Brian Meek is now at King's College London.

*With some differences in presentation, the paper was originally published
in French in Mathématiques et Sciences Humaines, No 25, pp13-23,
1969.*

- (A) The number of 'wasted' votes in an election (i.e, which do not contribute to the election of any candidate) is kept to a minimum.
- (B) As far as possible the opinions of each voter are taken equally into account.
- (C) There is no incentive for a voter to vote in any way other than according to his actual preference.

Candidates with more than *q* votes are elected, and have their surplus
votes transferred according to the next preferences marked; if there are no
such candidates, the bottom candidate is eliminated and all his votes so
transferred. Repeated application of these rules ensures that at the end of
the count *s* candidates have at least *q* votes each and so the
total wasted vote* w* satisfies *w* < *T/(s+*1*)*.

Given *s* and *T*, it is clear from the definition of *q*
that condition (A) is satisfied provided the next preference at each
transfer is always given. It is possible for the above inequality, and hence
condition (A), to be violated, if *w* is increased by the addition of
votes which are non-transferable because no next preference has been
indicated. In this paper we shall assume that this does not occur; it will
be shown in a second paper that it is possible still to satisfy (A) in such
cases by modifying the definition of *q*.

Within this obvious limitation, attempts have been made to eliminate possible sources of inequity of treatment by various modifications of the counting rules. Such sources include:

- the choice of which votes to transfer from the total for a candidate who has exceeded the quota
- errors introduced by taking whole-number approximations to fractions of totals for transfer - particularly in elections with small total vote
- calculation of the proportion for transfer from an elected candidate on the basis of the last batch of votes transferred to him, and not on his total vote.

Difficulty (1) is overcome by transferring the appropriate proportion of
each divided vote, while the method clearly reduces the errors involved in
(2) by the factor 1/*K*. If *K*=10***n* this is simply
working to *n* decimal places. The value of *K* has only to be
increased until the errors are too small to affect the result of the
election.[3] The method is equivalent to transferring the
*whole* vote at an appropriately reduced value, and it is this
interpretation we shall use from now on.

Difficulty (3) is slightly more technical, and warrants further explanation.
Suppose at some stage a candidate has obtained *x* (<*q*)
votes. By transfer from another (elected or eliminated) candidate he now
acquires a further *y* votes, where *x+y* >= *q*. His
surplus is now *z=x+y-q*. It would appear that his *x+y* votes
should now be transferred, with value reduced by the factor* z/(x+y)*.

It is, however, common practice for only the *y* votes to be
transferred, with value reduced by the factor *z/y*. The reason for
adopting this procedure is simply the practical one, in a manual count, of
reducing as much as possible the rescrutiny of ballots for later
preferences. However, neither this nor the argument that 'the difference is
unlikely to affect the result' are particularly relevant to a
decision-theoretic discussion, though we shall return to the practicability
problem later.

Of more importance here is the argument 'in STV a vote only counts for one
candidate at a time, and should count for the first preference where
possible'. If accepted, this would of course also render difficulties (i)
and (ii) irrelevant, and the Senate Rules unnecessary; the first part of it
is in fact sometimes used as a 'proof' that STV satisfies condition (B). But
even without the Senate Rules the statement is false; however the surplus
votes are chosen for transfer, *it is* *the existence of the
untransferred votes which makes the* *transferred votes surplus.* A
vote not only counts directly for one candidate; it can indirectly affect
the progress of the count, the pattern of transfers, and ultimately the
*election or* *non-election of other candidates.*[4]

It is this fact which is at the root of the failure of STV to satisfy condition (B).

In the specific situation described above, the candidate achieves election
not only because of the accession of the *y* new votes, but because of
the existence of the *x* previous votes; hence for condition (B) to be
satisfied, all *x+y* votes should be transferred at the appropriate
reduced value.

However, there is yet a fourth difficulty, one which does not seem to have been recognised hitherto.

- (4) In determining the next preference to which a vote is to be
transferred,
*elected as well as eliminated**candidates are ignored.*

Let us suppose that of *y* votes to be transferred, *y/2* are
marked next to go to candidate A, and *y/2* to candidate B. Let us
further suppose that A has already been elected; under STV the *y/2*
votes which would otherwise go to him are transferred to the next candidate
marked (assumed C in every case) provided that that candidate is not also
already elected. Thus *y/2* go to B, and *y/2* to C. The
inequities are plain; the votes for A which enabled the *y/2* to go to
C rather than A had no say in their destination, while C obtains these votes
at the same value as B receives his. Suppose these *y* votes were
originally first-preference votes for a candidate D, now eliminated; those
who voted for A next and then C at least have had their second choice
elected, while those who voted next for B have not - yet these votes go,
under STV, to both B and C at full value.

