The Draw Index: A Measure of Competition for Winner-Take-All Elections
Douglas D. Rose
Summary: Winner’s percentage, a common measure of electoral competition in
winner-take-all elections, measures the shift in vote shares required to produce
changes in election outcome. Thus, winner’s percentage of the vote cast is a logical
measure of winner-take-all competition. It treats equally shifts from higher to lower
ranking candidates. A related measure, the Draw Index, even more clearly measures
vote shifting needed to produce tied, then changed outcomes, with weighting by the
preceding ties. A final adjustment, yielding the Draw Plus measure, provides a
greater emphasis on the importance of a first change in outcome. Across all cases of
five or fewer candidates, five measures of competition – including closeness and
Rae’s Index of Fractionalization – are highly correlated. Methods of expanding these
measures to include non-voters or no preference are explored in an appendix.
Introduction
For winner-take-all elections, there has been no adequate measure of electoral
competition that can take account of contests with differing numbers of candidates.
When only the votes cast are important, measures focus on the percentage division
of the vote. With only one candidate, the results need no measure. With two
candidates, almost any measure will do, as there is only one degree of freedom –
once the winning candidate’s vote share is described, everything else is known.
When the number of candidates (n) is three or more, there are multiple (n-1)
unknowns. There is no agreed upon method for reducing the variety to a single
measure. When comparison is unavoidable, notably across jurisdictions or over
time, winner’s share of the vote or the more common party share of the two party
vote is used to reduce the data to the two-candidate situation. These are not
entirely satisfactory when there are third or fourth candidacies with large vote
shares or a significant impact on the outcome. Without an adequate measure of
electoral competition, analyses of causes and consequences of electoral competition
suffer from measurement error, confusion, and questions about validity.
This note proposes a measure of electoral competition for winner-take-all elections
that can compare contests with differing numbers of candidates. Several categories
of contests are not directly covered, though a similar logic can be developed for
them: contests with a turnout threshold or winner share of the eligible vote
(nonvote becomes a consideration), multi-seat contests, proportional
representation, and “elections” which are mere aggregations of individual decisions
among “candidates” of goods and services.
I first review an existing measure and its logic, then lay out a general approach to
competition. This provides a rationale for the winner’s share of the vote as a
measure of competition but also provides a critique of that measure. I introduce a
2
measure more focused on competition among the top candidates, the Draw Index,
and then a practical adjustment. These measures are compared for all cases of five
or fewer candidates involving whole percents, and I summarize the results.
Measuring Variation or Changing the Outcome
A measure of electoral competition is an instance of a measure of variation, so a
logical place to start is with variance. Indeed, one prominent measure, Rae’s Index
of Fractionalization, whose square is also known as the Herfindahl index of
economic competition, is exactly this.1 It takes the root of the sum of the squared
vote shares across candidates or parties, which, for a given number of candidates, is
a direct function of the standard deviation. This index has the interesting property
that it equals the chance that two randomly selected voters agree in their vote
choice. It equates contests with different numbers of candidates, implicitly by
assuming that non-candidates get zero percent of the vote. It helpfully decreases,
indicating greater competition, for every shift of votes from a higher-ranking
candidate to one with fewer votes.
As a measure of dispersion, however, Rae’s Index of Fractionalization decreases
most when votes are shifted more than one rank – from the first to the third-place
candidate rather than from first to second-place, for instance. This is a property of
summing squares. The question is whether this is what we mean by competition. Is
a .505-.495 contest less competitive than a .50-.48-.02 one? Yes, by fractionalization.
Possibly also in economics in which small differences in market share among
dominant firms matter little. But in winner-take-all elections, small differences can
mean a lot – you win it all or lose it all. Competition means closeness to changing
the outcome, and a 50%-48%-2% vote, with its 2% margin of victory, seems less
competitive than a 50.5%-49.5% vote with a 1% margin. So the Index of
Fractionalization does not seem to capture what we mean by electoral competition
in winner-take-all elections. Its problem does, however, suggest an alternative
approach.
For winner-take-all elections, competition concerns potentially changing the
outcome. While actual causal analyses of potential changes might involve many
factors, the competition measure that eases these analyses is the one that focuses on
potential vote shifts. The simplest measure is: what share of the vote would have to
shift in order to change the outcome? The measure is how close the second
candidate’s vote share is to the first, because a vote shift of one-half of this amount
could change the outcome. This closeness measure is also a commonly used
measure of competition.
•
1
Douglas Rae, A Note on the Fractionalization of Some European Party Systems,
Comparative
Political
Studies
October
1968
1:
413-418,
doi:10.1177/001041406800100305, Sage Publications.
