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Election politics

Here is an amusing differential equation that is relevant to the upcoming elections and/or primaries. Suppose that there are three candidates, x,y and z. Suppose that in an x,y race, x wins, in a y,z race, y wins, and in a z,x race, z wins. What happens in a three-way race? Here are the differential equations:

\begin{eqnarray*}x' &=& x(1-x-ay-bz) \\
y' &=& y(1-y-bx-az) \\
z' &=& z(1-z-ax-by)
\end{eqnarray*}


We choose a<1,b>1 to mimic the pairwise behavior. The XPPAUT file is called voter.ode and is:
x'=x*(1-x-a*y-b*z)
y'=y*(1-y-b*x-a*z)
z'=z*(1-z-a*x-b*y)
par a=.5,b=2
init x=.28,y=.281,z=.279
@ total=1000,njmp=5
@ xhi=1000,ylo=-.1,yhi=1.1
done
Run this program. Add the two curves y(t) and z(t) as follows. Click on \fbox{{\tt Graphics stuff}} \fbox{{\tt Add curve}} (tap \fbox{{\bf g}} \fbox{{\bf a}}) and fill in the dialog box as:
\fbox{\begin{tabular}{l\vert l\vert l}
\par {\tt X-axis:} t& {\tt Z-axis:} x & {\tt Line type:} 0 \\
\hline
{\tt Y-axis:} y & {\tt Color:} 1 & {}
\end{tabular}}
Repeat this for the variable z filling the dialog box as
\fbox{\begin{tabular}{l\vert l\vert l}
\par {\tt X-axis:} t& {\tt Z-axis:} x & {\tt Line type:} 0 \\
\hline
{\tt Y-axis:} z & {\tt Color:} 5 & {}
\end{tabular}}


You should see a red and yellow curve on the graph. Note how at first one wins the competition, then the next, and so on but at each stage, the amount of time that any given variable wins gets longer and longer. In fact, the solution is period in the logarithm of time. This means that in each cycle, each one stays active exponentially longer than it did the previous cycle! This is called the voter paradox.

You can sort of see what is going on by plotting in three-dimensions. Delete all the added curves by clicking \fbox{{\tt Graphics stuff}} \fbox{{\tt Remove all}} (tap \fbox{{\bf g}} \fbox{{\bf r}}). Click on \fbox{{\tt Viewaxes}} \fbox{{\tt 3D}} (tap \fbox{{\bf v}} \fbox{{\bf 3}}). Fill in the first three entries of the dialog box as:

\fbox{\begin{tabular}{l}
{\tt X-axis:} x \\
{\tt Y-axis:} y \\
{\tt Z-axis:} z
\end{tabular}}
You will see nothing particularly meaningful on the screen. Click on \fbox{{\tt Window/Zoom}} \fbox{{\tt Fit}} (tap \fbox{{\bf w}} \fbox{{\bf f}}) and the graphics window will choose a view that optimally fits the graph. (This is always the way I do three-d graphs, letting XPPAUT figure out the best way to show it.) You should see a triangular surface with a spiral coming out the center. The three corners of the triangle represent the fixed points (1,0,0), (0,1,0), and (0,0,1). There are no stable fixed points, nor a stable limit cycle for this model.


next up previous
Next: About this document ... Up: Fun with competition Previous: Fun with competition
G. Bard Ermentrout
1999-10-26