In section 6 we shall describe a counting mechanism which overcomes all these difficulties.

Let *T*=3599, *s*=3, *q*=900, and the unsophisticated
first-preference votes for the six candidates A, B, ... F be as follows:

A B C D E F 1020 890 880 589 200 20In this case the 120 surplus votes of A divide 60 to B, 20 to C, 40 to D and the elected candidates are A, B and C.

Suppose there are 170 voters who above voted A, D, C ... It is known that
the second-preference votes of F will go to C, and of E to D. Then the
sophisticated way for these 170 to vote is F, A, D, C,... *in order to
prevent A from being elected on the first count.*

A B C D E F 850 890 880 589 200 190On the elimination of F, his original 20 votes go to C, and the 170 sophisticated votes return to A. However, the 120 surplus is now taken entirely from this batch (see (3) in section 3) and goes to D. C having no surplus, E must be eliminated and D is elected.

A different type of sophisticated voting is given below:
*T*=239, *s*=2, *q*=80.

Unsophisticated case: C and A elected:

C,A,B... C,B,A... B,A.... A,B..... 120 80 31 8Sophisticated case: C and B elected:

C,A,B... C,B,A... E,B,A... B,A.... A,B..... 120 50 30 31 8It seems to be a new result that sophisticated voting is possible in STV, though it is well-known that it can occur in other voting systems and considerable work has been done on decision processes using a games-theoretic approach. Black [5] in his discussion of STV does mention the possibility of 'an organised minority (perverting) the use of the system' but only in connection with a candidate with just the quota on first preferences who is rated last by the rest of the electorate. STV supporters would claim that if a candidate can obtain a quota this

The conditions (A), (B), (C) discussed so far were chosen simply because they seem to be specific to STV among constituency-type systems in parliamentary elections. However, other conditions could be applied, notably those specified by Arrow in his General Possibility Theorem.[6]

As STV elections are multi-vacancy, the preferences between candidates
listed by the voters do not as they stand represent an ordering of
*independent* alternatives, and so Arrow's analysis is not directly
applicable. The deduction from the voter's ordering of candidates of his
ordering of the actual independent alternatives (the possible subsets of the
set of all candidates who might actually be elected) is by no means
straightforward. Nevertheless, at some stage of the count the process
reduces to electing one candidate to one remaining vacancy, and so the
consequences of the theorem, and the Condorcet paradox, cannot be escaped.
Using the alternatives as they stand, even though they are not independent,
STV clearly satisfies Arrow's conditions 1, 4, and 5. The condition 3 of
independence of irrelevant alternatives is not satisfied, nor is condition 2
(the positive association of social and individual values). This can be seen
from the above analysis.

A related point, and probably the strongest decision-theoretic argument against STV, is the fact that a candidate may be everyone's second choice but not be elected. This difficulty is not overcome by the feedback method, and it does not seem to the author to be possible to do so while retaining a system which would be recognisably a 'single' transferable vote.

Virtually all other discussion of STV, both for and against, seem to have been about political and not decision-theoretic considerations.

For example, Black[5] does discuss STV from what he terms
the 'statical' point of view, but although he does express some disquiet
about the 'heterogeneity' involved in STV (basically, that some votes count
for first preferences, others for second or later preferences), he does not
go into the problem in detail and concludes 'in spite of those drawbacks
(STV) has merits ... it is not difficult to see why many people,
*regarding it purely as a statical system*, (Black's italics) should
hold (it) in esteem'. The italicised phrase is to introduce other,
'dynamical' arguments against STV.[7] Black does not discuss
the conditions mentioned here; though the germ of the idea of inequity is
contained in the word 'heterogeneity'; in fact as section
3 shows, the heterogeneity which worries him is more apparent than real,
and the feedback method described in section 6 eliminates
what there is. Nor - oddly - does the 'everyone's second choice' problem,
even though this is closely connected with the doubts mentioned at the end
of the last section.

**Principle 1.**If a candidate is eliminated, all ballots are treated*as if that candidate had never stood*. [8]**Principle 2.**If a candidate has achieved the quota, he retains a fixed proportion of every vote received, and transfers the remainder to the next*non-eliminated*candidate, the retained total equalling the quota.