3
The difficulty with this simple measure is that it ignores additional candidacies and
therefore additional possible changes in outcome. A three-way tie is not measured
to be closer than a two-way tie, yet is more competitive because there are more and
equally easy potential changes. A four-way tie is even more competitive and so on.
For instances, the US Presidential elections of 1860 and 1912 were competitive
beyond what the closeness of the two leading candidates’ vote shares indicates.
Conceiving Changes in Outcome
One way of dealing with the varying number of competitive candidates in different
elections is to assume that all elections have an infinite number of candidates,
almost all of who get no votes, and then design a measure for which candidates with
no votes have no measured impact. For every contest we can consider how close
the outcome is to a tie between the top two candidates, between the top three
candidates, between the top four candidates, etc. by having a measure that simply
adds zeroes if there are no votes cast for additional candidates. With a twocandidate contest, the possibility of a three-way tie adds nothing, as there is no third
candidate.
For each competition among the first n finishers, the minimum vote share shift
producing a new winner is the difference between the average vote of the first n -1
finishers (the competition among whom has already been measured and included)
and the nth finisher. This is
(c1+c2… cn-1 –(n-1)cn)/n
Where n is the number candidate being considered, and c1 is the highest vote getter.
The sum of these terms is mathematically unwieldy. The sum of closeness terms is
the sum across i, from 2 to n, of
(c1 + c2 + …. – (i-1) ci)/i
Additionally, this sum of vote shifts producing ties does not constrain the sum in any
valuable way that reflects what we mean by competition. That is, it does not
consider the relationship between a two-way tie, for instance, and a three-way tie.
Let a equal the difference between the first two candidates, b equal the difference
between the second and third candidates, c equal the difference between the third
and fourth candidates, etc. The simple sum is then
a/2 + (a+2b)/3 + (a+2b +3c)/4 + (a + 2b + 3c + 4d)/5 +…
A shift in a vote from a higher ranking candidate to the next lower ranking candidate
– which always increases competition until it is large enough to reverse the ranking
– has the effect of decreasing one closeness gap and increasing the next. For
example, a shift in a vote from the winning candidate to the second place candidate
decreases a, the difference between them, and increases b, the difference between
the second and third place candidates. The mathematical effect is to decrease the
4
total more for later shifts, so that closeness among the top ranked candidates counts
less than closeness among the lower ranked candidates. 2
At a minimum, we would prefer that closeness among top candidates count no less
than closeness among the also-rans. For this to happen, the coefficients for each
closeness gap must be equal. There is an easy way to do this which is to divide each
closeness terms by i*(i-1) rather than simply by i.
Our revised sum is
(c1 + c2 + … -(i-1)ci) / (i*(i-1))
The sum of the coefficients for the first term is i/(i+1). As i approaches infinity, this
sum approaches 1. That is, the weight for the first candidate’s vote is 1. The
coefficients for all other votes are the above coefficient for winner’s percentage
minus 1, or zero.
Winner’s Percentage of the Vote
The winner’s percentage of the vote is the measure that treats equally all shifts in
closeness gaps, as just indicated. It considers simultaneously all shifts among
candidates and all possible ties, counting equally all vote shifts among candidates,
hence treating equally all possible ties.
This measure runs from 0 to 1. It is 1 when one candidate receives all the votes and
is zero when there is a tie among an infinite number of candidates. The competition
version of the index is 1 minus the winner’s share of the vote. The winner’s share of
the vote has been a measure of choice in recent competition literature.3
This measure has attractive properties. It is easy to calculate, makes some intuitive
sense, fits the election rules for winner take all, accommodates multiple candidacies,
and can be justified in terms of measuring changes in outcomes. For an n-way tie
among candidates, the measure is 1 – 1/n.
The disadvantage is its fundamental characteristic, that it treats as equally
competitive elections with vote shifts between the nth and nth + 1 candidates.
The decrease in the measure for a shift from the nth to (n+1)th candidate, is the
vote share shift times (n2 + n -1)/[(n+1)*(n+2)], which increases from 1/6 when n is
1 to approximate 1 when n is infinity.
3 For example, see “The Decline of Competition in U.S. Primary Elections, 19082004,” Stephen Ansolabehere, J. Mark Hansen, Shigeo Hirano and James M. Snyder,
Jr.) in The Marketplace of Democracy, Michael McDonald and John Samples (eds.),
Brookings Press, 2006.
2
5
Intuitively, this seems wrong to me, as each additional tie or change in outcome
ought to count less the more ties that have already occurred. That is, the winner’s
share of the vote measure treats equivalently a 50-50 election and a 50-25-25 or a
50-10-10-10-10-10 election. In practice, we want to treat getting to a two-way tie as
more important – indeed, a precondition – to getting to a three-way tie.
