& x_1 (x_1 - x_2 - x_1(x_1 - x_2)+x_2(x_2 - x_1))\\ \frac{dx_2}{dt} =& x_2 (-x_1 + x_2 - x_1(x_1 - x_2)+x_2(x_2 - x_1)) \end{align}\]which can be simplified 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 derivatives 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.
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