Note that the proportion of each of B's votes to be transferred is increased by this accession of support; B's supporters have a say in the transfer of the extra surplus, since it is their existence which has made it surplus. All support for B is now treated equally, being divided proportionately to leave him with exactly the quota.

Consider now the effect of Principle 2. The transfer of B's vote may lead to
another candidate, D, being elected. All votes, new and old, for D, have now
to be divided, leaving D with the quota and distributing the rest to the
next *non-**eliminated* candidate. Some ballots may have B,
another elected candidate, as next candidate. Under previous rules, only
continuing (i.e. non-eliminated and *non-elected*) candidates can
receive transfers. Now these votes are regarded as extra support for B: he
takes the proportion allotted him by D, *retains the proportion that he
keeps of all he receives*, and transfers the rest - now the *third*
marked candidate. Formerly the third candidate would get all of the
proportion transferred by D (see (iv) section 3).

It can be seen that B will once more have more than the quota if he does not
again reduce the proportion which he retains. However, the increased
proportion transferred may in part go to D who will therefore have to reduce
the proportion *he* retains. This will react back on B, and it is clear
that we have an infinite regression. However, it is also clear that the
proportions for transfer do not increase without limit, there being only a
finite total surplus available from B and D, who must each retain a quota.
The problem is in fact a mathematical one of determining the proportions to
be retained by each which will leave them both with a quota, taking into
account the extent of mutual support. If *pB* is the proportion B
transfers, and *pD* that which D transfers, supporters of both B and D
have their votes transferred to third preferences at value
*pB×pD*. Those putting B first have 1-*pB* retained by him
and *pB×(*1-*pD)* retained by D; those putting D first have
1-*pD* retained by him and *pD×(*1-*pB)* retained by B.

We now, as examples, give the formulae for the proportions for transfer in the cases of 1, 2, 3 and 4 elected candidates:

This is the same formula as before, except that *t1* now contains all
effective first-preference votes for the candidate, including those obtained
from eliminated candidates, who by Principle 1 are now ignored. The
proportion *p1* is recalculated every time *t1* is increased by
the elimination of a candidate.

*(t1*+*p2×t21)(*1-*p1)=q*

*(t2*+*p1×t12)(*1-*p2)=q*

Thus:

*[t1+p2×t21+p3×t31+p2×p3(t321+t231)](*1*-p1)=q*

Two similar formulae hold, obtained by cyclic permutation of the suffices.

where dashed summation indicates summation over all permutations of (234); there are three similar formulae.

The extension to any number of candidates is straightforward. It should be noted:

- The formulae for
*n*candidates may be reduced to those for*n*-1 candidates by eliminating the*n*th equation and putting*pn*=0 in the others; - Full recursion is not necessary on the elimination of a candidate if none of the totals or subtotals in the formulae in use at that stage are changed as a result.

*(t1+p2×t21)(*1-*p1)=q**(t2+p1×t12)(*1-*p2)=q*

The total vote for the two candidates is *t1+t2*; for them both to be
elected *t1+t2* >= 2*q*. Suppose the strict inequality holds;
in a non-trivial case *t12, t21* are both non-zero. Further, at least
one of *t1, t2* is greater than *q*; assume it is* t1*. If we
put* p2*=0 in (1) we can solve for *p1*, giving a value *p1*
> 0. This *p1* is the proportion to be transferred if candidate 1
were the only elected candidate; thus *t2+p1×t12* > *q*
or candidate 2 would not be elected. If the equality holds, candidate 2 only
just gets the quota and so *p2*=0 from equation (2); thus the equations
are solved.

If the strict inequality holds, we get a value of *p2 * > 0 which is
too small. Substituting in (1) increases the coefficient of
*(*1-*p1)* and hence increases *p1*; the new value of
*p1* is increased (but is still too low). Substitution in (2) gives
similarly an increased, but too low, value of *p2*. Thus the iterative
process gives monotonically increasing sequences of values *p1, p2*
bounded above, which hence tend to limits which are the solutions of the
equations. A cycle of iterations which leads to two successive sets of
values the same to the given accuracy is taken as the approximate solution
required. Note that the approximate values may be slightly smaller than the
exact ones, but this is exactly what we want; otherwise too much of the
support for the candidate concerned would be transferred and he would be
left with less than the quota. The process can also be easily shown to work
in the limiting case, *t1+t2*=2*q.*