The characteristic that justified the winner’s share of the vote, equal treatment of
closeness terms, is the one we wish to adjust, to produce more weight for the initial
closeness terms.
The Draw Index
One way to weigh more heavily closeness among top finishers is to discount the
later ties by the preceding ties. This can be accomplished by changing the divisor for
the closeness terms, this time to i!, leading to an updated sum of closeness terms,
the sum across I, from 2 to n, of
(c1 + c2 + …. – (i-1) ci)/i!
This runs from 0, for equal votes for an infinite number of candidates, to e-2, the
natural logarithm of 2.718281828 minus 2 or .718281828 when one candidate
receives all the votes.4 It is easier for calculation to create an index equal to one
minus the sum, divided by e-2, with e equaling the natural log. Hence a onecandidate election has competitiveness of zero and the infinite candidate tie has
competitiveness of one.
One way of conceptualizing this measure is to consider that the four-way tie term
and all subsequent n-way ties are discounted by ½ for the preceding three-way tie
implying two new winners. Each 5-way and later tie is additionally discounted
(divided) by 1/3 because of the preceding four-way tie. And so on. This produces
the i! divisor. The Draw formula weighs additional changes by what is left to
contribute.
In practice, this means that a two-way tie leads the measure to be 70% of its
maximum, a three-way tie is measured as 93% of maximum competition, a four-way
tie is 99% of the maximum, and a five-way tie is, in rounded percents, 100% of
maximum competitiveness. Put another way, the initial closeness gap is weighted at
1, the second at .61, the third at .22, the fourth at .06, the fifth at .01, and the sixth at
.00.
The natural logarithm may be expressed as a power series
e1 = 1/0! + 1/1! +1/2! + 1/3! +…
The coefficients for the winner’s share of the vote in the Draw Index are a subset of
this:
1/2! + 1/3! + 1/4! +…
or e-2. Hence, if the winner received 100% of the vote, all other candidate’s vote
shares would contribute zero, and the index would equal e-2.
4
6
Computationally, the index is awkward. Better is the competition format, which is 1
minus the sum, termed the Draw Index. The term “draw” refers to substituting a
new victor for the old by shifting votes, producing a tie or draw, which with a
further infinitesimal shift in vote share produces a new winner. To create this
measure, rank order the candidates by vote share, with c1 being the candidate with
the most votes. Stop calculating when no further candidates have votes.
Draw Index computation:
c2 + c3 + c4*5/6 + c5*3/4 +c6*29/40+ c7*259/360+ c8*1207/1680 + c9* 1609/2240 +
c10*260651/362880. 5
These multipliers start at 1 for higher ranked candidates and descend to
approximate e-2. To include lower ranking candidates, for instance the ith, the
multiplier is the preceding multiplier minus (i-3)/(i-1)!
Divide the sum by (e-2) to arrive at the Draw Index.
Draw Index = (c2 + c3 + c4*5/6 + c5*3/4 +c6*29/40+ c7*259/360+ c8*1207/1680 +
c9* 1609/2240 + c10*260651/362880)/ (e-2)
Where e = 2.718281828, c2 is the second place candidate’s vote share, c3 is the 3rd
place candidate’s vote share, etc.
The Draw Index is conceptually defensible and readily calculable on a spreadsheet.
It departs from the winner’s share of the vote that measures competition as the vote
shifts required to produce changed outcomes, treating all vote shifts producing ties
equally, by discounting the contribution of additional changes by the number of
preceding changes, where precedence reflects candidate order. It has desirable
measurement properties, such as a range from 0 to 1.
Draw Plus
A problem with the Draw Index is that vote shifts between the 2nd and 3rd place
candidate do not affect the measure. Hence, a 50%-50% contest is considered
exactly as competitive as a 50%-25%-25% contest. Without expecting consensus on
this point, I regard the initial contest as more crucial, hence a 50-50 result to be
more competitive that a 50-25-25 one. To have a measure reflecting this priority, I
create a Draw Plus index by shifting some weight between the 2nd and 3rd ties, so
that the 2nd place draw is weighted at 9/5 rather than 2 and the 3rd place draw is
weighted at 9 rather than 6. With this change, the weight for the 2nd place becomes
In the calculation leading to the Draw Index, the coefficient for the nth candidate is
the coefficient for the winner minus { (the sum from i from 2 to n-1 of 1/i!) plus [(1/
(n-1)!]}. To create the Draw Index, we divide by e-2, which is the coefficient for the
winner’s share, and subtract 1. Hence the Draw Index coefficient is the above term
divided by e-2.
