[EM] Directional Resistance
Forest W Simmons
fsimmons at pcc.edu
Thu Mar 22 12:53:45 PDT 2007
Michael Poole worried that it might be impossible to find circuit
elements with the precise properties needed.
As always, theoretical circuits have elements with idealized
properties. An idealized diode of the kind we need with reisitance R1
in one direction and R2 in the other direction could be constructed
out of some combination of linear resistors and perfectly rectifying
diodes.
Suppose that R2 is the larger resistance. Then put a linear resistor
of resistance R3=R1*R2/(R2-R1) in series with a perfectly rectifying
diode, and put the resulting compound circuit element in parallel with
a linear resistor of resistance R2.
His other question was what does resistance have to do with elections.
short answer: voters who rank candidate i over candidate j are
manifesting resistance to the election of j, at least in comparison
with candidate i.
Jobst correctly divined that the method is not clone free. Here's an
example.
3 ABC
2 BCA
The idealized diode resistances are
R(A,B)=2, R(B,A)=3, R(A,C)=2, R(C,A)=3, R(B,C)=0, R(C,B)=5
The respective circuit resistances are
r(A,B)= 1/(1/2+1/7)=14/9=1.5555...
r(B,A)= 1/(1/3+1/3)=3/2 = 1.5
r(A,C)= 1/(1/2+1/2)= 1
r(C,A)= 1/(1/3+1/8)=24/11=2.181818...
r(B,C)=0
r(C,B)=1/(1/5+1/5)=5/2=2.5
The min max circuit resistance of 1.5 ohms is that encountered by B
when the DC current flows from B to A.
>From A to B the circuit resistance is slightly larger: 1.555... ohms.
The max circuit resistance to C is 2.5 ohms.
So B is the winner. But removing B's clone C makes A the winner, as
you can easily check.
Note that the teaming effect is not quite as bad as Borda's, since here
B just barely wins, whereas in Borda B has a nice margin of safety.
Could this method be better than Borda for calculating tournament
winners in Round Robin sports contests?
Perhaps the Electrical Engineers could adopt it if they decide to have
a volleyball team playoff.
Forest
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