It is clear that the success of this iterative procedure depends on the fact
that all the quantities in the totals (the coefficients of*
(*1-*pi)* in each equation) are non-negative, and that therefore it
will work for any number of equations provided they are solved cyclically in
order of election - this condition being necessary to avoid getting negative
values of *pi*. Since the counting process can only increase the totals
of support for elected candidates, it is also clear that the *pi* for
those candidates can only increase as the count progresses;[10] thus it is safe to take as starting values of the
*pi* the ones obtained at a previous stage, putting *pi*=0
initially for newly-elected candidates only (in which case, as mentioned
above, the equations reduce to the ones at the previous stage and hence will
yield, at the beginning of the iteration, the same answers).

It can be shown fairly simply that the convergence rate of the iterative
process is likely to be unsatisfactory only when both of the following
conditions hold; that all the *pi** *are small, and the
cross-totals *tij* etc, are as large as possible. This would not cause
difficulty even on the rare occasions on which all these conditions were
satisfied, since the occurrence of slow convergence can be detected in
advance and allowed for, while at a later stage in the count some at least
of the *pi *are likely to rise sufficiently to accelerate to the true
convergence satisfactorily.

It may be argued that the actual results of any election would be different so infrequently that the additional complication is unnecessary. This is a matter for conjecture, or preferably, for further investigation. However, the method has been tried out in two cases, once using figures obtained by a quasi-random process, and once in an actual STV election. In both, there were differences in the candidates elected.[12] Particularly since STV supporters lay such emphasis on the criterion of equality of treatment (condition (B)), it would seem worthwhile in automated counting to adopt the feedback method.

To sum up, the feedback method does satisfy the criterion, subject to the limitations imposed by the basic STV system - i.e. the theoretical minimum of wasted votes, and the elimination of candidates. There is one further limitation not so far discussed, imposed by the voters themselves if they take advantage of the possibility allowed by STV of listing only some of the candidates in preference order. The extension of the feedback method to cover this is dealt with in Paper II; it turns out that the extension also, as a bonus, allows voters to express their views much more accurately than under previous STV methods.[13]

- For a complete description of STV see E Lakeman and J
Lambert: Voting in Democracies (Faber and Faber 1955).
*(The current edition in 1994 is E Lakeman: How Democracies Vote (4th edition, Faber and Faber 1974).)* - This is nevertheless more than can be said for some common voting systems, such as the simple majority system.
- This cannot, of course, cope with the case of exact equality, where some other method has to be used, if only drawing of lots.
- To argue, in connection with a
*transferable*system, that a vote should where possible not be transferable, seems inconsistent, particularly in view of the strong arguments put forward by STV supporters against the single non-transferable vote system, where an elector may choose only one from a list of candidates even though more than one are to be elected. See Lakeman and Lambert,*op, cit.*[1] - Duncan Black: Theory of Committees and Elections (2nd edition, Cambridge, 1963, pp 80-83).
- K Arrow: Social Choice and Individual Values (2nd edition, Wiley 1962).
- The case for the other side may be found in Lakeman and
Lambert,
*op cit.*[1] - The similarity of this principle to Arrow's condition of independence of irrelevant alternatives is obvious. However, the interdependence of the alternatives here means that the condition is not in fact satisfied.
- This innocent-looking assumption is open to major criticism. Full discussion is outside the scope of this paper; it is hoped to include this in the projected more general paper mentioned in section 5.
- Clearly Arrow's condition 2, the positive association of individual and social values, is now satisfied by the non-independent alternatives.
- For a feasibility study in general terms, see P Dean and B L Meek: the Automation of Voting Systems; Paper I; Analysis (Data and Control Systems, January 1967, p16); Paper II; Implementation (Data and Control Systems, February 1967, p22), and B L Meek: Electronic Voting by 1975? (Data Systems, July 1967, p12) - the date in the last source referring to the UK. For a description of the actual use of computers in STV elections in the United States, see Walter L Pragnell: Computers and Conventions (The Living Church, 20th August 1967, p12).
- For obvious reasons the work on the actual election cannot be made public!
- These papers are the result of a problem posed by Miss Enid Lakeman, Director of the Electoral Reform Society, London; the author wishes to thank her for her encouragement in the progress of the work. Thanks are due also to Professor W B Bonnor, Mr Robert Cassen, Mr Peter Dean, Mr Michael Steed and Professor Gordon Tullock for valuable discussions, correspondence and advice.