5
7
10/9 and the weight for the 3rd place candidate becomes 8/9, rather than both being
1. The exact weighting is not justified on any particular grounds except ease of
calculation, as a shift of a different amount will have nearly the same effect. Other
coefficients remain unchanged. With the Draw Index, the competitiveness of both
the 50%-50 and 50%-25-25 contests is measured at .696, while the Draw Plus index
changes the competitiveness of the 50%-50 to .773 while leaving unchanged the
other contest’s measured competitiveness. Only weights for the 2nd and 3rd place
candidates are affected by the Draw Plus adjustment.
Comparison of Measures
To examine their overall behavior, five measures – fractionalization, winner’s share,
1 minus the draw index, 1 minus the draw plus index, and closeness – are measured
for the 46,223 cases of five candidate distinct outcomes in whole percents. The
Draw and Draw Plus indices are subtracted from one to ease comparison with the
other measures. The vote divisions of the cases tend, on the whole, to be
competitive, with a median vote split of 44%-26-16-8-6.
All measures are responsive to vote shares. While the maximum for all measures is
100%, the range varies from 55% for fractionalization to 100% for the closeness,
draw and draw plus indices, with the standard deviation varying in parallel and the
average varying inversely. The mean and standard deviation range from 21% and
17% for closeness to 58% and 8% for fractionalization.
Mean
Standard dev
Maximum
Minimum
Range
Fractionalization
58%
7.9%
100%
45%
55%
Winner's
Share
47%
12.7%
100%
20%
80%
1Draw
29%
16.5%
100%
0%
100%
1Draw +
27%
16.5%
100%
0%
100%
Closeness
21%
17.5%
100%
0%
100%
The measures are closely connected, with correlation ranging from .836
(fractionalization – closeness) to .995 (winner’s share – draw index), with .997 for
draw and draw+ . Aligned on a proximity continuum, the order of measures is:
fractionalization – winner’s share – draw index – draw plus -closeness. The ranks
are somewhat less closely connected, with Spearman’s rho ranging from .72 to .99,
but otherwise reflect the interval pattern.
Pearson's r
Fractionalization
Winner's Share
1 - Draw
1 - Draw +
Closeness
Fractionalization
1.00*
0.961
0.939
0.921
0.836
Winner's
Share
0.961
1.00
0.995
0.989
0.936
1 - Draw
0.939
0.995
1.00
0.997
0.957
1 - Draw +
0.921
0.989
0.997
1.00
0.98
Closeness
0.836
0.936
0.957
0.977
1.000
8

Entries are r, the Pearsonian product-moment correlation coefficient.
The draw index can be accurately predicted from winner’s share and closeness. The
BLUE is :
Draw Index = 1.044 * (1- winner’s share) + .193 (1-closeness).
This reflects the Draw Index’s departure from equal treatment of ties to weigh more
heavily the closeness of the initial ties.
Draw plus is more influenced by closeness:
Draw Plus = .776 * (1- winner’s share) + .394 (1-closeness)
The difference from the Draw Index reflects the Draw Plus measure’s (twice)
greater emphasis on the closeness between the top two candidates.
Examples
The following sixteen examples indicate these more general patterns.
Examples
#1
#2
#3
#4
#5
#6
#7
#8
#9
#10
#11
#12
#13
#14
#15
#16
1st
candidate
100%
80%
60%
60%
50%
50%
50%
40%
40%
40%
40%
30%
30%
30%
30%
29%
2nd
candidate
0%
7%
15%
34%
15%
44%
49%
26%
23%
25%
26%
18%
22%
24%
27%
25%
3rd
candidate
0%
7%
9%
6%
12%
4%
1%
12%
22%
22%
23%
18%
18%
23%
18%
19%
4th
candidate
0%
4%
8%
0%
12%
1%
0%
11%
14%
11%
9%
17%
15%
20%
16%
17%
5th
candidate
0%
2%
8%
0%
11%
1%
0%
11%
1%
2%
2%
17%
15%
3%
9%
10%
While the measures generally agree, Fractionalization is most responsive to minor
candidates and Closeness depends only on the top two candidates. The Draw +
measure is more attuned to Closeness than the Draw measure, which closely tracks
winner’s percentage.
