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Voting matters - Issue 9, May 1998

Voter Choice and Proportionality

B A Wichmann and R F Maddock

At the Electoral Reform Society 1997 AGM, Hugh Warren produced an eye-catching diagram in which several electoral systems were plotted on a diagram in which the two axes were voter-choice and proportionality. The diagram was not intended to give precise measures of the characteristics of each electoral system, but merely their relationship. However, for (party) proportionality, the Rose Index is a reasonable approximate measure. For voter- choice, no existing measure appears to be available which would be necessary to provide a more accurate representation of the diagram.

A possible measure of voter choice is the information-theoretic value of the result of an election, which appears to be new. For instance, in a dictatorship which has mock elections, the result is known beforehand, and therefore the information-theoretic value is zero. On the other hand, if the electorate is given a choice between three candidates then, assuming that each outcome is equally likely, the information-theoretic value is log₂(3)=1.58. As the number of possible outcomes increases, so does this measure of voter choice.

For values of the Rose Index, Kestelman gives values for the major electoral systems. It must be acknowledged that the Rose Index as a measure of party proportionality, may not be appropriate for STV elections, as pointed out by David Hill.

We compute the values for a hypothetical election for a 600 seat assembly in which there are three parties. For the use of STV, we take 120 constituencies each electing 5 members. For the regional list, we take 10 regions electing 60 candidates each. For the additional member system, we assume 300 seats elected directly and 300 added by proportionality. Note that if n seats are to be filled with 3 parties, then the number of ways to do this is n²/2+3n/2+1. We assume that all possible outcomes are equally likely. The entries in the diagram are as follows:

First Past The Post (FPTP): Rose Index 70% (UK), voter choice is
600 log₂(3)=951.
Alternative Vote (AV): Rose Index 72% (Australia), voter choice is
600 log₂(3)=951.
Single Transferable Vote (STV): Rose Index 92% (Ireland), voter choice is
120 log₂(⁵C₁₅)=1386.
(We are assuming each party has five candidates and therefore could theoretically obtain all five seats; hence the number of possibilities is the number of ways of selecting 5 from 15.)
Additional Member System (AMS): Rose Index 98% (estimated), voter choice
300 log₂(3)+log₂(300²/2 +3×300/2+1)=491.
Party List (PL): Rose Index 98% (estimated), voter choice is
log₂(600²/2+3×600/2+1)=17.5.
Regional party Lists (RL): Rose Index 98% (estimated), voter choice is
10 log₂(60²/2+3×60/2+1)=109.

It is important to note that this diagram will change if the underlying assumptions are changed, for instance, if the number of parties was increased from 3 to 4. An alternative way to compute voter choice values would be to take into account the probability of the various outcomes, based upon appropriate statistical data. This was considered initially but rejected due to the difficulty of the calculation and the problems in finding appropriate statistical data. If the voting system was changed, then one can only guess at the future statistical data. (The diagram here has the x-axis reflected from Hugh Warren's version so that the Rose Index is increasing.)

The conclusion from this diagram is hardly unexpected: party lists do not give voter choice, and FPTP/AV do not give party proportionality, while STV can claim, to a reasonable degree, to provide both.

References

P Kestelman. Is STV a form of PR? Voting matters. Issue 6. p5-9.
I D Hill. Measuring proportionality. Voting matters. Issue 8. p7-8.

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