Examples
#1
#2
#3
Fractionalization
100%
81%
64%
Winner's
Share
100%
80%
60%
1 - Draw
100%
74%
49%
1 - Draw +
100%
74%
48%
Closeness
100%
73%
45%
9
#4
#5
#6
#7
#8
#9
#10
#11
#12
#13
#14
#15
#16
69%
56%
67%
70%
52%
53%
53%
54%
46%
46%
49%
48%
47%
60%
50%
50%
50%
40%
40%
40%
40%
30%
30%
30%
30%
29%
44%
37%
31%
30%
23%
20%
20%
19%
12%
11%
8%
9%
9%
40%
37%
25%
23%
21%
20%
19%
19%
12%
11%
8%
8%
8%
26%
35%
6%
1%
14%
17%
15%
14%
12%
8%
6%
3%
4%
For a highly accurate and meaningful measure, the Draw Index is preferred.
However, the estimate based on winner’s share and closeness is very easy to
calculate. The Draw Plus measure imports the conceptual power of the Draw Index,
while giving priority to the closeness of the top candidates. As indicated, the
winner’s share itself contains hidden power as a measure of competition as
potential change in outcome, and closeness focuses on the first and most important
of these, the approach of the second candidate to victory. The Draw Index combines
the focus on changed outcomes with the emphasis on the importance of the initial
changes.
Summary
The Winner’s Share of the vote cast measures competition in winner take all
elections as chances for changing the outcome by shifting vote shares, equally
weighted among two, three, and more potential winners. To place extra importance
on the initial changes in outcome, the Draw Index calculates competition as
potential changes with greater weight placed on the initial changes – a two-way tie
weighted as if a precondition for the three-way tie. A further refinement, creating
the Draw Plus measure, emphasizes the importance of the closeness of the vote
share between the top two candidates. Across all elections involving five or fewer
candidates, these measures generally reflect a combination of the winner’s share of
the vote and the closeness of the contest between the top two candidates. The Draw
Index and Draw Plus measures reflect more of the complexity of competition than
do the simpler winner’s share and closeness measures.
Appendix: Including No Opinion or Nonvoters
There is a range of situations in which it would be desirable to include all potential
voters in the measure of competition. These include polls leading up to the election,
over time data on elections with varying turnout, elections with different levels of
turnout, and elections with requirements for a winning outcome to achieve some
share of the eligible electorate’s support.
10
When uncommitted voters are included, a two-candidate race turns into a threeoption event, so, as in the case of a three-candidate contest, no single measure
captures or implies all aspects of competition. Every contest contains a reason to
consider the options for measuring competition.
The basic issue is whether to treat the uncommitted as supporters of a noncandidate alternative, so that this alternative is a potential victor. I believe that this
makes sense only in the situation where a single candidate is being voted up or
down, in which case, with only two alternative outcomes, virtually any measure will
do.
What remain are measures dividing the candidates’ vote totals by the number of
potential votes, rather than by the votes cast. In the case of pre-election polls, this is
the normal procedure. Now the candidate shares add to the total turnout, or
preferences expressed, rather than to 100%.
The definition of potential votes depends on the context. For instance, can
independent voters in a state cast votes in a party’s primary election?
The same five competition measures discussed above can be used with shares of
potential votes. For any specific result, a measure is the same, except multiplied by
the ratio of actual to potential votes, the turnout.
Over the course of an election, both the vote shares and the “turnout” may change.
For instance, the voters committed to an incumbent might be relatively stable while
the number committed to a challenger might increase, leading to an increased total
commitment by voters. In this case, the incumbent leader’s share of the committed
would decrease while his or her share of the potential vote would remain constant.
The challenger’s share of both committed and potential votes would increase. The
closeness would increase, and in general the election would become more
competitive, especially if measured in terms of potential vote. Hence, the stability
incumbent’s share of the potential vote – here assumed to equal the winner’s share
– misses entirely the changing dynamic of competition. If polls during the election
measured competition solely by winner’s share of the potential vote, the election
would seem as competitive at the beginning, when the challenger had no support, as
at the end, when the challenger’s support nearly equaled the incumbent’s.
Closeness would reflect the dynamic but would miss the aspect that the size of the
pool of the remaining uncommitted determines how competitive the contest
actually is. A split of 52%-48% with no uncommitted is less competitive than 40%36% with 24% uncommitted. The Draw Index reflects both the closeness and the
magnitude of committed, so is a better choice to track the election of time in this
respect.
Indeed, the candidates themselves might change as trial balloons are floated, some
hopefuls fail to gain support, party nominations are won, and third party candidates
11
are included in trial heats. These can be confusing changes to analyze. From a
competition perspective, the use of a potential voter base can be helpful, as only the
major developments are likely to change the results much, except for
Fractionalization. Using a single measure across this variety reduces the confusion,
increasing clarity perhaps at the cost of comprehensiveness. By reflecting the major
aspects of competition, the Draw Index may better combine clarity and complexity
better than winner’s share or closeness.